aerodynamics by karamchetti

Principles of Ideal-Fluid AerodynamicsKrishnamurty KaramchetiProfessor of Aeronautics and Astronautics Stanford University

Preface

The aim of this book is to explain the basic principles and analyticaJ methods underlying the theory of the motion of an ideal fluid (an inviscid incompressjble fluid) and the role of the theory in describing and predicting the flows associated with the motion' of certain bodies of aerodynamic: interest suc:h as wHigs and bodies of revolution. I have attempted to describe ideal-fluid aerodynamics, although restricted to certain problems, . .as a branch of theoretical physics. The subject is' developed from basic: principles showing clearly the complementary features of physical understanding and the mathematical handling of the theory.. The intention is to show the role of physical understanding in mathematical formulation, .to ~ring out the motivation for the mathematical language and, methods employed and the necessity for a!,plying a certain,amount of mathematical rigor in arriving at physically appealing solutions. The book is written to serVe as a sel~-contained text at the senior under.. graduate or first-year graduate level. The idea is not to give inadequately explained solutions to many special problems, but rather to present, for~ few selected practical problems, a unified treatment leading from ba.ic principles to practically meaningful results. A large part of the book deals with general concepts and mathematical methods, always related, however, to the solution of problems. In this way I hope that the book will perform the valuable function of teaching subject matter related to a broader methodology that will lead logically to more advanced topics and methods in fluid mech.anics; it should be of interest to students in various disciplines, such as applied mathematics, physics, and engineering. This book has grown out of lectures on aerodynamic theory which I have offered for the last decade and which have been received with considerable enthusia>m. It is because of the students' encouragement that I venture to publish thnm. I am greatly indebted to Professor Irmgard Fliigge Lotz for reviewing the manuscript and fOr many valuable suggestions and discussions. My special thanks are due to Dr. Maurice L. Rasmus~n who read the manu script and offered valuable critic'sm. Many students have helped me enthusiasticaUy with the preparation of the book, and my deep appreciation goes to all of them.

ali

face

I am very grateful to Professor O. O. Tietjeos for furnishing me with original prints of many of the fto.w photographs. . The original source of the- photographs for the plates and 9 is the National Physical Laboratory, Engl~nd, and I am gready obJiged to its Director for permission to reproduce the photographs which are Crown copyright. The original photographs for plates 3, 4, 6a, and 7 are all from prewar Germanpubiication$, and J wish to r~ord my indebtedness to their respective sources. Plates 3 and 4 are after F. Homann, Forschrmg auf dem Gebiete des Ingenieurwesens, 7 (1936). Plate 68 is after L. Prandtl, Handbucli cler Experimentalphysik, 4, Part } (Leipzig. J931). Plate 7 is after Piandtl, The Physics of Solids and Fluids (London, 1~30) The typing was capably handled by' Mrs. Katherine Bradley, Miss Gail Lemmond, and Mrs.' Elaine Morris. My sincere thanks to them, Finally I wish to express my appreciation to John Wiley and Sons for the understanding, patience, and encouragement they have extended me over the years. Krishnamurty Karamcheti

a

Contents

1. INTRODUCTION. . . . . . . .

.. - ...... - . , ..

1

Stanford, Caltf0miD August 1966

1.1 Fluid as a Continuous Medium . . . . . . . . . 2 1.2 Properties of a Fluid at Rest: Thermodynamic properties; Compressibility.; Incompressible fluid; : Heat conduction and the coefficient of thermal conductivity . . . . . . . 2 1.3 Properties of a Fluid in Motion: Friction or viscosity; Coefficient of viscosity; Compressibility; Heat transfer. 4 1.4 Laminar and turbulent motioos . . . . . . . . . . . . 10 1.5 Some Relevant Parameters: Relative magnitude of the foroes, Froude number, ReynOlds number, and M~h number; Par.ameters characterizing compressibility; PrandtJ number; Parameters on which force and heat transfer depend . . . 13 1.6 Range of Some Parameters . . . . . : . . . . ,23 1.1 Conditiong for Neglecting Compressibility' Effects; Case of liquids; Case of gases. . . . , . . .', . . . . 23 1.8 Conditions for ~eglecting Gravity EffeCts . . . " . . 25 1.9 Nature of the Problem when Compressibility Effects are Negligible. . . . . . . . . . . . . . . . . . . . . . 25 1.10 Variation of Fl(>w Patterns with Reynolds Num~; Flow past bluff bodies; Flow past stteamlined bodies . . . . . . ". 26 1.11 Variation of Flow Pattern with Mach number . . . . . . . 3S J.l2 Effects of Viscosity at High Reynolds Numbers:, The Boundary layer: Boundary layer concept; Some characteristiCs of the lami.nar boundary layer; Turbulent boundary layer; separauon; Wakes . . . . . . '. . . . . . . . . . . . '. . 1.13 Consequences of the Boundary-Layer Concept .. ' 1.14 Ideal Fluid Theory . . . . . . . . ' . . . . . . . . . .

2. ELEMENTS OF VEcrOR ALGEBRA AND ,u(x

+

dx, y) = u(x, y)~

+au

Fla. 1.12 Schematic representation of .the II-vclocity distribution in' the boundarylayer over a flat plate.

We then havev(x, y) = -

situated at x

+ tIz.

The rate of inflow through the face at x is equal top

LVu(x, y} dy+ dx is equal to + dx, y) dy

Note that au/ox is negative, At the plate v is zero. At the edge of the boundary layer y = 6(x) the v-component does not vanish but has a vaiue 'given byv(x, b(x

lo -ax dy

whereas the rate of outflow through face at xpLVu(x

=-

l

'CZ)

o

-

au

ax

dy

(1.30)

On the basis of this relation we may state thatv(x, b) __ U 6(x) x

SinCe u(x + tIz, y) is less than u{x, y), the outflow does not balance the inflow. This means that becl\use there is no accumulation of mass in the box, there must be Ii flow of fluid through the y-faces of the box to balance the net inflow through its x-faces. There must therefore be a v velocity. However, there is no flow through the y-face at y == 0, which is the plate itself. If v(z, y) is the y-component of the velocity at the point x, y, the

or(1.31)

We thus conclude that the v-velocity is small, ,the ratio vi U being of the same order of magnitude as blx. We next consider the variation of pressure across the boundary layer. The pressure, force in the y-direction, aplay per unit. volume, on a fluid element in the boundary layer is of the same order of magnitude as that of the inertial force, pv2/b per unit volume, on the element in that direction. We therefore have(1.32)

The'magnitude of the pressure change ll.p across the boundary layer is therefore given by [ b(x)] I U2 ll.p -- pv' __ put ' __ 12(1.33)xFIa.l.13 Illustrating the need for the v-velocity in the boundary la)~r.

R.,

We conclude that in the c:>undary layer l'patial variation of pressure in the The viscous force in that direction is of the same order of magnitude.

46

Ideal-Fluid Aerodynamics

Introduction

47

y-direction and the actual change in pressure across the layer are negligibly 5mall. As we shall see later, this conclusion has far-reaching implications. The solution for the problem of steady incompressible flow in a laminar boundary layer alonga flat plate, as formulated by Prand~l, was obtained by Blasius in 1908. His solution shows that" (defined as the value ofy when u is within 0.6 per cent of U) is gi\l'en by

The subscript e. signifies that the Q\l8iltity refers to the edge of the boundary l@yer.

r",.bIIk", lloIuuItuy Lqu. ObserYatiOQs shoW that at high Reynolds pumbers the flow in a boundary layer does not remain laminar all along the surface of a solid body but usu&lly becomes turbulent at some distance

b(z) ,5.2 --;- = ..jR"The local skin friction coefficient is given by

(1.34)

6.0

C = 'To(x) _ ~1-

ipU

I

-

..jR"

(1.35)5.0

R-+ 1.(8)( lOSct 1.82 )( 105 03.64)( 105 5.46)( 10& .728)( 105

The distribution of the velocity across the boundary layer is similar at different x-stations along the plate. This being the case, va:-iation ofu with x and y may be represented by a single curve. Such a curve, as obtained by Blasius, is !>ilo\\ '1 in Fig. 1.14, where ulV is plotted against the parameter 7] = y..j V{vx. Als( shown in the figure are experimental results obtained by N!kuradse (194: t. It is seen that the theory and the experiment are in excellent agreeme' t. This agreement is also found for the local skin friction coefficie' .t, shown in Fig. 1.15. The experimental results are obtained by Liepmann and Dhawan (1951, 1953) by direct measurement of the skin frictio!", coefficients. So far we have co~sld,ered tbe features ofa laminar boundary layer along a plane surface. Similar results also hold for the boundary layer along a curved surface if certain conditions are met. It is necess"-ry that the radius of curvature of the surface is everywhere large compared to the boundarylayer thickness and that there is nOne of the variation in curvature that would occur near sharp edges. Under such conditions the flat-plate results may be applied to the prol'!em of boundary layer along a curved surface if. we now regard the coordinates x, y as defining a suitable system of curvilinear coordinates given by curves parallel to the given surface and straight lines normal to the surface. The surface itself is giyen by y = 0 (see Fig. 1.16). For flow past a curved surface the velocity component parallel to the wall at the edge bfthe boundary layer is not a constant, as it is' in the case of the flat plate, but a function of ~he distance x. Similarly the pressure at the edge of the boundary layer is also a function of 2:. We therefore introduce the following notation:

~: 4.0

""

.

3.0

2.0

1.0

1.0

I'll. 1.14 Velocity distribution in the laminar boundary layer on aExperimental re,suhs from Nikuradse (1942).

ftat plate.

V, = V.(x) = u(x,

Thus a vector function of- a vector is equivalent to a system of three , independent sealar functions of three scalar variables. In a similar maAner, the functi,ons rf, ~ rf,(r. t) a~d A = A(r, t) can be express~d in terms of scalar functions of scalar vanables.

