research article inviscid uniform shear flow past a smooth...
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Research ArticleInviscid Uniform Shear Flow past a Smooth Concave Body
Abdullah Murad
Department of Mathematics University of Chittagong Chittagong 4331 Bangladesh
Correspondence should be addressed to Abdullah Murad murad-mathcuacbd
Received 1 February 2014 Accepted 30 June 2014 Published 23 July 2014
Academic Editor Shouming Zhong
Copyright copy 2014 Abdullah Murad This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Uniform shear flow of an incompressible inviscid fluid past a two-dimensional smooth concave body is studied a stream functionfor resulting flow is obtained Results for the same flow past a circular cylinder or a circular arc or a kidney-shaped body arepresented as special cases of the main result Also a stream function for resulting flow around the same body is presented for anoncoming flow which is the combination of a uniform stream and a uniform shear flow Possible fields of applications of this studyinclude water flows past river islands the shapes of which deviate from circular or elliptical shape and have a concave region orpast circular arc-shaped river islands and air flows past concave or circular arc-shaped obstacles near the ground
1 Introduction
Shear flow is a common type of flow that is encountered inmany practical situations Milne-Thomson [1] has discussedinviscid uniform shear flow past a circular cylinder It shouldbe noted here that in practical situations objects past whichflows occur deviate from regular geometric shapes In thepresent paper we have examined two-dimensional incom-pressible inviscid uniform shear flow past a smooth concavecylinderWe have obtained a stream function for the resultingflow It is found that the stream function given in [1] (obtainedby using Milne-Thomsonrsquos second circle theorem [1]) for theresulting flow due to insertion of a circular cylinder in auniform shear flow of an inviscid fluid is a special case of thatof the resulting flow past the concave body presented in thispaperMoreover flow around the same body has been studiedfor an oncoming flow that is a combination of uniform streamand uniform shear flowThe stream function for each of shearflow past a circular arc or a kidney-shaped two-dimensionalbody has been calculated from the main result as specialcases
The mathematical results for inviscid fluid flows holdgood for flows of common fluids like water and air the resultis valid for the whole region of a flow field except in thethin layer called boundary layer adjacent to the body aroundwhich the flow occurs The results of the present study mayhave applications in many areas of science engineering and
technology Here we would like to mention a few particularareas for applications of the present theoretical work inwater flows around river islands the shapes of which deviatefrom circular or elliptical shape and have a concave regionnoting that rivers have shear flow across and from riverbed tosurface it should be mentioned here that the present resultsfor oncoming flow parallel to line of symmetry of the concavebody (ie parallel to horizontal axis) can easily be extendedto the cases where oncoming flows make arbitrary angleswith the line of symmetry The result for inviscid shear flowpast circular arc obtained here as special case may find itsapplication in flows past circular arc-shaped islands in riversAs air flow near the ground is shear flow the present studymay also have applications in scientific investigation of airflow past concave or circular arc-shaped obstacles near theground
2 The Shape of the Body
The shape of the body forms owing to inversion of the oblateellipse by transformation [2]
119909 =
1199091minus 119888
1198772
1
119910 =
1199101
1198772
1
1198772= 1199092+ 1199102 (1)
where
1198771=
1
119877
= ((1199091minus 119888)2
+ 1199102
1)
12
(2)
Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 426593 7 pageshttpdxdoiorg1011552014426593
2 International Journal of Engineering Mathematics
which defines geometrical inversion with respect to the unitcircle centered at the point (119888 0) in the 119911
1equiv (1199091 1199101) plane
Transformation (1) can be conveniently expressed incomplex form as
1
119911
= 1199111minus 119888 (3)
The two-dimensional body in the inverse plane (119911 equiv (119909 119910)
plane) in general has a concave region facing the fluidHere the boundary in the 119911
1plane is an oblate ellipse given
by100381610038161003816100381610038161199111+ 12058221199111
10038161003816100381610038161003816= (1 + 120582
2) (1 minus 120582
2) 0 le 120582 le 1 (4)
A parametric form of (4) can be written as
1199111= 1199091+ 1198941199101= 120577 minus
1205822
120577
(5)
where 120577 = 119890119894120601 0 le 120601 le 2120587 in the 120577-planeThe inverse transformation of (5) is expressed by
120577 =
1
2
1199111+ (1199112
1+ 41205822)
12
(6)
in order that the exterior of the unit circle |120577| = 1maps ontothe exterior of the ellipse Moreover for 119888 gt 1 minus 1205822 and
1199111(119889) = 119889 minus
1205822
119889
= 119888
119889 gt 1 (the radius of the unit circle in 120577-plane) (7)
the exterior of the ellipse inverts into the exterior of the closedcurve in the 119911 plane and vice versa The equation obtainedfrom (4) by using transformation (3) that represents theclosed curve in the 119911 plane is
