on variable laminar convective flow properties … · on variable laminar convective flow...

ON VARIABLE LAMINAR CONVECTIVE FLOW PROPERTIESDUE TO A POROUS ROTATING DISK IN A MAGNETIC FIELD

EMMANUEL OSALUSI, PRECIOUS SIBANDA

School of Mathematics, University of KwaZulu-Natal Private Bag X01,Scottsville 3209, Pietermaritzburg, SA

Received April 14, 2006

The hydromagnetic flow of a steady, laminar conducting viscous fluid due toan impulsively started rotating porous disk is studied taking into account the variablefluid properties (density, ρ, viscosity, μ, and thermal conductivity, κ). These fluidproperties are taken to be dependent on temperature. The system of axisymmetricnonlinear partial differential equations governing the MHD steady flow and heattransfer are written in cylindrical polar coordinates and reduced to nonlinear ordinarydifferential equations by introducing suitable similarity parameters. The resultingsteady equations are reduced to an initial valued problem and solved numericallyusing a shooting method. A parametric study of all parameters involved wasconducted, and a representative set of results showing the effect of the magnetic field(M), the uniform suction parameter (W < 0) and the relative temperature differenceparameter (ε) on velocities, temperature, skin-friction and Nusselt number areillustrated graphically and tabularly to show typical trends of the solutions.

1. INTRODUCTION

The problem of hydrodynamic stability of flow due to a rotating disk hasbeen the subject of study by several investigators since the pioneering work ofvon Karman [14]. Research into this type of flow has been spurred on by boththeoretical imperatives as well as the practical applications of such flows, forexample, in industrial machinery and lately, in computer disk drives,Herrero et al. [6]. The early study by von Karman has since been considerablyextended starting with the work of Cochran [4] to include, inter alia, the effectsof: (1) impulsively starting the flow from rest (Benton [3], Roger and Lance[12]), (2) an axial magnetic field applied to the fluid without Hall effects(El-Mistikawy et al. [5]), (3) an axial magnetic field with Hall effects (Attia andAboul-Hassan [2]) and (4) variable fluid properties (Maleque and Sattar [11]).

The effects of variable properties on laminar boundary layers has beenconsidered by, among others, Herwig [7] and Herwig and Klemp [8]. Malequeand Sattar [11] have extended the consideration of the effects of variable fluid

Corresponding author: [email protected]

Rom. Journ. Phys., Vol. 51, Nos. 9–10, P. 937–950, Bucharest, 2006

938 Emmanuel Osalusi, Precious Sibanda 2

properties, namely the density, ρ , the viscosity μ and the thermal conductivityκ to flow due to a porous rotating disk. They found, among other things, that forfixed values of the suction parameter and Prandtl number, the momentumboundary layer increased considerably. Earlier work by Stuart [13] showed thatthe effect of suction is to thin the boundary layer by decreasing the radial andazimuthal components of the velocity while at the same time increasing the axialflow towards the disk at infinity. The recent study by Attia [1] considered theeffect of temperature dependent viscosity on the flow and heat transfer along auniformly heated impulsively rotating disk in a porous medium. In this study weextend the work of Maleque and Sattar [11] to include the effects of a magneticfield on flow due to a rotating disk in an electrically conducting fluid withtemperature dependent density, viscosity and thermal conductivity.

2. GOVERNING EQUATIONS

The description of the physical problem closely follows that of Maleque andSattar [11]. We use a non-rotating cylindrical polar coordinate system, (r, ϕ, z)where z is the vertical axis in the cylindrical coordinates system with r and ϕ asthe radial and tangential axes respectively. The homogeneous, electricallyconducting fluid occupies the region 0z > with the rotating disk placed at 0z =and rotating with constant angular velocity Ω. The fluid velocity components are(u, v, w) in the directions of increasing (r, ϕ, z) respectively, the pressure is P,the density of the fluid is ∞ρ and T is the fluid temperature. The surface of therotating disk is maintained at a uniform temperature Tw. Far away from the wall,the free stream is kept at a constant temperature T∞ and at constant pressure P∞.

The external uniform magnetic field is applied perpendicular to the surfaceof the disk and has a constant magnetic flux density B0 which is assumedunchanging with a small magnetic Reynolds number (Re 1).m

Following Jayaraj [9] (and more recently, Maleque and Sattar [11]), weassume that the dependency of the fluid properties, viscosity μ and thermalconductivity κ coefficients and density ρ are functions of temperature alone andobey the following laws;

[ ] [ ] [ ]a b cT T T T T T∞ ∞ ∞ ∞ ∞ ∞μ = μ / , κ = κ / , ρ = ρ / , (2.1)

where the a, b and c are arbitrary exponents, κ∞ is a uniform thermalconductivity of heat, and μ∞ is a uniform viscosity of the fluid. As in Malequeand Sattar [11] the fluid under consideration is a flue gas with a = 0.7, b = 0.83,and c = –1.0. The case c = –1.0 is that of an ideal gas.