"


Elements of Vector Algebra and Calculus

are parallel vectors. Recalling that the vector product of two parallel vectors is zero, we write tis )( A == 0 (2.2~) This, the~, is the (differential) equation that determines (at any instant) field lines of the vector field A - A(r, I). In determining the streamhnes of the velocity field of a fluid in motion we shall have occasion to study the integration of an equation oftbe type (2.26). Using the p,inciple of decomposition of a vector into components, Eq. (2.26) can be readily specialized to any chosen coordinate system. Since the function A - A(r, t) is equivalent to three scalar functions of position and time, any vector field may be regarded as equivalent to three scalar fields.t~e

"

FI~. 2.26 Velocity field of a rotating rigid body .as.seen in a plane normal to the axisof rotation.

2.21

DHrenatiatioo of'. Vect6r Faac:tioa of. Scalar Variable

If a vector A ehanges from a value A, to a value A., the increment in A denoted by &A is simply the vector difference between A. and Ai' That is,

&A::oo A. - AiA change in ~ ~cct~r ~y be brought about by a change in its magnitude or by a change ID Its dm:ctlon or by a change in both magnitude and direction. If a vector A is a function of a scalar variable'l, the increment M in A corresponding to an increment &1 in I from I to I + &t is given by

lines remains unchanged with time; otherwise tbe picture changes from instant to instant. To cOnstruct analytically the field lines of a vector field" :: A(r, I), we proceed as follows. Consider the field line passing through a point r at some instant oftime. Let tis ~enote an element of the line through r. By definition tis has the same direction as that of the vector '" associated with the point r at the instant considered. That is, tis and A

M - A(I

+

&/) - A{I)

If the ratio &A/ &1 (i.e., the average variation of A with respect to t in the iQterval &1) tends to a limit when &t tends to zero, that limit is called the d'erivative of A with respect to I (compare the definition of the derivative of a scalar function of a scalar variable). Fo1l9wing the usual convention of differential calculus. we denote this derivative by dA/dt and writedA(r) = lim ~A = lim A(t dt ~I-O ~t . ~I"'O

+ ~t) ~t

A(t)

(2.27)

Let us now look at the geometrical interpretation of this derivative. If we represent the various values of the continuously varying ~ector A by means of arrows drawn from a common origin, denoted by 0, the terminus of the vector will describe a curve, denoted by ~; in space (seeFig. 2.28a). i..et 'Qp represent A at time t and OQ represent it at time t + M (Fig. 2.2gb). Then the increment ~A is represented by the vector

chollliQ ef the curve~. Thus we have~A

.... 1..l7 Field lines of the. veloci'Y field of Fig. 2;26.

-

chord PQ~t

--"

90

Ideal.F1uid Aerodynamics


'1

and

point P. Denoting bye, a unit tangent vector at P (sec figure). we write~

dA .Ii M.. I' PQ - - m - - Imdt A'''O fl.t A'''O fl.t

---'

(2.28) With this, Eq. (2.28) becomes

A.~O

lim PQ

as

== e.

To interpret the'limit we proceed as follows. A point such as P .or Q on the curve t'6 can be specified by giving either the v~tor A or the distance s measured along the curve from some initial point taken as ~lerence (sec . figure). As t varies. $ will change justas A doeS; so $ - $(1). and A may be

dA dtexpre~sing

== - e.

ds dt

(2.29)

the derivative as the product of a magnitude and a direction. As an example of the above consloerations, let us consider the motion of a mass particle. At any instant let its position be denoted by r == r(1) measured from a point fixed in a chosen frame of reference. The path (or the trajectory) of the particle is given by the curve j traced by the vector r as r varies. The velocity V of the particle at any instant (i.e., at the position r) is given by the derivative dr/dr. Thus we have

v == dr(t)o(t;J) (6)(b)

dt

= ~e dt'

==

Ve

FIo 1.21 (a) Space curve traced by A(t):

illustrating the differentiation oC A(t).

considered as depending on s. Let /).S denote the increment in s'from P to Q. Therefore fl.s = length of the arc PQ Introducing /).s, we rewrite Eq. (2.28) as

dA "-' lim PQ = (lim (lim /).s) dl 4t~O fl.B fl.t 41-0 /).s At~O /).1~

~~

PQ)

= ds lim PQ dt A.-O asNow, PQ/fl.s is a vector along iQ with a magni!'\) ~ equal to---'

(2.28a)

stating that the velocity is tangential to the tnrjectOlY at the instant considered and that the magnitude V of the velocity is equal to the rate of change of distance along the trajectory (i.e., to the speed). As a simple result following Eq. (2.29) we note that the direction of thl! derivati,'e dAidt when A is of constant length but of changing direction is perpendicular to the rector A. We consider next the differentiation of the sums and products of vector functions a!l of which depend on the same scalar variable. In all such case!. the formal methods of differentiation as employed in scalar calculus. are equally Hpplicable except that in cases involving vector products, Ihe order of the vectors must be preserved. This is. of course, a natural consequence of the fact that vector products are not commutati,Ve. ACL'Ordingly, we have the following results. . The higher derivatives of the function A = A(:) uc cl'nstructed by successive differcliti.ilion just as in scalar calculus. If U =~ 11(t) is the sum of two fUnCli,)ns such thattJ( t) =-= A(t)

length of the (:hord PQ length of the arc PO As /).S -+ 0,

+

B(f)

we JlIl.\'-:JU dA dB -=_.-1--

I~I-land the direction of PQ. becomes that of the tangent to the curve ~ at the Thedircctio,~

titof the tangent vector is

ril' d tt;~en

in the direction of increasing s.



93

If, however, the unit vectors are also changing with the scalar t, we have

If u = U(/) ami A = A(/), we have

d dA du -(uA) = u - +-Adt . dt dt

dA dAI A del dA 2 4 de2 - = el + 1 - + ez + 2 i

dt

dt

dt

dt

dt

+ dA3 e3 + dt

A dea 3 dt

(2.3Oc)

Here the order of the factors involved need not be preserved. If A = A(t) and B = B(t), we obtaindt (A B) - A

d

di + di . B

dB

dA

Since the scalar product is commutative, the order of the vectors in this differentiation need not be preserved. The derivative of the cross product' A(t) )( B(t) is given bydt

~ (A )(

B) = A )( dB dtdB dt

+ dAdt

)( B

To illustrate the case in which the reference unit vectors are also chanJlOg let us consider the description in cylindrical coordinates of the motion of a mass particle. Accordingly, we denote at any instant the position of the particle by r, 0, z and bye" e" e. the corresponding unit vectors. We ask for the .velocity, V, of the particle at the instant considered. By definition, the velocity of the particle is equal to the rate of change of its position. Thus if R = R(t) gives the position of the particle with respect to a fixed point, we have dR V = Vet) = dt

and is not equal to

In cylindrical coordinates

A )( -

dA + B x -', etc.dt

Therefore we obtainV = - (re, dt

Since the vector product is not commutative, here the order of the vectors should be preserved. Considering triple products, we have

d

+ ze.)

d dA dB de - (A B)( C) = - . B )( C +A - )( C + A . B )( -dlit dt . dt t

Since the direction of the unit vector e, changes with change of location of the nass particle, this equation expands to

and

~ [A x (8 x C)] = dt

dA x (B x C) + A x (dB xdt dt

C) + A x (B x ddC ),. t

dr V=-e dt'

+r-+-e

de, dt

dz. dt'

Here again the order of the vectors has to ~ pr~served. , In concluding this section we rdate the derivative of the vector A(t) Wlt~ the derivatives of its components. To do this we choose a system of UOit vectors el> ea, e, and expressA(t) = A!el

To evaluate> the rate of change of the unit vector e, we proceed in the same way as in deriving Eq. (2.29) and obtain de, dO -=-e. dt dt With this relation, the velocity expressed In cylindrical coordinates becomes dr dO dz . V = - e + r - e + - e. (2.31) dt' dt' dtIf we did not recognize that e, is changing. we woutd have arrived at the incorrect result that the velocity of the particle is equal to

+ A 2e Z + Ases'dt

We therefore obtain dA d -- = - (AIel)dt dl

d +-

(A i e 2)

d + -d (A 3e3)t

(2.30a)

If the unit vectors are constant, this reduces to

~~dt

= dAl e

at

1

+ dAdt

2

e2

+ dAs e3dt

(2.30b)-eT

}.. .:hange in a unit vector in a particular direction and refer to it as a directional deriL"!1til'e. According to these ideas it would appear that to describe completely the spatial variation at any puint of a function 4> = 4>(r), we may have to specify the derivatives of cb in all possibie directions at thElt point. Fortunately, however, this is not necessary. It turns out, as we shali sce, that all that is necessary is to give the derivatives of 4> in three indepcnuent directions. These three derivatives are then sufficient to dterminc the variation of =

o~ - dx

ox

o~ o~ + -oy dy + -oz dz\)f

Recalling that the scalar product of two vectors is the sLim

the products

1()(}



101

of their corresponding components. we rewrite Eq. (f44) as

dcp = (acp i ax ax

+

ay

acp j

+

acp

az

k) . (dxi + . ds

dyj

+ dzk)(2.45)

acp . acp . at/J) = ( -I+-J+-k ay

where hI> h 2 ha may be referred to as scale factors. These factors may vary from point to point; in other words they are. in general, functions of position. They can be determined in terms of ql, q2' q3' This is shown in Section 2.45 where further details are given. We may now write Eq. (2.47) as

az

dt/J = = =

.!. ot/J hi dql + 1. at/J hi dql + !. at/J ha dq3hi Oql _hi oqlha aqa'

Dividing Eq. (2.45) by ds we obtain

dcpds

= (acp i + at/J j + ot/J k) .dsax oy

az

.!. at/J l5s1 + .!.. ot/Jhi aql hi OQl1

ds

hi aq,h z aql

l5s1

+ 1. ot/J bS 3ha aqs

= (at/J i

ax

+ at/J j + ot/J k) .eoy oz

(2.46)

(1. at/J e + .!.. ot/J ez + .!. :ot/J es )ha 0'13

(!5s l e l

+

!5s ze z + oSse 3 ) (2.48)