100381610038161003816100381610038161003816100381610038161003816
119911 + 1205822119911
119911119911
+ 119888 (1 + 1205822)
100381610038161003816100381610038161003816100381610038161003816
= (1 + 1205822) (1 minus 120582
2) (8)
Since corresponding to different values of the parameter 120582 (4)will represent distinct oblate ellipses consequently (8) willalso yield distinct smooth closed two-dimensional objects inthe 119911 plane for a given fixed 119888 It is mentioned in Ranger[2] that 120582 = 0 120582 = 1radic2 and 120582 = 1 correspond to inthe same order a circle a concave body and a circular arcin 119911 plane a figure of an object which is a smooth closedcurvewith a concavity is given in [2] withoutmentioning anymathematical equation for it We note that a kidney-shapedbody may be obtained for 120582 = 1radic2 and 119888 = 1 [3] (Figure 1)
3 Mathematical Formulation and Solution
A uniform shear flow parallel to the 119909-axis in the 119911 planein absence of any boundary may be expressed by the streamfunction
Ψ (119911 119911) sim minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin (9)
where 120596 is the constant vorticity
The stream function (9) can bemapped onto the 1199111-plane
using transformation (3) which yields
Ψ1(1199111 1199111) sim minus
1
8
120596(
1
1199111minus 119888
minus
1
1199111minus 119888
)
2
as 10038161003816100381610038161199111
1003816100381610038161003816997888rarr 119888 (10)
Again the mapping of the stream function (10) by utilizingtransformation (5) onto the 120577-plane leads to
Ψ2(120577 120577) sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
(
1
120577 minus 119889
minus
1
120577 minus 119889
)
2
as 10038161003816100381610038161205771003816100381610038161003816997888rarr 119889
(11)
where 1199111is given by (5) and prime (1015840) stands for differentia-
tion with respect to 120577Now we insert a circular cylinder of radius unity with its
centre at the origin represented by 120577120577 = 1 Since the streamfunction (11) does not have constant vorticity we cannotuse circle theorem [1] or second circle theorem [1] in orderto obtain the resulting flow In this situation we propose aformula that will give the resulting flow and it is
Ψ119877
0(120577 120577) = Ψ
0(120577 120577) minus Ψ
0(
1
120577
1
120577
) + Ψotimes(120577 120577) (12)
where Ψ1198770(120577 120577) and Ψ
0(120577 120577) are resulting and basic stream
functions respectively and Ψotimes(120577 120577) is a perturbation stream
functionSince on the boundary of the circle 120577120577 = 1 therefore
Ψ0(120577 120577) minus Ψ
0(1120577 1120577) becomes zero on the boundary of the
unit circle moreover Ψ0(120577 120577) minus Ψ
0(1120577 1120577) becomes the
same as Ψ0(120577 120577) as |120577| rarr 119889 If we assume that all the
singularities of Ψ0(120577 120577) lie at a distance greater than unity
from the origin then all the singularities of Ψ0(1120577 1120577)
lie inside the circle of radius unity Regarding Ψotimes(120577 120577) we
assume that all the singularities lie inside the unit circle andΨotimes(120577 120577) rarr 0 or a constant on |120577| = 1 and for |120577| rarr 119889
Thus the stream function Ψ1198770(120577 120577) in (12) possesses all the
properties to represent the resulting flowIn the light of (12) the resulting flow for the present case
may be written as
Ψ119877
2(120577 120577) sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
(
1
120577 minus 119889
minus
1
120577 minus 119889
)
2
minus(
120577
1 minus 119889120577
minus
120577
1 minus 119889120577
)
2
+ Ψotimes(120577 120577)
(13)
The function Ψotimeswill be evaluated afterwards in this paper
and we will show that the function satisfies all the conditionsthat the functionmust fulfill in accordance with the proposedformula (12)
International Journal of Engineering Mathematics 3
15
1
05
0
minus05
minus1
minus15
04020minus02minus04
y1
x1
(a)
1
05
0
minus05
minus1
0minus05minus1minus2 minus15
y
x
(b)
Figure 1 (a) Oblate ellipse in 1199111equiv 1199091+ 1198941199101plane and (b) kidney-shaped body in 119911 equiv 119909 + 119894119910 plane (found by putting 1205822 = 12 and 119888 = 1 in
(8))
The flow (13) around the circular boundary can bemapped by using transformation (6) onto the region outsidethe oblate ellipse in the 119911
1-plane which yields
Ψ119877
1(1199111 1199111)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) (1199111+ (1199112
1+ 41205822)
12
) minus 119889
minus
1
(12) (1199111+ (1199112
1+ 41205822)12
) minus 119889
2
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)12
)
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)
12
)
2
]
]
]
]
+ Ψotimes(
1
2
1199111+ (1199112
1+ 41205822)
12
1
2
1199111+ (1199112
1+ 41205822)
12
)
(14)
Again the flow given by (14) around oblate ellipse can bemapped onto the region outside the smooth concave body
given by (8) in the 119911 plane by using transformation (3) whichleads to
Ψ119877(119911 119911)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
minus
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
2
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
2
]
]
]
]
+ Ψotimes(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
(15)
4 International Journal of Engineering Mathematics
4 Uniform Shear Flow around a FixedCircular Cylinder or a Circular Arc ora Kidney-Shaped Cylinder
41 Uniform Shear Flow past a Circular Cylinder For 120582 = 0(8) represents a circle in the 119911 plane the circle is given by