The physical model and geometrical coordinates are shown in Fig. 1.

3 On variable laminar convective flow 939

Fig. 1. – The flow configuration andthe coordinate system.

The equations governing the motion of the MHD laminar flow of thehomogeneous fluid take the following form

( ) ( ) 0ru rwr z∂ ∂ρ + ρ = ,∂ ∂

(2.2)

( ) ( ) ( ) 22 oBu u P u u uu w ur r z r r r r r z z

σ⎛ ⎞∂ ν ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ − + + = μ + μ + μ − ,⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ⎝ ⎠(2.3)

( ) ( ) ( ) ( ) 2oBv u v v v vu w v

r r z r r r r z zσ∂ ν ∂ ∂ ∂ ∂ ∂ ∂ρ + + = μ + μ + μ − ,

∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ(2.4)

( ) ( ) ( )1 ( )w w P w wu w wr z z r r r r z z

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ + + = μ + μ + μ ,∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2.5)

( ) ( ) ( ) ,pT T T T TC u wr z r r r r z z

∂ ∂ ∂ ∂ κ ∂ ∂ ∂ρ + = κ + + κ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2.6)

where σ is the electrical conductivity and Cp is the specific heat at constantpressure. The appropriate boundary conditions for the flow induced by aninfinite disk (z = 0) which is started impulsively into steady rotation withconstant angular velocity Ω and a uniform suction/injection W through the diskare given by

0 at 0

0 0 asw

w

u v r w W T T z

u v T T P P z∞

= , = Ω , = , = , =→ , → , → , → →∞.

(2.7)


3. SIMILARITY TRANSFORMATION

The solutions of the governing equations are obtained by introducing adimensionless normal distance from the disk, 1 2( )z /

∞η = Ω/ν along with thevon-Karman transformations,

12( ) ( ) ( ) ( )

2 ( ) and ( )

u rF v rG w H

P P p T T T∞

∞ ∞ ∞

= Ω η , = Ω η , = Ων η− = μ Ω η − = Δ θ η ,

(3.8)

where ∞ν is a uniform kinematic viscosity of the fluid and .wT T T∞Δ = −Substituting these transformations into equations (2.2)–(2.6) gives the nonlinearordinary differential equations,

12 (1 ) 0H F cH −′ ′+ + θ + θ = ,ε ε (3.9)

1 2 2 1(1 ) [ (1 ) ](1 ) 0c aF a F F G HF MF− − −′′ ′ ′ ′+ + θ θ − − + + + θ + θ = ,ε ε ε ε (3.10)

1 1(1 ) [2 (1 ) ](1 ) 0c aG a G FG HG MG− − −′′ ′ ′+ + θ − + + + θ + θ = ,ε ε ε ε (3.11)

1(1 ) Pr (1 ) 0c bb H2 − −′′ ′ ′θ + θ + εθ − θ + θ = ,ε ε (3.12)

where Pr pC k∞ ∞= μ / is the Prandtl number, 20M B ∞= σ /ρ Ω is the magnetic

interaction parameter that represents the ratio of the magnetic force to the fluidinertia and T T∞= Δ /ε is the relative temperature difference parameter, which ispositive for a heated surface, negative for a cooled surface and zero for uniformproperties. These equations differ from those in Maleque and Sattar [11] by wayof the additional terms that involve the magnetic parameter M. The transformedboundary conditions are given by;

0 1 1 at 0

0 at

F G H W

F G p

= , = , = , θ = , η == = θ = = , η→∞,

(3.13)

where W w ∞= / ν Ω represents a uniform suction (W < 0) or injection (W > 0)at the surface.

The skin friction coefficients and the rate of heat transfer to the surface aregiven by the Newtonian formulas:

12

0

1 (1 ) Re (0)at

z

v w Gz r ∞

=

⎡ ⎤⎛ ⎞∂ ∂ ′τ = μ + = μ + Ω ,⎜ ⎟⎢ ⎥∂ ∂φ⎝ ⎠⎣ ⎦ε

and

( ) 12

0(1 ) Re (0)a

rz

u w Fz r ∞

=

∂ ∂⎡ ⎤ ′τ = μ + = μ + Ω .⎢ ⎥∂ ∂⎣ ⎦ε


Hence the tangential and radial skin-frictions are respectively given by12(1 ) Re (0)

t

afC G− ′+ = ,ε (3.14)

and12(1 ) Re (0)

r

afC F− ′+ = .ε (3.15)

Fourier’s law

( )12

0(1 ) (0)b

z

Tq Tz ∞

= ∞

⎛ ⎞∂ Ω ′= − κ = −κ Δ + θ ,⎜ ⎟∂ ν⎝ ⎠ε

is used to calculate the rate of heat transfer from the disk surface to the fluid. TheNusselt number Nu is obtained as

12(1 ) Re Nu (0)b− ′+ = −θ ,ε (3.16)

where 2Re ( )r ∞= Ω /ν is the rotational Reynolds number.