We thus see that the directional derivative of t/J(r) in any chosen direction is equal to the component in that direction of a particurar vector. The components of that vector are the partial derivatives of t/J with respect to distances along the three coordinate axes (see Fig. 2.30). We arrive at the same conclusions if instead of Cartesians we choose any orthogonal, curvilinear coordinates. To see this, let us denote a point P i,l space by a set of orthogonal, curvilinear coordinates ql> qz, qs (see Section 2.17). Then the scalar function of position t/J is expressed as t/J = t/J(r) = t/J(q1> qa. q3) Note that qh q2, qa are coordinates and are not necessarily the components of r. As before, let ds = dse denote a differential increment in r in some direction e from the point P. If dq1> dq2, dq3 are the corr~sponding increments in the coordinates, the differential increment de/> over the divided distance ds is given by (2.47) The partial derivatives appearing in Eq. (2.47) are not derivatives with respect to distances. Also, dq1> dql' dqs are not components of ds. To express Eq. (2.47) in aform similar to Eq. (2.45), we proceed as follows. Let S1> S2. Sa denote distances measured from P along the ql' qz, qs curves respectively. Let OSlo !5s., bs, denote the distances along ql' qz, q3 curves corresponding to the increment's dql' dql, dq3' Let us say that the dq's and the !5s's are related as follows

where e1 ea. e a are the unit 'leCtors at P in the directions of the coordinates ql. qa, qa (See Section 2.17 an.d Fig. 2.21). Now since I5S1> bs 2 !5s s are differential lengths along the ql, qz, q3 curves, we have ds = !5s1e l

+ bsze z + oSSe 3ez +

(2.49)

By using this relation, Eq. (2.48) may be rewritten as

dt/J =

(.!. ot/J

hi aql

e1

+ 1. ot/J

hI aq,

.!.. acb

h3 aq3

e3 )

d"

(2.50)

which has the same form as Eq. (2.46). Dividing Eq. (2.50) by ds we obtain the spatial'derivative of t/J in the direction e as'

(2.51 )

We thus see that to every scalar function of position there corresponds at each poin~ a particular vector which determines at that point the spatial variation of the scalar function. We call such a vector the gradient of the function and denote it by the word grad. With this notation Eq. (2.50) can be put in the form (2.52) dcp = grad cp ds and (2.51) in the formd cp = grad cp f ds

bS I = hi dql bS a = hI dqz bS 3

(2.53)

== h. dqa

The gradient is then.defined by stating that in any orthogonal, curl'ilinear

101


Elements of Vector Algebra anJ Calculus

I(JJ

coordinate system the components of the gradient of a scalar function of position are simply the partial derivatives of the function with respect to distances in the directions of the respective coordinate axes. Symbolically, we writegrad t/> _ (at/> e lOSI

+ at/> ~ + at/> ~)as. ass

but the field of the gradient of the scalar function in question. With this field interpretation, an interesting result follows from Eq. (2.53). Consider any level surface ofthe scalar field t/>(r). By definition", is a constant on such a surface. That is, at any point on a level surface

=

(1. at/>

hi Qql

el

+ 1. at/> ~ + 1. at/>h. oq.

hi oqa

ea)

(2.54)

dt/> -0 dsfor every direction lying in the surface. Then from Eq. that

(i. 53)

it follows

The components of grad t/> in Cartesians are given by

at/> at/> at/>

ox' oy' 0%in cylindrical coordinates r, 8,%

by

at/> lot/> at/> ar '; a8 '

az

and in spherical coordinates r, 8,

qJ

by q, =constantsurface

at/> 1 at/> 1 at/> or ' ; 08 ' r sin () aqJFurther significance of a gradient can be gathered from Eq. (2.53). If, at any point, the components ofgrad t/> are constructed in different directions, the component that is numerically the greatest will be in the direction of the gradient itself and will be of the same magnitude as that of the gradient. This means (from Eq. 2.53) tb.at the greatest value of the derivative dt/>Ids at a given point occurs in the direction of grad t/> and equals the magnitude of the gradient. Conversely, we may state that at any point the gradient of a scalar function of po~ition t/>, is equal in magnitude and direction to the greatest derivative of t/> with respect to distance at that point. In general, the gradient ofa scalar function of position varies from point to point of the region of space in which the function is defined. Thus. to every scalar function of position t/>(r) there corresp(r), which describes the spatial variation of rP at all points of the space concerned. ' Since a scalar function of position describes a scalar field, the concept of a gradient is directly applicable to a scalar field. Thus the spatial variation of a scalar field is given by a certain vector field, which is nothing Although for convenience we use the representation given in the top line of (2.54), we always mean by that lhe repre~litaticn given in the lower line of (2.54).

oFla. 2.31 Level surface.

when e lies in a level surface. Since a vector has no component in a direction normal to itself, we conclude that grad t/> at any point is normal to the lhel surface passing throQ'gh tha, point (Fig. 2.31). If we m~p the scalar field t/> by means of its level surfaces and draw at the. same time the field lines of grad t/>, we find that the field lines. intersect the level surfaces orthogonally. We have defined here the gradient by means of a differential operation. A definition of the gradient by means of an integral operation will be given in Section 2.31.I

2.25 Differentiation of a Vector Function of a VectorConsider a vector function of position A = A(r). Here again the independent variable is a vector. Therefore, when we speak a~o~t the spatia! variation of A, we must say in which direction that vanatlon IS constructed. To see what is required to describe completely the spatial rate

101


Elements

or Vector

Algebra and Calculus

105

of change at any point of the vector function A(r), let us choose Cartesians and write A(r) == A) + A.j + A.k As before, let ds == dse be a differential increment in r in some direction e. Let dA be the differential increment in A over the directed distance ds, and let dA",. dAw. and dJlf. be the corresponding increments in the scalar components of A. We therefore write (2.55) dA = dA) + dA.J + dA)I Since A." Av. A. are scalar functions of position, according to Eq. (2.52), the increments dA,., etc., are given bydA,

If instead of Cartesians we choose a system of orthogonal, curvilinear coordinates, we would arrive at the same conclusion. In this case, however, the elements of the tensor describing the spatial variation of A(r) are not simply the partial derivatives of the components of A with respect to distances along the coordinate axes. They now include additional terms that arise due to the fact that the directions of the reference unit vectors change with change of position. To see this, we choose some orthogonal, curvilinear coordinates"!t> q., q. and express A(r) as A(r) == Aiel + A.el + A.e.wnere elt el, e. are' the reference unit vectors associated with the point r andA It AI' A. are the components of A with respect to the system e l. et, e,. The compOnents and the unit vectors .are functions of position. The differential increment in~A over a directed distance tis == dse is then given by dA - {(dAI)e1

== ds grad A,

(2.56)

where the subscript rmay.be z, y, or z. By using (2.~6), Eq. (2.55) may be rewritten as dA = (tis. grad A~)i

+ (tis. grad A.,)j + (ds grad A.)k

(2.57)

+ (dAJe. + (dA.)e.} + {AI(del ) + A.(de,J + A.(de,J}

(2.59)

Dividing this equation by ds we obtain, in Cartesians, the derivative of A with respect to distance in any direction easdA = (c. grad A.,)i ds

where dA I , etc., are the changes in the components over the distance cis, and del' etc., are the corresponding changes in the unit vectors. Now, as before, we can write (dAI)el

+ (e. grad A.)j + (e grad A.)k

(2.58)

+ (dAJe. + (dA.)e.== (tis grad AI)e1

Equations (2.51) and 0.58) show that to determine the sPatial variation of A(r) in any direction from a given point, we need to kn~w at that point a set of three vectors associated with 4., namely the gradients of A. z' A A v' '" Equivalently we need to know a set of nine numbers that constitute these three vectors. In Cartesians, the set of nine numbers is given by the array

+ (tis grad AJe. + (tis grad A,)e.+ (cis XJe.+ (tis XJe.

(2.60)

After evaluating del' etc., (see Section 2.45) it is possible to ..-rite AI(deJ

+ A.(de,J + A.(de,J.. (tis. Xl)el (2.61)

eM,. aA,. oA.,.

ax oA. ox. oA. ox

oy

oz

oy az oA. oA.. oy oz

~ ~

where Xi, X., X. are vectors. involving AI' A., A. and components of the vectors del' de., and de~. Verify this for cylindrical and spherical coordinates. Combining the relations (2.60) and (2.61) and introducing the notationWI

== grad Al + Xl(2.62)

+ X2 WI == grad As + X,W t := grad A2

The elements of the array are the various partial derivatives of the components of A with respect to distances along the X, Y. Z axes. The elements obey the same rules. as the elements of whai is known as a second-order tensor. We may, therefore, describe the array of nine numbers as a second-order tensor and state that the spatial variation of A(r) is specified completely by a second-order tensor. .

equation (2.5 0 ) may be or, dividing by ds, as

~ewritten

as(2.63)

dA = (ds WI)e l

+ (ds Wz)e;! + (ds W,)e,

dA - = (e W1)eI + (e W 2 )e2 + (e Wa)eads

(2.64)

106



1(17

where e is the direction of tis. Thus we see tnat in any orthogonal, curvilinear, coordinate system the spatial variation at any point of A(r) is given by three vectors or by a second order tensor. As seen from relation (2.62), the elements of the tensor are not simply the partial derivatives of the components of A with respect to distances along the coordinate axes. The tensor that specifies completely the spatial variation of A(r) is known as the tensor ~radient of A, usually denoted by the symbol grad A. We thus have (2.65)

This tensor is usually known as the 'rotation of A. As seen, it is antisymmetrical about the diagonal formed by the elements that are zero. The strain of A, being symmetr:ic, actually contains six independent elements. The rotation of A, being antisymmetric, actually cont~ins three independent elements. In other words, the rotation of A is actually specified by a set of three numbers. This set of three numbers, as may be verified, obeys the same rules as a set specifying a vector. This means the rotation of A, although it is a second-order antisymmetric tensor, oan be represented by a vector. Denoting ~uch a vector by B, its components in the directions of the reference unit vectors el' e a, e l are given by e l B = I(WII-

Wu )WIl)

where the element ~; (i andj may take any of the values 1,2, or 3) denotes the jth component (i.e., m the, direction of e;l of the vector WI' again i being I, 2, or 3. The vectors etc., are defined by the relation (2.62), In general, the tensor gradient varies from point to point. Hence we say that the spatial variation of a veetor field is given by a tensor field. In the vector description of physical problems~ we are not directly concerned with the complete tensor gradient of A(r). Only certain combinations of the elements of the tensor are significant. Three such combinations are particularly important. One of them is ~ scalar quantity obtained by summing the diagonal terms Wu , W n and Was. This sum is known as the divergence of A and is denoted by div A. We thus have

ez ' B

=:

I(Wl l

WI'

e. B = l(WIi - Wu ) For reasons that will become evident later, a vector equal to 2B is known as the curl of A denoted by curl A. The physical significance of the names divergence, rotation, and curt will become apparent later on when we shall define divergence and curl of a vector by means of certain integral Qperations. For the significance of the name strain, reference may be made to Section 9.1. The reader n!ay verify that in Cartesians the follo,wing results are obtained:

div A = Wu

+

WII

+

Was

The other two combinations are second-order tensors. One of them is given by the arrayWlI

2 oA,!

oxstrain of A = -

oA", oy

+ oA.ox oy

oA", oz oA. OZ

+ oA.ox

+ +

WZl

W+ WII)lI

2Wu Wu Wu

Wu

+

Wit

loA. 2 ox oA. ox.

oAr +-. oy OZ

2 oA.