10038161003816100381610038161003816100381610038161003816
119911 +
119889
1198892minus 1
10038161003816100381610038161003816100381610038161003816
=
1
1198892minus 1
since for 120582 = 0 we have 119888 = 119889
(16)
We put 120582 = 0 in the stream function (15) to obtain the streamfunction for flow around the circle (16) as
Ψ119877
3(119911 119911)
sim minus
1
8
120596[(119911 minus 119911)2
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(17)
The transformation
119885 = 119911 minus (minus
119889
1198892minus 1
) (18)
gives us the equation of the circle (16) as
119885119885 = (
1
1198892minus 1
)
2
(19)
Under transformation (18) the stream function (17) takes theform
Ψ119877
4(119885 119885)
sim minus
1
8
120596[
[
(119885 minus 119885)
2
minus
1
119885(1198892minus 1)2minus
1
119885(1198892minus 1)2
2
]
]
+ Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
(20)
Since there can be no change in the value of vorticity nearthe cylinder therefore
41205972Ψ119877
4(119885 119885)
120597119885120597119885
= 120596 (21)
Utilizing (21) on calculation it is found that in (20)
Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
=
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(22)Therefore the result (20) represents uniform shear flow
past a circular cylinder which is in agreementwith the knownresult [1] for the same flow
The relation (22) implies that
Ψotimes(120577 120577) = Ψ
otimes(120577 120577)
=
1
8
120596(
1
1199111015840
1(119889)
)
2
2
(1198892minus 1)2
(120577 minus 119889) (120577 minus 119889)
(119889120577 minus 1) (119889120577 minus 1)
(23)where
1199111015840
1(119889) = 1 +
1205822
1198892 (24)
It is clear from (23) that all the singularities of Ψotimes(120577 120577) lie
inside the unit circle in 120577-plane (since 119889 gt 1) andΨotimes(120577 120577) rarr
0 as 120577 rarr 119889 andΨotimes(120577 120577) rarr (14)120596(1119911
1015840
1(119889))
2
(1(1198892minus 1)
2
) (aconstant) on the circle |120577| = 1 Thus the function Ψ
otimessatisfies
all the assumptions that we have made in proposing formula(12) which therefore effectively gives the resulting flow dueto insertion of a circular cylinder in the flow (11) of whichvorticity is not constant
42 Uniform Shear Flow past a Circular Arc The streamfunction for uniform shear flow past a circular arc can beobtained by putting 120582 = 1 (and when 120582 = 1 119888 = (1198892 minus 1)119889)in the stream function (15) which yields
Ψ119877
5(119911 119911)
sim minus
1
8
120596(
1198892
1198892+ 1
)
2
times[
[
[
(
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
minus (
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
2
International Journal of Engineering Mathematics 5
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
(25)
43 Uniform Shear Flow past a Kidney-Shaped Two-Dimensional Body The stream function for the uniformshear flow past a kidney-shaped cylinder can be obtained byputting 1205822 = 12 and 119888 = 1 in the stream function (15) whichyields
Ψ119877
6(119911 119911)
sim minus
1
8
120596(
2 + radic3
3 + radic3
)
2
times[
[
[
(
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
minus (
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
2
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
times ((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
)
(26)
5 Flow Consisting of a Uniform Stream ofConstant Velocity 119881 Parallel to 119909-Axisand a Uniform Shear Flow Parallel to theSame Axis with Constant Vorticity 120596 past aConcave Body
Here the basic flow in the 119911 plane is
Ψ7(119911 119911) sim minus
1
2
119894119881 (119911 minus 119911) minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin
(27)
Now if we insert the two-dimensional concave body givenby (8) into the flow (27) the resulting flow following ananalogous procedure that we have adopted to obtain streamfunction (15) may be expressed as
Ψ119877
7(119911 119911)
sim minus
1
2
119894119881(
1
1199111015840
1(119889)
)
times[
[
((
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
minus(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
)
minus (
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Engineering Mathematics
which defines geometrical inversion with respect to the unitcircle centered at the point (119888 0) in the 119911
1equiv (1199091 1199101) plane
Transformation (1) can be conveniently expressed incomplex form as
1
119911
= 1199111minus 119888 (3)
The two-dimensional body in the inverse plane (119911 equiv (119909 119910)
plane) in general has a concave region facing the fluidHere the boundary in the 119911
1plane is an oblate ellipse given
by100381610038161003816100381610038161199111+ 12058221199111
10038161003816100381610038161003816= (1 + 120582
2) (1 minus 120582
2) 0 le 120582 le 1 (4)
A parametric form of (4) can be written as
1199111= 1199091+ 1198941199101= 120577 minus
1205822
120577
(5)
where 120577 = 119890119894120601 0 le 120601 le 2120587 in the 120577-planeThe inverse transformation of (5) is expressed by
120577 =
1
2
1199111+ (1199112
1+ 41205822)
12
(6)
in order that the exterior of the unit circle |120577| = 1maps ontothe exterior of the ellipse Moreover for 119888 gt 1 minus 1205822 and
1199111(119889) = 119889 minus
1205822
119889
= 119888
119889 gt 1 (the radius of the unit circle in 120577-plane) (7)