4. METHOD OF SOLUTION

Equations (3.9)–(3.12) are solved numerically using a shooting method fordifferent values of suction, 0W < and parameters Pr, ε and M. To reduce theequations to first order equations we set 1,F y= 2 ,G y= 3 ,H y= 4 ,yθ =

5 ,F y′ = 6 ,G y′ = 7y′θ = to get;

1 5 1

2 6 21

3 1 3 7 4 3

4 7 71 2 2 1

5 7 5 4 1 2 5 3 1 4 4(5)

51 1

6 7 6 4 1 2 3 6 2 4 4

(0) 0

(0) 1

2 (1 ) (0)

(0) 1

(1 ) [ (1 ) ](1 )

(0)

(1 ) [2 (1 ) ](1 )

c a

y y y

y y y

y y cy y y y W

y y y

y ay y y y y y y My y y

y s

y a y y y y y y y My y y

−

− − −

− −

′ = , = ,′ = , = ,

′ = − − + ε ε, = ,′ = , = ,

′ = −ε + ε + − + + + ε + ε

= ,

′ = − ε + ε + + + + ε + ε(6)

62 1 (7)

7 7 4 3 7 4 7

(0)

(1 ) Pr (1 ) (0)

c a

c b

y s

y b y y y y y y s

−

− −

,

= ,

′ = − ε + ε + + ε , = ,

(4.17)

where (5) ,s (6)s and (7)s are determined such that 5( ) 0,y ∞ = 6 ( ) 0y ∞ = and

7( ) 0.y ∞ = The essence of this method is to reduce the boundary value problemto an initial value problem and then use a shooting numerical technique to guess


(5) ,s (6)s and (7)s until the boundary conditions 5( ) 0,y ∞ = 6 ( ) 0y ∞ = and

7( ) 0y ∞ = are satisfied. The resulting differential equations are then easilyintegrated using the initial value solvers lsode and fsolve available in GNUOctave. To establish the validity of our numerical code, a comparison of ourcalculated results with those of Maleque and Sattar [11] and Kelson andDesseaux [10] for 0M = is shown in Table 1. The three set of results comparefavourable for 0W < .

Table 1

Comparison of current and recent numerical values of the radial and tangential skin -frictioncoefficients and the rate of heat transfer coefficient obtained for Pr = 0.71, 0M and ε = 0.

Present ( 0)M Maleque & Sattar [11] Kelson & Desseaux [10]W (0)F′ (0)G′− (0)′−θ (0)F′ (0)G′− (0)′−θ (0)F′ (0)G′− (0)′−θ

0 0.42406 0.65140 0.53873 0.51015 0.61596 0.32576 0.51023 0.61592 0.32586–2 0.23239 2.06872 1.51962 0.24251 2.03911 1.44212 0.24242 2.03853 1.43778–4 0.12462 4.00649 2.85201 0.12477 4.00537 2.84470 0.12474 4.00518 2.84238–5 0.09990 5.00289 3.55414 0.09996 5.00297 3.55411 0.09992 5.00266 3.55122

–10 0.04999 10.00033 7.10016 0.05059 10.00156 7.10202 0.05000 10.00015 7.10015

5. RESULTS AND DISCUSSIONS

Following Maleque & Sattar [11], the numerical solutions displayed inTables 2–3 and Figs. 2–4 are relevant for a flue gas, that is, when Pr = 0.64. Wehave confined our analysis to the case when we have suction velocity only, thatis, when W < 0.

Table 2 shows the effect of increasing the magnetic field strength on theradial and tangential skin-friction coefficients and the rate of heat transfer forvariable property ε = 0.1 and fixed suction coefficient W = –1. For moderateincreases in the strength of the magnetic field the effect is a gradual monotonicdecrease in the values of ,F′ G′− and .′θ

Table 3 shows that cooling the surface while holding constant the magneticfield strength and the suction parameter has the effect of increasing the radialand tangential skin-friction and the rate of heat transfer coefficients.