+ oA.oy

2Wuand

+ oA",

oA. oyj

+ oA.ozk

2 oA. OZ

This tensor is usually known as .the strain of A. As seen, it is symmetrical about the diagonal formed by the elements Wao W 22 , and Wu. The other second-order tensor is given by the array

curl A =

0

0

0OZ

ox oy

oRotation of A is simply

A", A. A.

1 curl A.

1012.16 Del, the Vector DUl'ereatial Operator


Elements of Vector Algebra and Calculus and

109

Consider the expression (2.54) for the .gradient of #,r) in any orthogonal, cllrvilinear, coordinate system: grad

t/> - e1 ~aa + e. at/> + e. at/>SI

as.

as,

This expression may be rewritten as

This means to

'a a -a t/>= ( ela + e'-a + e. a ) t/> -SI s. s. obtain the gradient of t/> we operate on t/> by thegrad

In carrying out these operations in coordinate systems other than the Cartesians, we should take proper account of the fact that the system of reference units el , e t , e. 'changes with change of position. It may be verified that in Cartesians we obtain the .following results:

operatorand

V A _ oA", + oA. + oAr. Ox 011 0%

el

-

+ e.. os. + e.OSI os,

000

tVxA==

This is a vedor differential operator and is usually denoted b)' the symbol V, called del. We thus define

ox

0

011

a

az

0

V .== el - 0

oSI

+' e- + e.-a a a

==

el

--

1 a 1 a + e.-- +-hi Oql . h. aq. h.aq. 1 0

os.

.A", A. A.(2.66) These results are identical, respectively, with the divergence and curl of the vector A (see Section 2.25). Thus it is usual to set V.A and V x A defined as

S.

The expression for e.:t'in Cartesians is

== divA == curl A

(2.69) (2.70)

V _,I.!in cylindrical coordinates r, 0,r % is

ax

+ J1. + j! oy a%r

Tile Operator B V. If B is any vector. an operator B. V can be

o 1a a V=e -+e,--+ear ao 0%and in spherical coordinatesr, 0, pis

B . V == (el BI + e.B. + e.B.) . ( el .i. +OSI

e .i.OSt

+ e3.i.)

010 V == er - + e,-ar r 00

1 a + e.--r sin 0 ap

a 0 0 == Bl - + B,- + B.as, as.OSI

os.

(2.71)

TluOperators V tUUl V x. Since deLis a vector operator, the operators V. and V x may be introduced and applied on any vector field. If A(r) is any vector field, the scalar product V A and the vector product V x A are formed as follows:

It is seen that B V is a scalar differential operator. ,In Cartesians we have

-~+B .BV=B'" ox 1.+B! oy , ozApplying the operator to a scalar function #.,r) we have

V A = (el

OSI

0 + a + e. -a (e AI + e.A, + e.A.) 0)el-

as,

Sa

i

(2.67)

Recall that although for convenience we use the representation in the top line of (2.66), we always mean by that the. representation in the lower line of (2.66).

(2.72)

110Applying it to a vector function A(r) we have



III

(B. V)A

==

(Bl ,:>0+ B, ,:>0 + B. ,:>0_ '(Aiel + Aat, + AA)tlSI tiS, tlsJ.

(2.73)

In carrying out the operation, account must be taken of the tact that the unit vectors el , e t , e. change with change of position. Of particular significance, are the operators V and e V, where ds == dse is, at any given point, a small increment in some direction e of the pOsition vector r .. We recall [see Eq. (2.52)] that the differential change in a function t/>(r) over the directed distance tis ftom a given point is given by

as.

with respect to distance)in the direction e. In applying Eqs. (2_76) and (2.77) to coordinate systems other than .Ca~sians we should take note of the fact that the reference unit vectors change with c~ange of position. . The utility of the operator V lies in the fact that in working out problems we can treat V formally as a vector and apply to it the rules of vector al~ebra and calculus. In doing this, however, we should bear in mind that V is an oper-ator and not an actual vector. Therefore, in the formal application of vector rules to V, it is necessary. to preserve the order in which del appears with respect to the otJ:!er factors involved. For instance, even though A B = B "A, the operation V A is not equal to the operator A V. 2.27 Integration of Vector Function of Scalar

dt/> == ds grad t/>This may be rewritten as

dt/> == (ds grad)t/>

.== (ds V)t/>

(2.74)

If a vector A is a function of a scalar variable t, we can form the so-called indefinite integral

This means that the o~rator ds V, when applied to a scalar function t/>(r), yields the differential chan,ge in t/> over the distance ds. Similarly, we have

f

A(t) dt_

(2.78)

dt/> == (e. V)t/> ds

(2.75)

showing that the operator e . V when applied to t/>(r) yields the derivative of t/> with respect to distance in the direction e. Consider now the expression (2.57) we have obtained, in Cartesians, for the change dA in the function A(r) over the directed distance ds = dse from a given point. We have . dA

in the same. manner as is done in scalar integration (i.e., integration of a ~calar function of a scalar variable). The result of the integration (2.7.8) IS another vector function of the scalar t and is determined to within an additive crostant which, in general, is a vector. We thus write

fAct) dt =It follows that

B(t)

+C

==

(ds grad AJi

+ (ds. grad Aw)l + (ds grad A.)k

dBdt

== A(t)

This may be rewritten (noting that i, j, k are fixed directions) as dA

== (ds grad)(A) + (ds grad)(AJ) + (ds grad)(A.k)= (ds grad)(A)

If the variable t changes ~ntinuously from a particular value I to another particular value .t~, the integral 1.

+ Awj + A.k)(2.76)

= (ds grad)A = (ds. V)A

i

ts

A(t) dr

11

is the definite integral of A between the limits 11 and I,. 2.28 Uoe Integrals: Circulation Consider a scalar function of position t/> = t/>(r) and the field 'described by it. In such. a field let'" r:epresent a space curve drawn from point a to another pomt b. We asSign a direction to the curve as that of travel along the curve from a to b. Let r denote the position from some origin' of a point P on the curve and ds an element of length along the curve froln

This means the operator ds V when applied to a vector function A(r) yields,just as wher applied to a scalar function t/>(r), the differential change in A over the directed distance ds. Similarly, it can be seen that dA -=(eV)A ds(2.77)

a

showing that the operator e V applied tC' A(r) yields the derivative of A

111



113

the point P (sec Fig. 2.32). If e, is a unit vector tangential to the curve at the point P, we have tis == dse,. The integral

or, equivalently,

f f

c/>(r) d.

of the integral (2.79) depends only on the endpoints and, becomes independent of the path that joins them. We shall look into those conditions later. We note that another line integral or A(r) may be formed as follows: fA(r) )( d. or fbA(r( e. ds

c/>(r)e, ds The result of this integral is a vector.Circulation. Line integrals of the type des~ribed above may be formed around closed curves. Of particular interest is the integral

taken along the curve ~ is called the line integral of t/> along the path ~. The value. of this integral is a vector.b

fA. ds or

fA. e,d, or

fA. cos at ds

around a closed space curve~. Such an integral is known as the circulation of the vector A ~round the curve~. In general, the value of the circulation is nonzero and depends on the function A(r) aDd the closed curve~. In certain circumstances, however, the circulation vanishes and becomes independent of the curve. We shall look into these details later ..2.29 Surface Integrals

Fic. 1.31 Line integral.

Consider an open surface S drawn in the field described by a scalar function of position c/>(r). Let the surface be divided into a number 0.( infinitesimal elements. Each of the su.face elements may be considered as a plane area and denoted as a vector

dSConMter now a vector runctlon of position A(r). I(~ is a.spaec curve as before, we-:can form the ~ine integral

== odS

'c'A. ds 1.

or

fA. e, ds

(2.79)

along the curve ~ between the given endpoints (sec Fig. 2.32). th.e integral is simply the, integral along ~ of the component of A tangential to the curve_ 1f, as shown in tile figure, at is the angle between e, and A. the integral (2.79) may bewrittelJ as

o is a unit vector normal to tlfe surface element (sec Fig. 2.33). The unit vcC"+or 0 is drawn arbitrarily from one side or the other of the surface S. If, however~ a direction of travel is first assigned along the boundary curve ~ of the surface, the direction of 0 is chosen according to the right,. hand rule with respect to the direction of travel along~. Using these notationsweform the integral

II.f]

t/>(r) dS

or

II8

t/>(r)o ds

fb A cos

at

ds.