the exterior of the ellipse inverts into the exterior of the closedcurve in the 119911 plane and vice versa The equation obtainedfrom (4) by using transformation (3) that represents theclosed curve in the 119911 plane is
100381610038161003816100381610038161003816100381610038161003816
119911 + 1205822119911
119911119911
+ 119888 (1 + 1205822)
100381610038161003816100381610038161003816100381610038161003816
= (1 + 1205822) (1 minus 120582
2) (8)
Since corresponding to different values of the parameter 120582 (4)will represent distinct oblate ellipses consequently (8) willalso yield distinct smooth closed two-dimensional objects inthe 119911 plane for a given fixed 119888 It is mentioned in Ranger[2] that 120582 = 0 120582 = 1radic2 and 120582 = 1 correspond to inthe same order a circle a concave body and a circular arcin 119911 plane a figure of an object which is a smooth closedcurvewith a concavity is given in [2] withoutmentioning anymathematical equation for it We note that a kidney-shapedbody may be obtained for 120582 = 1radic2 and 119888 = 1 [3] (Figure 1)
3 Mathematical Formulation and Solution
A uniform shear flow parallel to the 119909-axis in the 119911 planein absence of any boundary may be expressed by the streamfunction
Ψ (119911 119911) sim minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin (9)
where 120596 is the constant vorticity
The stream function (9) can bemapped onto the 1199111-plane
using transformation (3) which yields
Ψ1(1199111 1199111) sim minus
1
8
120596(
1
1199111minus 119888
minus
1
1199111minus 119888
)
2
as 10038161003816100381610038161199111
1003816100381610038161003816997888rarr 119888 (10)
Again the mapping of the stream function (10) by utilizingtransformation (5) onto the 120577-plane leads to
Ψ2(120577 120577) sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
(
1
120577 minus 119889
minus
1
120577 minus 119889
)
2
as 10038161003816100381610038161205771003816100381610038161003816997888rarr 119889
(11)
where 1199111is given by (5) and prime (1015840) stands for differentia-
tion with respect to 120577Now we insert a circular cylinder of radius unity with its
centre at the origin represented by 120577120577 = 1 Since the streamfunction (11) does not have constant vorticity we cannotuse circle theorem [1] or second circle theorem [1] in orderto obtain the resulting flow In this situation we propose aformula that will give the resulting flow and it is
Ψ119877
0(120577 120577) = Ψ
0(120577 120577) minus Ψ
0(
1
120577
1
120577
) + Ψotimes(120577 120577) (12)
where Ψ1198770(120577 120577) and Ψ
0(120577 120577) are resulting and basic stream
functions respectively and Ψotimes(120577 120577) is a perturbation stream
functionSince on the boundary of the circle 120577120577 = 1 therefore
Ψ0(120577 120577) minus Ψ
0(1120577 1120577) becomes zero on the boundary of the
unit circle moreover Ψ0(120577 120577) minus Ψ
0(1120577 1120577) becomes the
same as Ψ0(120577 120577) as |120577| rarr 119889 If we assume that all the
singularities of Ψ0(120577 120577) lie at a distance greater than unity
from the origin then all the singularities of Ψ0(1120577 1120577)
lie inside the circle of radius unity Regarding Ψotimes(120577 120577) we
assume that all the singularities lie inside the unit circle andΨotimes(120577 120577) rarr 0 or a constant on |120577| = 1 and for |120577| rarr 119889
Thus the stream function Ψ1198770(120577 120577) in (12) possesses all the
properties to represent the resulting flowIn the light of (12) the resulting flow for the present case
may be written as
Ψ119877
2(120577 120577) sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
(
1
120577 minus 119889
minus
1
120577 minus 119889
)
2
minus(
120577
1 minus 119889120577
minus
120577
1 minus 119889120577
)
2
+ Ψotimes(120577 120577)
(13)
The function Ψotimeswill be evaluated afterwards in this paper
and we will show that the function satisfies all the conditionsthat the functionmust fulfill in accordance with the proposedformula (12)
International Journal of Engineering Mathematics 3
15
1
05
0
minus05
minus1
minus15
04020minus02minus04
y1
x1
(a)
1
05
0
minus05
minus1
0minus05minus1minus2 minus15
y
x
(b)
Figure 1 (a) Oblate ellipse in 1199111equiv 1199091+ 1198941199101plane and (b) kidney-shaped body in 119911 equiv 119909 + 119894119910 plane (found by putting 1205822 = 12 and 119888 = 1 in
(8))
The flow (13) around the circular boundary can bemapped by using transformation (6) onto the region outsidethe oblate ellipse in the 119911
1-plane which yields
Ψ119877
1(1199111 1199111)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) (1199111+ (1199112
1+ 41205822)
12
) minus 119889
minus
1
(12) (1199111+ (1199112
1+ 41205822)12
) minus 119889
2
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)12
)
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)
12
)
2
]
]
]
]
+ Ψotimes(
1
2
1199111+ (1199112
1+ 41205822)
12
1
2
1199111+ (1199112
1+ 41205822)
12
)
(14)
Again the flow given by (14) around oblate ellipse can bemapped onto the region outside the smooth concave body
given by (8) in the 119911 plane by using transformation (3) whichleads to
Ψ119877(119911 119911)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
minus
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
2
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
2
]
]
]
]
+ Ψotimes(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
(15)
4 International Journal of Engineering Mathematics
4 Uniform Shear Flow around a FixedCircular Cylinder or a Circular Arc ora Kidney-Shaped Cylinder