Table 2

Numerical values of the radial and tangential skin-friction coefficients and the rateof heat transfer coefficient obtained for ε = 0.1, W = –1 and Pr = 0.64

M (0)F′ (0)G′− (0)′−θ0.00 0.31241 1.10349 0.815830.03 0.31206 1.10476 0.815800.05 0.30670 1.12461 0.81525


Table 3

Numerical values of the radial and tangential skin-friction coefficients and the rateof heat transfer coefficient obtained for M = 0.1, W = –1 and Pr = 0.64

ε (0)F′ (0)G′− (0)′−θ–0.5 0.42374 3.47019 2.42976–0.25 0.37768 1.89574 1.313200.00 0.32210 1.28933 0.907770.25 0.27229 0.98531 0.71281

Figs. 2(a)–2(d) show the effects of ε on the velocity (radial, tangential andaxial) and temperature profiles. The primary purpose of these figures is to give acomparison between the constant property and variable property solutions whena magnetic field is present. The results are qualitatively similar to those given byMaleque and Sattar [11] except that the effect of the magnetic field is depress themotion of the fluid. In Fig. 2(a), it is seen that due to the existence of acentrifugal force the radial velocity attains a maximum value close to the disk forall values of ε . However, in contrast to the observations in Maleque and Sattar[11], where the maximum velocity is larger at the surface of the disk in the caseof constant property, the effect of a moderate increase in ε is not seen to depressthe fluid motion at the disk surface. For most part of the boundary layer, at anyfixed position η, the radial velocity increases with the increase of the relativetemperature difference parameter ε. The tangential velocity, as observed inFig. 2(b) is found to increase with increasing values of ε at a fixed point of theboundary layer while in Fig. 2(c) that axial velocity decreases with an increase inthe relative temperature differences ε. The results Fig. 2(d) shows that thenon-dimensional temperature increases with increasing values of ε, but the rateof increase is very small and hence confirming the findings in Maleque andSattar [11] that the thermal boundary layer does not vary with ε.

The effects of fluid suction ( Ws ) for 0Mε = = and Pr = 0.64 on the radial,the tangential, the axial velocity profiles and temperature profiles are shown inFig. 3(a)–(d). Strong suction has a stabilizing effect on the axial velocity; theradial velocity attains its maximum near the surface while the tangential velocityand temperature decay rapidly away from the surface. The radial velocityreaches its maximum very close to the surface and decreases monotonicallyaway from the boundary layer. Similar effects of W are also observed in case ofthe tangential velocity. It is noticed from Fig. 3(d) that the thermal boundarylayer increases gradually for a decrease in suction velocity.

Fig. 4 shows the effect of the magnetic field on velocity and temperatureprofiles. Imposition of a magnetic field generally creates a drag force that has thetendency to slow down the flow around the disk at the same time increasing fluidtemperature. This is shown by the decreases in the radial, tangential and axial


velocity profiles as M increases as shown in Fig. 4(a)–(d). The increases in thetemperature profiles as M increases are accompanied by a correspondingincrease in the thermal boundary layer.

(a)

(b)

Fig. 2. – (a), (b).


(c)

(d)

Fig. 2. – (a) Effect of ε on the radial velocity profiles, (b) effect of ε on the tangential velocityprofiles, (c) effect of ε on the axial velocity profiles (d) effect of ε on the temperature profiles:

for M = 0, W = –1, Pr = 0.64, a = 0.7, b = 0.83, c = –1.


(a)

(b)

Fig. 3. – (a), (b).


(c)

(d)

Fig. 3. – (a) Effect of W on the radial velocity profiles, (b) effect of W on the tangential velocityprofiles, (c) effect of W on the axial velocity profiles (d) effect of W on the temperature profiles:

for M = 0, ε = 0, Pr = 0.64, a = 0.7, b = 0.83, c = –1.


(a)

(b)

Fig. 4. – (a), (b).


(c)

(d)

Fig. 4. – (a) Effect of M on the radial velocity profiles, (b) effect of M on the tangential velocityprofiles, (c) effect of M on the axial velocity profiles (d) effect of M on the temperature profiles:

for M = 0, W = –1, Pr = 0.64, a = 0.7, b = 0.83, c = –1.


6. CONCLUSION

In this paper we have extended the work of Maleque and Sattar [11] on theeffects of variable properties on the problem of a steady laminar flow due to arotating disk to include the effects of an applied magnetic field. The studyconfirms that the radial velocity reaches its maximum value close to the surfaceof the disk. In the presence of a magnetic field the radial velocity increases atconstant rate for all values of ε so that the largest maximum value is not obtainedfor constant property ε = 0 close to the disk surface. The effect of the magneticfield for fixed suction rate and Prandtl number is to accelerate the rate ofdecrease of the radial and tangential skin-friction coefficients and the rate of heattransfer coefficient.

Acknowledgement. This material is based upon work supported in part by the NationalResearch Foundation (NRF).

REFERENCES

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9. S. Jayaraj, Thermophoresis in laminar flow over cold inclined plates with variable properties,Heat Mass Transfer, 40, 167–174, 1995.

10. N. Kelson and A. Desseaux, Note on porous rotating disk flow, ANZIAM J., 42(E), C847–C855, (2000).

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