'where A is the magnitude of A. The result ~f th.is integral is a scalar. In general, this line integral, like any other hne ~nte~ral, d~pends on the 'function A(r) the path 'alol1g which the integration IS camedout and 011 the endpoint; of the path. Under certain conditions. however, the value

over the entire surface S. Such an integral is called the surface integral of t/> over the'surface S. The result of the integral is a vector. Consider next a vector funqtion of position A(r) and let S be an open surface drawn in its field. Then the integral(2.80)

s

8

111


Elements of V~or Algebra and Calculus For a scalar field ~ = t/>(r}, we have the integral

lIS

taken over the surface S is called the surface integral of A over the surface S. Since A 0 is the component of A in the direction of the normal to the surface element (see Fig. 2.33). the integral (2.80) is simply the surface integral of this component. The value of the integral is thus a scalar. The quantity A dS or A 0 dS is usually callt"d the outflow of vector A through the surface element dS. By outflow of A we me- .t the flow of A

ff~ dSsA.

or

ff~n dS8

(2.81)

The result of this integral is a vector.Il

FII. 2.33 Surface integral.

FII. l.34 Integral over a closed surface.

into the region that contains the normal to dS. This is the case when the component A 0 is positive. With this interpretation, the surface integra! (2.80). is called the outflow of vector A through the surface S .. Since A 0 may be positive at some points and negative at other points of the surface S, by outflow of A through S we mean actually the net outflow of A . . For the vector field A(r}, w.e can form another surface integral expressed by

For a vector field A = A(r), we have two integrals .. On.e of'them is

fjA.dS8

or

fj A.odS8

the result of which is a scalar. The other integral is

II8

A(r} x dS

or

IIs

ifs

Ax dS or

ifs

A x

0

dS

(2.83)

A(r) x odS

the result of which is a vector. The three integrals (2.81), (2.82), and (2.83) appear frequently in.the analysis of physical problems. The int~gral (2.82) is called the outflow of A through the surface S. It actually gtves the net outflow of A through S from the region enclosed by S.

The result of such an integral is a vector. Surface integrals of the type described above may be formed also with closed surfaces. As shown in Fig. 2.34, let S bea closed surface and, as before, let dS = 0 dS denote an element of S. For a closed surface, we shall always draw the normal so /IS to point outward from the region enclosed by the surface and refer to it as the outward normal. Using this convention we form the following surface integrals.

2.30 Volume IntegralsConsider a regi~n of spa~e.1I in. the field of a scalar function of position ep(r}. Let the regIOn be dlvlced IOto a number of infinitesimal volume

116



117

elements of magnitude dT. Then the integral

III4>(r)91

To see, the significance of the vcctor U, we ~k at the point P the component of U in some direction e. We have

dT

u e-lim 1. Jt4>(D e) dS.'''04T Jj' , AS

(2.85)

taken throughout the volume 91 is known as the volume integral of cP over the region 91. The result of the integral is a sCalar. Similarly, if 91 is a region of space in the field of a vector function of position A(r), the integral

since e is ~ given direction. The right side 'of Eq. (2.85) is evaluated as follows. Sancethe shape of the volume element dT is arbitrary, we set it

IIfIII

A(r)dT

is known as the volume integral of A over the region fJI. The result of such an integral is a vector. In subsequent chapters we will find many examples of the line, surface, and volume integrals introduced in the preceding tJtree sections. There are certain transformation relations that enable us to convert these integrals into one another. These relations follow directly from the integral definitions of gradient, divergence. r.lld curl that will be given in the following section's:, The operations involved in those definitions are the ones that usually arise in the setting up of physical problems. 1.31 IDtegral DefiDitiOD ofthe GradieDt Consider a point P in a scalar field described by cp = 4>(r). Surround the point by a small closed surface dS. ,Let dT be the volume enclosed by dS (see Fig. 2.35). The shape or the volume element is arbitrary. If dS = D dS is an elemental area on the surface dS (D, according to our convention, is the outward normal), the quantity cpa dS is a vcctor at the element dS of magnitude t/J dSpointing in the direction D. We form the integral

FI8. 2..15 Illustrating the integral definilion of a gradient.up at ~he point P as a cylinder with axis along e, base du, and length 6.s (see Fig. 2.36). Then fonhe cylinder we hav'!' jfc/>{D. e) dS

==

II

c/>{D e) dS

+

If

4>(0. e) dS

+

If

4>(D e) dS (2.86)

divide it by the volume dT and take the limit of the resulting ratio for vanishing dT. This limit, when it exists, represents a certain vector associated with the point P. The vector, as obtained, is derived from the scalar functio.n c/>(rj. Denoting this, vector tentatively by U we write U = Jim.! i=(t/Jo dS Ar-O dTlfA8

A8

wall

race 1

ral:e I

Since, at every point of the cylinder wall D is normal to e, n vanishes on the wall,. and consequentJy

e

(2.84) On the face I,D

.aU

II

tp(n e) ds = 0

(2.87)

= -:-e. We assume that cp is uniform OVf'r the face and is

118

Ideal-Fluid

A~rodynamics

Elements of Vector Algebra and Calculus Using Eqs. (2.91) and (2.85) we have U e

119

equal to the value of t/> at the point

P, that is, at r. We then have(2.88)

Ifface 1

== lim ..!... d(n

e) dS = - t/>(r) Au

ATds

On facc 2, n == e and assume that t/>(r

t/> is uniform over the racc with the valuet/>(r)

(2.92) . This s~ow~ that ~he vector at !he point P is such that its component any. duc:ctlon e gIves th~ derIvative at P of t/> with respect to distance in that dlrec:tJon. Such a vector has been named previously the gradient of t/> (see S~ctlon 2.24). Ther.efore we identify grad t/> with U and give the followmg integral definition:10

+ Ase) =

+ dt/> Asds

(2.89)

l!

e

grad

t/> == lim 1.. t( t/>o d.f:A, ..

oATlfAS

.

(2.93)

In closing this section we draw attention to the fact that small like AT (see Eq. 2.91);

Ift1S

t( t/>n dS

. IS

2.32 Divergence of a Vector FieldConsider next a vector field A == A(r). Let P be a point in that field and let tlS be a small .closed .surface surrounding the point and enclosing a volu~e AT (see ~Ig. 2.37). As explained before (see Section 2.29), the quantIty A 0 dS IS the outflow of A through the elemental area 0 dS of

Fit. 2.36 Volume element to compute the gradient.

where dt/>/ds is the derivati.e at P of t/> with respect to distance in the direction e. In relation (2.89) terms involving higher derivatives are neglected. This and the assumption of uniformity of t/> over the faces are permissible in view of the ensuing limit .Jperation. Using (2.89) we obtain

II

, div A and curl A in Cartesian, cylindrical, and spherical coordinates show that in terms of the differential operator V we have grad c/> = Vc/> div A == VA and curl A == V x A2. Using the operator V, the integral definitions (2.93). (2.94), arid (2.105) for the gradient, divergence, and curl, respectively, may all be grouped in the form

The approximation implied in (2.116), (2.117), and (2.118) may be made as close as we please, !>ince dT may be taken as small as we please. 4. Consider a surface element 0 !:is in a vector field A(r). Let en denote the boundary curve of the surface element, and let ds denote a differential element of the Curl A curve. The direction of ds is that of righthand rotation about n (see Fig. 2.43). According to' the integral definition (2.111) for curl a A, we have

o curl A == hm

AS-o!:iS

J.. ,(

Y A dsCA

(2.119)

v(.:)X

If the surface element n dS is taken equal to n uS, a differential surface element, Eq. (2.119) can be written in the approximate formn curl A =

= limA,-O

A

J_..A:( :. !:iT 11'AS

)0

dS

(2.115)

-A x

dS .Je.

~~

ds

A ds

(2. L20)Fig. 2.43 Outflow of curl' thr0ugh a surface element is equal tu the cin;uiJ tion around its boundary.

or in the form

3. If the volume element in Eqs. (2.93), (2.94), and (2.105) defining the gra -tient, divergence, and curl is taken equal to dT, a differential 'volume

ndS.curlA=~

Ads (2.120a)

... en

130


Elements of Vector Algebra and Calculus grad q, in the region 91. We thus have lim I(grad q,h {notl: ... co

131

The approximation implied in Eqs. (2.120) may be made as close as we please, since dS may be taken as small as possible. 2.36Relat~oDS

between Surface and Volume Integrals

The integral definitions of grad q" div A, and curl A give rise to some important relations between surface and volume integrals that occur frequently in the analysis of scalar and vector fields. We'shall now derive these relations. Let us first consider a scalar field q,(r). In this field let 91 denote the volume of a finite region of space enclosed by a closed surface S, Subdivide 91 into a number of small volume elements. Let the volume of element k be denoted by aT" and its surface by 6.S". Then, fcr the clement OT", according to Eq. (2.116a), we have

"-1

=

III

grad q,'dT

(2.124)

With the relations (2.123) and (2.124) Eq. (2.122) becomes

if

q,n dS =

IIIR

grad q, dT

(2.125)

if

rP o" dS" = (grad q,)" OTI;

(2.121)

which gi:ves a relation between surface and volume integrals in a scalar field. Equation (2.125) is sometimes called the gradient theorem. . Consider next a vector field A(r) and, as before, let S be a closed surface enclosing a finite region of space, denoted by 91. Using relations (2.117a) and (2.118a), and proceeding on the same lines as above, we arrive at the following relations:

where 0" dS~ is an element of area on the surface 6.S", and (grad 0" dS" over [he common surface for the two elements vanishes. In this way, considering all the elements, we will be left for the sum with contributions from only the surface elements that lie on the surface S. This result is independent of the way the region f!l is divided into elements OTk We therefore obtain

and

if ifsS

A n dS =

.IIIR

div A d

(2.126)

-A x n dS

=

IIIR

curl A dT

(2.127)

The relation (2.126) is usually known as the divergence theorem or the theorem of Gauss. It states that the outflow of a veeier field A through a closed surface S is equal to rhe volume integral of the divergence of the vector field over the region enclosed by S. Using the operator V, the integral relations (2.125), (2.126), and (i.127) may be grouped in the form

if ( :.S

)ndS =

-A x

III v(.:)'R

dT

(2.128)

x A

2.37

Theorem of Stokes

=

Jjs

1>0 dS

(2.123)

The sum ill the right side of Eq, (2.122) is by definition the integral of

Equation (2.120a) gives rise to a relation between a line integral and a surfa,;e integral in a vector field. Let ~ be a closed curve in a vector field A(r) and let 5 be an arbitrary (in general, curved) surface bounded by the curve (see Fig. 2.44). We assign arbitrariiy one or the other side of th~ surface as the positive side and divide that side of the surface into a largl! number of elements oS" by a network of small curves C". At each element the normal Ok is set up from the positive side of S. A direction of travel along any Ck is chosen according to the rule of right-hand rotatiull about

131


Elements of Vector Algebra and Caklulus

13J

the corresponding normal Ok. This procedure also fixes the direction of travel along the curve~. For the surface element Dt tJSk , according to Eq. (2.120a), we have "

jet A dSt = (curl A)k' Ok tJSk

(2.129)

where ds k is a differential length along Ck , a.nd (curl A)k is curl A taken at an aOrbitrary point within the element tJSk The approximation in Eq. (2.129) becomes closer as tJSk becomes smaller.