41 Uniform Shear Flow past a Circular Cylinder For 120582 = 0(8) represents a circle in the 119911 plane the circle is given by
10038161003816100381610038161003816100381610038161003816
119911 +
119889
1198892minus 1
10038161003816100381610038161003816100381610038161003816
=
1
1198892minus 1
since for 120582 = 0 we have 119888 = 119889
(16)
We put 120582 = 0 in the stream function (15) to obtain the streamfunction for flow around the circle (16) as
Ψ119877
3(119911 119911)
sim minus
1
8
120596[(119911 minus 119911)2
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(17)
The transformation
119885 = 119911 minus (minus
119889
1198892minus 1
) (18)
gives us the equation of the circle (16) as
119885119885 = (
1
1198892minus 1
)
2
(19)
Under transformation (18) the stream function (17) takes theform
Ψ119877
4(119885 119885)
sim minus
1
8
120596[
[
(119885 minus 119885)
2
minus
1
119885(1198892minus 1)2minus
1
119885(1198892minus 1)2
2
]
]
+ Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
(20)
Since there can be no change in the value of vorticity nearthe cylinder therefore
41205972Ψ119877
4(119885 119885)
120597119885120597119885
= 120596 (21)
Utilizing (21) on calculation it is found that in (20)
Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
=
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(22)Therefore the result (20) represents uniform shear flow
past a circular cylinder which is in agreementwith the knownresult [1] for the same flow
The relation (22) implies that
Ψotimes(120577 120577) = Ψ
otimes(120577 120577)
=
1
8
120596(
1
1199111015840
1(119889)
)
2
2
(1198892minus 1)2
(120577 minus 119889) (120577 minus 119889)
(119889120577 minus 1) (119889120577 minus 1)
(23)where
1199111015840
1(119889) = 1 +
1205822
1198892 (24)
It is clear from (23) that all the singularities of Ψotimes(120577 120577) lie
inside the unit circle in 120577-plane (since 119889 gt 1) andΨotimes(120577 120577) rarr
0 as 120577 rarr 119889 andΨotimes(120577 120577) rarr (14)120596(1119911
1015840
1(119889))
2
(1(1198892minus 1)
2
) (aconstant) on the circle |120577| = 1 Thus the function Ψ
otimessatisfies
all the assumptions that we have made in proposing formula(12) which therefore effectively gives the resulting flow dueto insertion of a circular cylinder in the flow (11) of whichvorticity is not constant
42 Uniform Shear Flow past a Circular Arc The streamfunction for uniform shear flow past a circular arc can beobtained by putting 120582 = 1 (and when 120582 = 1 119888 = (1198892 minus 1)119889)in the stream function (15) which yields
Ψ119877
5(119911 119911)
sim minus
1
8
120596(
1198892
1198892+ 1
)
2
times[
[
[
(
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
minus (
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
2
International Journal of Engineering Mathematics 5
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
(25)
43 Uniform Shear Flow past a Kidney-Shaped Two-Dimensional Body The stream function for the uniformshear flow past a kidney-shaped cylinder can be obtained byputting 1205822 = 12 and 119888 = 1 in the stream function (15) whichyields
Ψ119877
6(119911 119911)
sim minus
1
8
120596(
2 + radic3
3 + radic3
)
2
times[
[
[
(
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
minus (
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
2
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
times ((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
)
(26)
5 Flow Consisting of a Uniform Stream ofConstant Velocity 119881 Parallel to 119909-Axisand a Uniform Shear Flow Parallel to theSame Axis with Constant Vorticity 120596 past aConcave Body
Here the basic flow in the 119911 plane is
Ψ7(119911 119911) sim minus
1
2
119894119881 (119911 minus 119911) minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin
(27)
Now if we insert the two-dimensional concave body givenby (8) into the flow (27) the resulting flow following ananalogous procedure that we have adopted to obtain streamfunction (15) may be expressed as
Ψ119877
7(119911 119911)
sim minus
1
2
119894119881(
1
1199111015840
1(119889)
)
times[
[
((
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
minus(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
)
minus (
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 3
15
1
05
0
minus05
minus1
minus15
04020minus02minus04
y1
x1
(a)
1
05
0
minus05
minus1
0minus05minus1minus2 minus15
y
x
(b)
Figure 1 (a) Oblate ellipse in 1199111equiv 1199091+ 1198941199101plane and (b) kidney-shaped body in 119911 equiv 119909 + 119894119910 plane (found by putting 1205822 = 12 and 119888 = 1 in
(8))
The flow (13) around the circular boundary can bemapped by using transformation (6) onto the region outsidethe oblate ellipse in the 119911
1-plane which yields
Ψ119877
1(1199111 1199111)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) (1199111+ (1199112
1+ 41205822)
12
) minus 119889
minus
1
(12) (1199111+ (1199112
1+ 41205822)12
) minus 119889
2
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)12
)
minus
(12) (1199111+ (1199112
1+ 41205822)
12
)
1 minus (12) 119889 (1199111+ (1199112
1+ 41205822)
12
)
2
]
]
]
]
+ Ψotimes(
1
2
1199111+ (1199112
1+ 41205822)
12
1
2
1199111+ (1199112
1+ 41205822)
12
)
(14)
Again the flow given by (14) around oblate ellipse can bemapped onto the region outside the smooth concave body
given by (8) in the 119911 plane by using transformation (3) whichleads to