This is known as Stokes' theorem. It states that the circulQtion of a vector A around a curve ~ is equal to the outflow of curl A through an arbitrary surface S bounded by the curve~. Note that if we consider different surfaces drawn with the same boundary curve, the outftow of curl A js the same through all the surfaces. In terms offtuid ftow, if A represents the velocity field V, curl A becomes the vorticity and Eq. (2.131) states that the circulation around a curve ~ is equal to the out flow of vorticity through an arbitrary surface S bounded by~o

A simple result tbat follows immediately from Eq. (2.131) is that if we consider a closed surface, the integral taken over the boundary curve vanishes givingo

ifB

curl A D dS

=0

(2.132)

1.38 Further OperationsIt may be recalled .that gradient. divergence, and curl are operations that involve first-order partial derivatives with respect to space. Repeated application of gradient, divergence, and curl lead to expressions that involve spatial derivatives of an orde, higher than the first. We shall now look into those cases th8t involve only the second-order derivatives. Consider first a scalar field r/l{r). Associated w~th ", there is only one first-order differential expression .. namely grad",. Since grad", is a vector field, we can form from it ihe following two second-order differential expressions: div (grad",) or V V", and curl (grad ",) or V)( V ~ Consider n~xta vector field A(r). The first-order differential expressions associated with A are only two, the div A and the curl A. From these we can form the following second-order differential expressions:

VIFig. 2.44 Illustrating the theorem of Stokes.

We sum (2.129) for all the elements on the surface S and proceed to the limit as their number becomes indefinitely large. We thus have (2.130) It is easily verified that lim6S.-.k-oo

1, A. dS k 1:-1 Yet

i

=

krl

Yet 1,

A.

USk

=

1, A. ds YW'

grad (div A) or Vrv A)div (curl A) or V V )( A and curl (curl A) 9r V)( (V )( A)

where ds is an elemental length along "C. The right of Eq. (2.130) is, by definition, the integral of curl A n over the surface S. Therefore Eq. (2.130) becomes

.

A ds =

IIS

1.39 Laplace OperatorIn terms of the del operator, for div grad", we can write divgrad ",

curl A . n dS

(2.131)

== V Vt/> = (V V)t/>

134 The operator



135

v . V == div gradVI

is a scalar second-order differential operator. It is known as the Laplace operator or Laplacian and denoted by the symbol VI. We thus have

mathematical physics. Its applications are numerous and may be found in the theory of gravitational fields, fluid mechanics, electrodynamics and optics, and in the theory of differential and integral equatioJls. To derive Green's theorem we start with the divergence theorem, Eq. (2.126):

==

V. V

== div grad

(2.133)

The expression for the Laplacian in general orthogonal, curvilinear coordinates is given in Section 2.4~ In Cartesians x, y, z it has the formI

IfIR

div A dT =

fJ8

A n dS

V

= ox. + oy. + Oz.

o

a

a

Let 'I' = 'I'(r) and 4> = t/>(r) be two scalar functions of position. Form the vector fieldtp

(2.134)

grad tP,

== 'I'V4>

In cylindrical coordinates r, (), z it has the form

and substitute it for the vector field A(r) in the divergence theorem (2.126).

VI=~~(r~) +l~+~r

or

Or

rl O()I

oz'

(2.135)

We thus have

IffR

div(lp grad

4 dT =

~ 11' grad 4>.n dS82.S

(2.138)

In spherical coordinates r, (), rp it has the form VI =

-l-[~(rlsin ()~) + ~(sin ()~) + 0 sin () ~)J ~(_1 0 r2 sin () or or o() o()

The integrand in the left side of Eq. (2.138) may be expanded (2.136) div (11' grad

4 = 11' div grad 4>

+ grad 11"

grad4>

The Laplacian being a scalar operator may be applied to a vector field, A(r), and \te can speak of VIA. In obtaining the expression for VIA !O an orthogonal, curvilinear, coordinate system, account must be taken of the fact that not only the components 'of A but also the reference unit vectors are functions of position. In Cartesians, since the unit vectors are constant, the components of V2A are simply the Laplacians of the corresponding components of A (i.e., V2A." etc.). This, however, is not true in the case of a general orthogonal, curvilinear system. To avoid any possible misinterpretation, in general, of the components of VIA as the Laplacians of the ~orresponding components, we express VIA in a form involving grad, dlV, and curl only. Such a form is the vector identity~ay

With this, Eq. (2.138) takes the form

fffR

('I' div grad 4>

+ grad 11' grad 4dT =

dT =

if8

'I' grad 4> n dS

or, using the Laplacian, in the form

fffR

('I' yl4>

+ grad 'I' grad 4

If8

(2.139)'I' gra4 4> n dS

Introducing o4>/on to denote the derivative of 4> with respect to distance in the direction of the outward normal n, we have grad 4>. n =

VIA = grad div A - curl (curl A) = V(V A) V )( (V x A) (2.137) This id~ntity may be readily. verified by expansion. in Cartesians or by expandmg V x (V x A) according to the formula for a vector triple product.

04> on

Equation (2.139) therefore may also be written in the form

IffR

('I' V I 4>

+ grad 11' grad 4

dT =

ifS

'I'

:~ dS

2.40 Green's TheoremHaving introduced the Laplacian, we shall now derive another integral relation involving it and known as Green's theorem. Green's theorem occupies an important position in mathematics and in various branches of

Equation (2.139) is known as Green's theorem in the first form. Now, consider the vector function

11' grad 4> - 4> grad 'I'

136

Ideal-Fluid Aerodynamics Elements of Vector Algebra and Calculus137

and substitute it for the vector A in the divergence theorem (2.126). We thus have

IIIR

div('P grad t/> - t/> grad, 'P)dT ==

if8

('I' grad t/> - t/> grad ,'1'). n dS

(2.140) The integrand in the I~ft side of Eq. (2.140) may be expanded and shown to be equal to 'I' Vlt/> - t/> VItp Therefore Eq. (2.140) takes the form

identity (2.143) may be demonstrated by working out the differential operation V x Vt/> in any orthogonal, curvilinear coordinate. In particular, the identity is easily verified in Cartesians. Now, suppose that a vector field is such that in certain regions of space its curl vanishes. Then in those regions, on the basis of Eq. (2.143), the vector field may be represented as the gradient of a scalar field. Thus when curl A

== 0(2.144)

we can writeA=gradt/>' grad t/> - t/> grad '1'). D dS (2.141) ('I' o,

JJJ - t/> Vltp) dT =

if

on

- t/>

0,,) dS on

This equation is known as Green's theorem in the second form .

2.41 Irrotatlooal FieldLet 0 dS be a differential surface element containing a point P in a scalar fieldtMr). Consider the vector field grad ~andapplytoit, at P, Eq. (2.120a). We thus have

DdS curl (grad t/ ==

t.

where t/> == t/>(r) is some scalar field. . If A is known, ~ can be determined from Eq. (2.144), which is a first-order partial differential (vector) equation. When A is not known, an equation for ~ is to be developed from the equations that govern A. In physical problems, replacing an unknown irrotational vector field by an unknown scalar field reduces the number of scalar unknowns from' three to one and consequently introduces some simplification in the analyses of the problems. If the curl of a vector field vanishes in certain regions of space, we say it is an irrotational field in those regions. The scalar field, the gradient of which represents an irrotational vector field, is usually known as a scalar potential or simply a potential. This name results from the fact that the scalar field representing an irrotational force fiel~ is simply (except perhaps for the sign) the potential energy of the force field.

grad t/> ds

(2.142)

2.42 Solenoidal FieldLet d-r be a differential volume element containing a point P in a vector field V = VCr). Consider the vector field curl V and apply to it, at P, Eq. (2.117a). We thus have

where e" is the boundary curve of the surface element 0 tiS. Since grad t/>. ds is equal to d.p, the total differential along e", we have

fsince

c.

grad t/> d.

==

i Yea dt/> ==0

0

dT div (curl V)

==

if

curl V n dS

(2.145)

en

4B

is a closed curve. Therefore Eq. (2.142) becomes odS curl (grad~)

where /l.S is the surface of the vrlume element dT. According to Eq. (2.132), the right side of Eq. (2.14:;) is zero. Therefore Eq. (2.145) becomes

Since this equation is true fo" any arbitrary surface eJement.o ds through P, it follows that curl (grad~)

dT div (curl V)

=0

=0

(2.143)

This states that the curl or rotation of any gradient vector is zero. The The 'nomenclature regarding the first or second form is not uniform. The nomenclature used here seems to be preferred in America.

Since this relation is true for any arbitrary volume element dT enclosing P, 'it follows that div (curl V) = 0 (2.146) This states that the divergence of any curl vector is zero. The identity (2.146) may be demonstrated by working out the differential operation V (V x V) in any orthogonal curvilinear coordinates. In particular, it is easily verified in Cartesians.

138



139

Now, suppose that A(r) is a vector field such that in certain regions of space its divergence vanishes. Then in those regions, on 1he basis of Eq. (2.146), A(r) may be represented as tile curl of some other vector field, say B(r). Thus when div A = 0 we can write A = curl B (2.147) In the analysis of physical problems, replacing a divergenceless vector A by curl B leads to certain advantages. A vector field whose divergence is zero is known as a solenoidal field. In analogy with a scalar potential a vector such as B, defined by Eq. (2,147), is known as' a vector potential.

2.44 Poisson's EquationLet us consider the situation' where the rotation of a vector field A(r) is zero but its divergence is not zero. In fact..,. 'letdiv A be described by a scalar field q(r). We thus have curl A = 0 div A = q(r) We set A = grad (2.1S7) thus identically satisfying Eq. (2.1SS). Using Eqs. (2.1S6) and (2.1S7) we obtain (2.1S8) 'VI == div grad = q(r) as the equation to determine (r). Equation (2.158) is an inhoPogeneous Laplace equation. Such an equation is called poisson's equation. Consider next a vector field VCr) whose dinergence is zero but whose rotation is not zero. Let curl V be described by a vector l1eld nCr). Thus V(r) is characterized by the equations divV = 0 curl V = nCr) We set V = curl B (2.161) (2.159) (2.160) (2.1SS) (2.1S6)

2.43 Laplace's EquationSuppose A(r) is a vector field whose divergence and curl are both zero. We then have div A = 0 (2.148) curl A = 0 (2.149) On the basis of Eq. (2.148) we can represent A as the gradient of a scalar function, say (r). Alternatively, on the basis of Eq. (2.149), we may represent A as the curl of a vector function B(r). First let us write A = grad (2.1S0) satisfying identically Eq. (2.149). Substituting Eq. (2.1S0) into Eq. (2:148) we have 'VI =(ql' q., q.) is readily obtained by substituting grad r/> for A ar d the corresponding compon~nt~ of grad r/> for At> AI. and AI in the relatk'l (2.183). We thus obtainVr/> =

di~ grad

r/> =

_._l_[~(hlh. or/hlh.ha 41ql hI 41ql(2.185)

(II)

(c) (a)

+ l..(hahlO.r/ + .E..(hlh.Or/]oq. h. oq. oa. ha oqsIt follows that the Laplacian opera4lr is given by V

FIe. 1.45 Volume element in curvilinear coordinates:cylindrical coordinates; (c) spherical coordinates.

general coordinates; (b)

Gradient. The expression for the gradi,ent?f a ~caHu function +(qh q q.) is conveniently obtained as deSCrIbed m SectIOn (2.24). We

==

div grad

have

- hlhzh s OqI

__l_[.i.(hzha.i.) + .i.(hshl.i.) + 1... (hlh2 .!_)lhI Oql

oqz

hz Oq2

OQ3

113 iJqa

J

Clulnges in the Referenee LJnit Vectors. The relation (2.173) t. nr bles us to determine the change in any of the unit vectors Cl e 2 , e 3 n::~ult:'\g \fom

1.'1. ' . "

ldal-Fluid Aerodynamics

Elements of Vector Algebra and Ca1culus

141

change in the coordinates qlt q q. Such a determination. however. leaves the result in terms of the i. j, k systems of unit vectors, but we are usually iriterested in having the result in terms of the coordiaates qh q q. and the corresponding unit vectors. Such a result is readily obtained by he following method. Denote by e l' ' ' the unit vectors at the point ql + &il' q. + lJq .. and q. +- &i where &il' &i 6q. are infinitesimal changes (for clarity we use temporarily the symbol 6 instead of 4). Denote by 6. the change in position corresponding to the coordinate changes &ilt &il' &i. The unit vectors e,.' ' ' may be regarded as resulting froin a translation of the over the directed distance 6s and a rotation of them vectors through an infinitesimal.angle ~ about Ii certain axis passing through their common origin .. ~ote that we are concerned here with orthogonal curvilinear coordinates only. Let ~ represent vectorially the infinite~imal angular rotation: Once ~ is known. the changes in the unit vectors are given (since translation causes no change)

and vectors are all functions of posit:on.

V(f1tI;) III: ",V", + ",VV' V (",A) - ",V A + V",. A V x (",A) ." ",V )( A + V", x A V(A B) .. (A V)B + (B. V)A + A x (V x B) + B )( (V )( A) V (A )( B) - B V )( A - A V x B . V x (A x B) = A(V B) + (B. V)A - B(V A) - (A V)B V x (V x A) - V(V. A) - V'A

'1. '., '.

~ - e,.' - e,. - ~ )( e,.

". - e.' - e. - ~ x e. 6.. - e.' - '" ." ~ X

(2.187)

Now, we know that the angular displacement"" and the associated displacement h are related. Iii faci, we have (2.188) To express ." in tenns of the COOrdinatesqb q.,.f. and the unit vectors

e,., el, e. we write-

h .. 6s1e,.

+ 6s~. +. 6s... hi 6qle,. + h.6q ... + he &i ...hie,. hae.

Using this and (2.184). we express relation (2.188) as

h.e.

~-~

1

2h t h.h.

aqlhi 6ql

a

aq.

a

aq.6q.

a

(2.189)

h.' 6q. hal

1046 Some V.1uI R.latloas 'Jlte following formulas are of frequent use .in applications. They can be verified either by expansion in cartesian coordinates or by treating v as If vector (while retaining its actual meaning as an operator) in the appropriate vector formulas for products. In the following the scalars

Stre,ss in a Fluid

149

Chapter 3

Stress in a Fluid

Theory of the motion of a fluid, like the theory of the motion of a system of masses, is based on Newton's laws of mechanics. Similarly, the study of the state 'of rest of a fluid is based on the laws of static equilibrium as applied to a mechanical system of masses. In adapting these laws to a fluid we usually choose as our syste~ a certain finite or elemental region of the fluid. Thus for a fluid in motion we require, according to Newton's second law, that the rate of change of momentum of the fluid contained within any chosen region he equal to the resultant of all the forces acting on it. If the fluid is at rest, the momentum is zero and we require that the resultant of all the forces acting on any portion of the fluid be zero. If the fluid is in a state of uniform motion (i.e., all fluid elements have the same velocity for all timeli), the rate of change of its momentum is zero. Therefore the laws of static equilibrium also apply to a'state of uniform motion. To the law of change of momentum we add the companion law of moments that the sum of all the moments of the forces acting on any part of a fluid at rest is zero. The moments are all taken with respect to a single reference point. As a preliminary step in the formulation of these laws, we shall consider in this chapter the nature of the fort:es that act on any region of a fluid and the method of their specification. Thili involves the concept of stress in a fluid. FollOWing fnese considerations, we shall develop the law of static equilibrium for a fluid. To formulate completely the law of motion for a fluid we must first decide about a method for describing fluid motion. This we do in Chapter 4. Finally,~n Chapter 5, we shall formulate the law of motion along with other equatibns necess!!~y for the analysis of fluid motion.

forces are in the nature of actions and reactions. Such forces are commonly referred to as internal forces. Since they act across a surface that is imagined to ~par~te the fluid, they are also called surface forces. The co~plete specification of these forces is based on the concept of stress, which we shall take up presently (see Section J.2). In addition to the surface forces the fluid may, in general, be SUbjected to forces that act throughout the body of the fluid as such. The simplest ex~mple of such a force is the force due to gravity, that is, the so-called weight ~f the fluid. Forces o!" this type are called body forces. They are proportIOnal to the volume or mass of the fluid considered. Thus the body for~es can be specified as so much per unit volume or per unit mass of the fl~ld, that ~s, as an inte~sity. In general, the body force may vary from pomt to pomt of the fluid, and at any point it may have different value~ at differ~~t instances of time. Thus the body force is a vector function of position and time. .

3.2 Concept of Stress and the Spedficatioa of Stress at PointThe concept of stress and the method of Its specification, as described below, are applicable to any continuous deformable medium whether it be a fluid or a solid. ~ . ~t us con~ider a plane surface S drawn through some point F in a flUid. The flUid may be either in motion or at rest. We denote by n the nor~al to the su.rface .and. consider t~e forces exerted across S by the portion of the flUid whIch hes on the Side of n. These internal forces are not, in. general, distributed uniformly across S. We represent these fo~Ces as equivalent to a force F acting at the point P and a moment M about some axis t.hrough P (Fig. 3. Ca). If we gradually shrink the area of the surface to the point P, both F and M will tend to zero. However for a vani~hingly small area the equivalent lorce .,' may be assumed to ~ pro, portlo,nal to t~e a,:a: Then the ratio of force to area may be assumed to tend t~ a defimte hmlt as the area shrinks to a point. It can be shown that the ratIO ofth~ moment t.o the area vanishes as the area goes to zero. This means the action of th~ Internal forces in the immediate vici.nity of point p. ~cross the surface n (I.e., .Mrmal to n) can be specified by the limit of the" ratio of force to area, that IS, by a force per unit area at that point. This is called the stress at the point P across a plane rl. It is a vector and its direction is, in general, different from that of n. Denoting such' a stress rector by O'n we define symbolically0'"

3.1 Surface Forces and Body ForcesIn the space occupied by a fluid, whlch is in motion or at rest, let us imagine a sbrface enclosing some part of the fluid. The portions of the fluid close to the surface on its two sides exert forces on each other. These148

. F = I1m s.~o Sn

(3.n

In this connection see Love (1944).

IJO


St~ss

in a Fluid

where Sit is a plane area normal to' D. The definition of a stress vector involves two directioM-that of the normal to the surface and that of the stress itself. Now if dS is an elemental area normal to D, t~ force exerted across dS by the ftuid that is on the side of D is (Fig. 3.lb) .G"ds

lSI

The stress at a point P across an elemental surface D can be specified either by a stress vector G" or, equivalently, by its three components(II)

~""dS dS(a)

t1"

Fie. 3.2 . Decomposition ora stress vector.