Ψ119877(119911 119911)
sim minus
1
8
120596(
1
1199111015840
1(119889)
)
2
times
[
[
[
[
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
minus
1
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
) minus 119889
2
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
minus
(12) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
1 minus (1198892) ((1119911 + 119888) + (1119911 + 119888)2+ 41205822
12
)
2
]
]
]
]
+ Ψotimes(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
(15)
4 International Journal of Engineering Mathematics
4 Uniform Shear Flow around a FixedCircular Cylinder or a Circular Arc ora Kidney-Shaped Cylinder
41 Uniform Shear Flow past a Circular Cylinder For 120582 = 0(8) represents a circle in the 119911 plane the circle is given by
10038161003816100381610038161003816100381610038161003816
119911 +
119889
1198892minus 1
10038161003816100381610038161003816100381610038161003816
=
1
1198892minus 1
since for 120582 = 0 we have 119888 = 119889
(16)
We put 120582 = 0 in the stream function (15) to obtain the streamfunction for flow around the circle (16) as
Ψ119877
3(119911 119911)
sim minus
1
8
120596[(119911 minus 119911)2
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(17)
The transformation
119885 = 119911 minus (minus
119889
1198892minus 1
) (18)
gives us the equation of the circle (16) as
119885119885 = (
1
1198892minus 1
)
2
(19)
Under transformation (18) the stream function (17) takes theform
Ψ119877
4(119885 119885)
sim minus
1
8
120596[
[
(119885 minus 119885)
2
minus
1
119885(1198892minus 1)2minus
1
119885(1198892minus 1)2
2
]
]
+ Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
(20)
Since there can be no change in the value of vorticity nearthe cylinder therefore
41205972Ψ119877
4(119885 119885)
120597119885120597119885
= 120596 (21)
Utilizing (21) on calculation it is found that in (20)
Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
=
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(22)Therefore the result (20) represents uniform shear flow
past a circular cylinder which is in agreementwith the knownresult [1] for the same flow
The relation (22) implies that
Ψotimes(120577 120577) = Ψ
otimes(120577 120577)
=
1
8
120596(
1
1199111015840
1(119889)
)
2
2
(1198892minus 1)2
(120577 minus 119889) (120577 minus 119889)
(119889120577 minus 1) (119889120577 minus 1)
(23)where
1199111015840
1(119889) = 1 +
1205822
1198892 (24)
It is clear from (23) that all the singularities of Ψotimes(120577 120577) lie
inside the unit circle in 120577-plane (since 119889 gt 1) andΨotimes(120577 120577) rarr
0 as 120577 rarr 119889 andΨotimes(120577 120577) rarr (14)120596(1119911
1015840
1(119889))
2
(1(1198892minus 1)
2
) (aconstant) on the circle |120577| = 1 Thus the function Ψ
otimessatisfies
all the assumptions that we have made in proposing formula(12) which therefore effectively gives the resulting flow dueto insertion of a circular cylinder in the flow (11) of whichvorticity is not constant
42 Uniform Shear Flow past a Circular Arc The streamfunction for uniform shear flow past a circular arc can beobtained by putting 120582 = 1 (and when 120582 = 1 119888 = (1198892 minus 1)119889)in the stream function (15) which yields
Ψ119877
5(119911 119911)
sim minus
1
8
120596(
1198892
1198892+ 1
)
2
times[
[
[
(
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
minus (
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
2
International Journal of Engineering Mathematics 5
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
(25)
43 Uniform Shear Flow past a Kidney-Shaped Two-Dimensional Body The stream function for the uniformshear flow past a kidney-shaped cylinder can be obtained byputting 1205822 = 12 and 119888 = 1 in the stream function (15) whichyields
Ψ119877
6(119911 119911)
sim minus
1
8
120596(
2 + radic3
3 + radic3
)
2
times[
[
[
(
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
minus (
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
2
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
times ((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
)
(26)
5 Flow Consisting of a Uniform Stream ofConstant Velocity 119881 Parallel to 119909-Axisand a Uniform Shear Flow Parallel to theSame Axis with Constant Vorticity 120596 past aConcave Body
Here the basic flow in the 119911 plane is
Ψ7(119911 119911) sim minus
1
2
119894119881 (119911 minus 119911) minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin
(27)
Now if we insert the two-dimensional concave body givenby (8) into the flow (27) the resulting flow following ananalogous procedure that we have adopted to obtain streamfunction (15) may be expressed as
Ψ119877
7(119911 119911)
sim minus
1
2
119894119881(
1
1199111015840
1(119889)
)
times[
[
((
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
minus(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
)
minus (
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Engineering Mathematics
4 Uniform Shear Flow around a FixedCircular Cylinder or a Circular Arc ora Kidney-Shaped Cylinder
41 Uniform Shear Flow past a Circular Cylinder For 120582 = 0(8) represents a circle in the 119911 plane the circle is given by
10038161003816100381610038161003816100381610038161003816
119911 +
119889
1198892minus 1
10038161003816100381610038161003816100381610038161003816
=
1
1198892minus 1
since for 120582 = 0 we have 119888 = 119889
(16)
We put 120582 = 0 in the stream function (15) to obtain the streamfunction for flow around the circle (16) as
Ψ119877
3(119911 119911)
sim minus
1
8
120596[(119911 minus 119911)2
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
minus
(1 + 119889119911)
(1198892minus 1) 119911 + 119889
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(17)
The transformation
119885 = 119911 minus (minus
119889
1198892minus 1
) (18)
gives us the equation of the circle (16) as
119885119885 = (
1
1198892minus 1
)
2
(19)
Under transformation (18) the stream function (17) takes theform
Ψ119877
4(119885 119885)
sim minus
1
8
120596[
[
(119885 minus 119885)
2
minus
1
119885(1198892minus 1)2minus
1
119885(1198892minus 1)2
2
]
]
+ Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
(20)
Since there can be no change in the value of vorticity nearthe cylinder therefore
41205972Ψ119877
4(119885 119885)
120597119885120597119885
= 120596 (21)
Utilizing (21) on calculation it is found that in (20)
Ψotimes(
1
119885 minus 119889 (1198892minus 1)
+ 119889
1
119885 minus 119889 (1198892minus 1)
+ 119889)
=
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(22)Therefore the result (20) represents uniform shear flow
past a circular cylinder which is in agreementwith the knownresult [1] for the same flow
The relation (22) implies that
Ψotimes(120577 120577) = Ψ
otimes(120577 120577)
=
1
8
120596(
1
1199111015840
1(119889)
)
2
2
(1198892minus 1)2
(120577 minus 119889) (120577 minus 119889)
(119889120577 minus 1) (119889120577 minus 1)
(23)where
1199111015840
1(119889) = 1 +
1205822
1198892 (24)
It is clear from (23) that all the singularities of Ψotimes(120577 120577) lie
inside the unit circle in 120577-plane (since 119889 gt 1) andΨotimes(120577 120577) rarr
0 as 120577 rarr 119889 andΨotimes(120577 120577) rarr (14)120596(1119911
1015840
1(119889))
2
(1(1198892minus 1)
2
) (aconstant) on the circle |120577| = 1 Thus the function Ψ
otimessatisfies
all the assumptions that we have made in proposing formula(12) which therefore effectively gives the resulting flow dueto insertion of a circular cylinder in the flow (11) of whichvorticity is not constant
42 Uniform Shear Flow past a Circular Arc The streamfunction for uniform shear flow past a circular arc can beobtained by putting 120582 = 1 (and when 120582 = 1 119888 = (1198892 minus 1)119889)in the stream function (15) which yields
Ψ119877
5(119911 119911)
sim minus
1
8
120596(
1198892
1198892+ 1
)
2
times[
[
[
(
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
minus (
1
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
) minus 119889)
minus1
2
International Journal of Engineering Mathematics 5
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
(25)
43 Uniform Shear Flow past a Kidney-Shaped Two-Dimensional Body The stream function for the uniformshear flow past a kidney-shaped cylinder can be obtained byputting 1205822 = 12 and 119888 = 1 in the stream function (15) whichyields
Ψ119877
6(119911 119911)
sim minus
1
8
120596(
2 + radic3
3 + radic3
)
2
times[
[
[
(
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
minus (
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
2
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
times ((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
)
(26)
5 Flow Consisting of a Uniform Stream ofConstant Velocity 119881 Parallel to 119909-Axisand a Uniform Shear Flow Parallel to theSame Axis with Constant Vorticity 120596 past aConcave Body
Here the basic flow in the 119911 plane is
Ψ7(119911 119911) sim minus
1
2
119894119881 (119911 minus 119911) minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin
(27)
Now if we insert the two-dimensional concave body givenby (8) into the flow (27) the resulting flow following ananalogous procedure that we have adopted to obtain streamfunction (15) may be expressed as
Ψ119877
7(119911 119911)
sim minus
1
2
119894119881(
1
1199111015840
1(119889)
)
times[
[
((
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
minus(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
)
minus (
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 5
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
minus
1
2
((
1
119911
+
1198892minus 1
119889
) + (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
)
times (1 minus
119889
2
((
1
119911
+
1198892minus 1
119889
)
+ (
1
119911
+
1198892minus 1
119889
)
2
+ 4
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
1
2
1
119911
+
1198892minus 1
119889
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
(25)
43 Uniform Shear Flow past a Kidney-Shaped Two-Dimensional Body The stream function for the uniformshear flow past a kidney-shaped cylinder can be obtained byputting 1205822 = 12 and 119888 = 1 in the stream function (15) whichyields
Ψ119877
6(119911 119911)
sim minus
1
8
120596(
2 + radic3
3 + radic3
)
2
times[
[
[
(
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
minus (
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
minus (
1 + radic3
2
))
minus1
2
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
minus
1
2
((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
)
times (1 minus
1 + radic3
4
times ((
1
119911
+ 1) + (
1
119911
+ 1)
2
+ 2
12
))
minus1
2
]
]
]
+ Ψotimes(
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
1
2
1
119911
+ 1 + ((
1
119911
+ 1)
2
+ 2)
12
)
(26)
5 Flow Consisting of a Uniform Stream ofConstant Velocity 119881 Parallel to 119909-Axisand a Uniform Shear Flow Parallel to theSame Axis with Constant Vorticity 120596 past aConcave Body