(b)

FIt. 3.1

Representation of internal forces in a ftuid.

forc: s at a pO\~t P across only a certain plane D. For the complete s cificatlOn of the ,mtcrnal forces at P we should know the stress vector': P across al~ the planes, infini.te i?number, that can be drawn through P. It IS easily seen, by consldenng the equilibrium of an infinitesimal ~etrahedron_su~h as shown in Fig. 3.3, that if the stress I:ector across three

tension ~r a ~ile stress. The components 0'", and 0'". are known as the tangential st~~sses at P on the D surfate. They are shearing stresses. The defirlltJ~ri (3.1) of a stress vector enables us to specify the internal

referred to a system of three unit vectors. Denoting the unit vectors by elo e., ea and the respective components of a" by.a nlo a ... , a"a, we write

The meaning of the double subscript is apparent. The first subscript denotes the plane across which the stress component acts, wher::as the second subscript denotes the direction of the component. The decomposi~ion of a stress vector is usually done with respect to an orthogonal system of unit vectors. Generally the unit vectors selected are those associated with the (orthogonal) coordinate system employed (Fig. 3:2a). Another way of choosing the unit vectors is to select one of them in the direction of the normal D and the other two in two mutually perpendicular directions lying'in the surface D. Denoting the latter directions L e, ~nd e. we write (3.3) 'fhe comoonent 0' nn is known as the normal stress at P on the D surface (FiS' 3.2b). Its positive direction is that of n. It is then known as a

pomt IS "omplet~ly specified by glvmg a set of three stress vectors. Equivalently we can give the components of these stress vectors. Thus if a C'm, (1 n Jenote the stress vectors at P across the planes e e e d ,If' " m' n' an I

Independent planes passing through a point 'is given, the stress vector across any other plane p~ssing through that p'1int is determined. It thus follows that . th~ stat: of the mterna~ forces, ~l~o known as the '3late of stress, at any

Fig. 3.3 Equilibrium of a tetrahedron.

Stress in a Fluid152

151


eb et, e. are a set of unit vectors at P, the stress components can be givenin the form(0'11 0'"0'...

O"S)0'",3

O'"d0',,1

(3.4)

For the complete specification of the state of stress in the region of interest of a ftuid we must give at each point the stress tensor. The tensor will generally be different at different points, and at any point it may vary from instant to instant. This means thelt the state of stress within a ftuid is specified by a tensor function of position and time, that is, by a tensor field. 3.3 Stress in Fluid at Rest: Hydrostatic: Pressure The stress tensor for a ftuid takes on a particularly simple form when the ftuid is at rest. It is a fact of experiellce that tangential stresses do not exist in a fluid at rest. This means the stress vector at any point of the fluid at rest is wholly normal to any surface element passing through that point (see Eq. 3.3). Symbolically we write that (3.7) In such a case the stress tensor takes the form

0'",

O'"s

For purposes of calculation it is convenient to take the unit vectors el> et , e3 as those defined by the (orthogonal) coordinate system used and the

(3.8).

FIa- 3A

State of stress at a point.

oeaWith For such a state of stress we can draw a fundamental cl~nclusion. Considering the equilibrium of an infinitesimal tetrahedron we can show that when tlTT! stress vector at a point is wholly normal in all directions, its magnitude is lhe same for all elemental planes passing through the point. In such a case all that is required to specify the ~tress at a pomt is simply a single number. Denoting this number b) (J we write(3.9)

I planes e" em, eft as the corresponding coordinate panes eb e., this notation the stress components (3.4) become

(::: ::: ::)0'11

(3.5)

O"SI

O'ss

This array of nine numbers is known as a stress tensor: It specifi~ completely the state of stress or simply the stress ~t a pomt. !he diagonal terms an, etc., represent the normal stresses, whI~e the nondiagonal terms 0"11' etc., represent tangential or shear stres~s (~lg ..3.4). The equilibrium of the moments on an mfimteslmal cube shows that0'11 '123 0'11

for all directions of n. If c: is positive, an represents a tensile ~l ess. But from experi~nce we find t,hal' no tensile stresses occur in the inrerio. :)f a fluid. t This nlean:; i~ would be morl! appropriate to rcpre~ent e~ ;,$ a compression. This we do by replacing the number rJ by a neg2!ive number, say -po Then IhF slrt!ss ill a fluid at rest is represented bya"

= 0'810'13

(3.6)

=

-pn

0"31 -

That is, tbe stress tensor is symmetrical about its diagonal. This ~eans to specifY'completely the state of stress at a point we actually need SIX stress components.

... This conclusion 15 known as Pascats fa..... rn this Cilse the str,!ss vector des.::ribes a sphere. t Tensile ~tresses are exhib!ted at the so-called "free surfaces" and in "thin flu:d film,,"

154


Stress in a Fluid

Jjj

for all directions ofn. The corresponding stress tensor is given by

(7 ~p ~) o0-p

(3.11)

The scalar p is called tho pressure or, more preciseiy, the hydrostatic pressure at the point considered. Equation (3.10) or (3.11) can be taken as the definition of a ftuid. Note that pis a positive number. The state of s~ress within the whole region of a ftuic;l at rest is specified by giving the prdssure as a function of position and time, that is, by a scalar field,.denoted by F == p(r, t). The hydrostatic pressure is generally related to the density and temperature or-the ftuid. For instance, we know that in gases that obey Charles' and Boyle's laws the pressure, the density p, and the temperature T are connected by the relation p == p]l..T, where. R is a constant for the gas conSidered.

they disappear when the rates of strain disappear, thus leaving the stress tenso~ as that of.a ~niform pressure in all directions. Similarly, if the velOCity of the ftuld IS zero everywhere, the viscous stresses become zero and we realise again uniform pressure at a point. Because of these inter~ pr~tations, the splitting of the stress tensor in a mOVing ftuid into ~ umform pressure p and the viscous stresses. is convenient. A~rdi~g ~o these ideas, the state of stress within the whole region of a movmg ftuld IS. expressedby the combination of a scalar field and a tensor field. In developi~g the. theo~es of ~uid motion we assume generally that t~e~odynamlc co.nslderatlor.s, which are strictly applicable to equilibrium SituatIons, are valId for moving ftuids and uSe them. In such a case we must talk about a thermodynamic pressure that is related to the density and temperature at a point in the ftuid. It is then assumed that the thermodynamic pressure is the same as the pressure p occurring in the stress tensor. The justification for these assumptions is that the theories based on them seem to give results that are in good agreement with experiments..

3.4 Stress in a Fluid in MotionWhen a ftuid is in motion the phenomenon of viscosity or of inte~l friction manifests it:;e)f, and at any point; in tbe fluid both tangential and normal.stresses occur. In this case we deal with the complete stress tensor and express itas made up of two paJ1s-One that represents viscous or fricti01lll1 stresses only, and the other thaJ represents compressive stresses equal ioall directions, that is, a pressure and thus not related to friction. This pressure, also denoted by p, is sin:tilar but not identical to the hydlo~ static; pressure. The viscous stresses, which occur both as la,ngential and normal compenents, a~ usually denoted by Tlb -r1l, etc. Thus the state of stress, at any point in a moving ftuidis g\ven in the form

3.5 State of Stress in a NoaUscous Fluid

ia

Modoll

. A large part of the theory .offtuid motion, as pointed out in Chapter I. IS developed on the assumptIon that the fluid is frictionless or nonviscous.In such a case the coefficients of viscosity, and conSlequently the viscous stresses, are set t~ zero ~nd. one ~akes the state of stress as that of a uniform. pressure. The u.ltlmate justIficatIOn for such an assuD"ption lies again in the comparISon of ItS consequences with experiments. The study of 'fluid motion treated in ihis book, as previously stated, it based on the assumption of a nonviscous fluid,3.6 Pressure, Distribution in a Fluid at RestForthe static e~uilibrium of a fluid, the sum of all the forces acting on an~ part o~ the ~UI~ should. be zero. To apply this law we consider at any

(7 ~p ~) + (:: ::: ::) o0-p'-ral -ralTU,

(3.12)

pOlDt r an mfimteslmal regIOn of the fluid and write, that the resultant of the boc!y forces a~ting on the region+the resultant of the surface forces acting on it == 0

The viscous stresses are assumed to be proportional to the Nites of straint occurring at the point considered. The proportioJ;lality constants, known ~s viscosity coefficients, depend on the nature of the fluid. For any ,given fluid the viscous stresses are small when the rates of :>tr ain are small, and "HydrostatiC" signifies a ftuid atrest~

(3.13) Let .Or denote the volume of the region considerc;.d and bS the su rlace t'. . enc IoSlng It. If r = f(r) represents the distribution of the body forces . f h fl 'd . per umt mass 0 t e UI ,the body force acting on the element OT is equal topC b1' (3.14)

t The rates of strain at a point arc in turn given by certain combinations of the partialderivatives of the components of the velocity. at that point (sec Section 9_1).

where p = per) IS the density of the fluid. The density may vary fror') point to point of the fluid.

{56

Ideal fluid Aerodynamics

Stress in a Fluid

157

T~ determine the resultant of the surface forces let us first consider dn infinitesimal area a dS on the surfa~ lJS (Fig. 3.5). Since the state of stress in a fluid at rest is given by a pressure p = p(r), the surface force acting on thefluiJ within lJS across the surface element a dS is

Equivalently we may wrik this equation as gradp = pC

(3.17)

-podSThe resultant of the surface forces is then obtained simply by adding vectorially the pressure forces acting on all the ele.mental areas of the surface lJS. It is thus equal to

-fi18.

This equation is thus the analytical form of the condition for static eqUIlibrium of a fluid. Therefore it is the basis for amllysing the statics of incompressible and compressible fluids. The content of Eq. (3.17) is significant. It states that static equilibrium of a fluid is possible only if the body force (pf) per unit L'olume can be expressed as the gradient of a scalar function. This implies that pf should be irrotational. If the density of the fluid is uniform throughout space, we have grad (

po dS

(3.15)

,p

l!.) = f

Equation (3.17) assures us that the resultant of the forces acting on any part of a fluid at rest is zero. For the fluid to be at rest the condition that the resultant of the. moments acting on a fluid element should also be zero. That this condi.tion is

aerodynamics by karamchetti

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