Here the basic flow in the 119911 plane is
Ψ7(119911 119911) sim minus
1
2
119894119881 (119911 minus 119911) minus
1
8
120596(119911 minus 119911)2 as |119911| 997888rarr infin
(27)
Now if we insert the two-dimensional concave body givenby (8) into the flow (27) the resulting flow following ananalogous procedure that we have adopted to obtain streamfunction (15) may be expressed as
Ψ119877
7(119911 119911)
sim minus
1
2
119894119881(
1
1199111015840
1(119889)
)
times[
[
((
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
minus(
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
minus 119889)
minus1
)
minus (
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Engineering Mathematics
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
minus
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
times (1 minus 119889
1
2
(
1
119911
+ 119888) + ((
1
119911
+ 119888)
2
+ 41205822)
12
)
minus1
)]
]
+ Ψ119877(119911 119911)
(28)
where Ψ119877(119911 119911) is given by (15)The resulting flow due to insertion of a circular cylinder
represented by the circle (16) into the oncoming flow (27) canbe obtained by putting 120582 = 0 (and when 120582 = 0 119888 = 119889) in (28)as
Ψ119877
8(119911 119911)
sim minus
1
2
119894119881[(119911 minus 119911) minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)]
minus
1
8
120596[(119911 minus 119911)2minus (
1 + 119889119911
(1198892minus 1) 119911 + 119889
minus
1 + 119889119911
(1198892minus 1) 119911 + 119889
)
2
]
+ Ψotimes(
1
119911
+ 119889
1
119911
+ 119889)
(29)
Using transformation (18) (ie considering the centre ofthe circle (16) as the origin of the coordinate system) andwith the help of relation (22) the stream function (29) canbe written as
Ψ119877
9(119885 119885)
sim minus
1
2
119894119881[
[
(119885 minus 119885) + (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)]
]
minus
1
8
120596[
[
[
(119885 minus 119885)
2
minus (
(1 (1198892minus 1))
2
119885
minus
(1 (1198892minus 1))
2
119885
)
2
]
]
]
+
1
8
120596(2
(1 (1198892minus 1))
4
119885119885
)
(30)
The resulting flow due to inclusion of the circular arc intothe basic flow (27) can be obtained by putting 120582 = 1 (andwhen 120582 = 1 119888 = (1198892 minus 1)119889) in (28) as
Ψ119877
10(119911 119911)
sim minus
1
2
119894119881(
1198892
1198892+ 1
)
times[
[
[
((
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
minus (
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
minus 119889)
minus1
)
minus(
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+ ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
minus
1
2
(
1
119911
+
1198892minus 1
119889
) + ((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
times(1 minus 119889
1
2
(
1
119911
+
1198892minus 1
119889
)
+((
1
119911
+
1198892minus 1
119889
)
2
+ 4)
12
)
minus1
)]
]
]
+ Ψ119877
5(119911 119911)
(31)
where Ψ1198775(119911 119911) is given by (25)
The stream function for resulting flow due to insertion ofthe kidney-shaped two-dimensional body into the oncoming
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 7
flow (27) can be obtained by putting 1205822 = 12 and 119888 = 1 in(28) which leads to
Ψ119877
11(119911 119911)
sim minus
1
2
119894119881(
2 + radic3
3 + radic3
)
times[
[
((
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
minus(
1 + radic3
2
))
minus1
)
minus (
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times (
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
minus
1
2
(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
times (1 minus (
1 + radic3
4
)
times(
1
119911
+ 1) + ((
1
119911
+ 1)
2
+ 2)
12
)
minus1
)]
]
+ Ψ119877
6(119911 119911)
(32)
where Ψ1198776(119911 119911) is given by (26)
The function Ψotimesis given by (23) Therefore the stream
functions (15) (20) (25) and (26) explicitly give resultingflow for the uniform shear flow past a concave body a circularcylinder a circular arc and a kidney-shaped body respec-tively And stream functions (28) (29) (31) and (32) explicitlygive the resulting flow around a concave body a circle acircular arc and a kidney-shaped body respectively wherein each of the cases the oncoming flow is the combination ofthe uniform stream and the uniform shear flow
6 Conclusions
In this work we have obtained a stream function in com-plex variables for inviscid uniform shear flow past a two-dimensional smooth body which has a concavity facing thefluid flow It is found that the result for the same flow pasta circle that is deduced from the central result as a specialcase agrees completely with the known result Also streamfunctions for the same flow past a circular arc or a two-dimensional kidney-shaped body are evaluated
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author would like to thank Professor Dr S K Sen(retired) of Jamal Nazrul Islam Research Centre for Math-ematical and Physical Sciences University of ChittagongBangladesh for valuable discussions during preparation ofthis paper
References
[1] L M Milne-ThomsonTheoretical Hydrodynamics MacmillanLondon UK 5th edition 1972
[2] K B Ranger ldquoThe Stokes flow round a smooth body with anattached vortexrdquo Journal of EngineeringMathematics vol 11 no1 pp 81ndash88 1977
[3] J M Dorrepaal ldquoStokes flow past a smooth cylinderrdquo Journal ofEngineering Mathematics vol 12 no 2 pp 177ndash185 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of