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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.123-131

Thermal Effects on Stokes’ Second Problem for Couple Stress Fluid Through A Porous Medium K. Srinivasa Rao1, B. Rama Bhupal Reddy2 and P. Koteswara Rao3 1

Research Scholar, Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur, A.P., INDIA. 2 Associate Professor, Department of Mathematics, K.S.R.M. College of Engineering, Kadapa, A.P., INDIA. 3 Professor, Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur, A.P., INDIA. (Received on: June 21, 2013) ABSTRACT In this paper, we investigated the thermal effects on Stokes’ second problem for couple stress fluid through a porous medium.

The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied in detail with the help of graphs in detail.

Keywords: Thermal Effects, Fluid Flow, Porous Medium.

1. INTRODUCTION The flow induced by a suddenly accelerating plate on the fluid above it, usually referred to as Stokes’ first problem (Stokes, 1851), and the flow due to an oscillating flat plate, usually referred to as Stokes’ second problem (see Raleigh, 1911) are amongst a handful of unsteady flows of a Navier–Stokes fluid for which one can obtain an exact solution. Such exact solutions serve a dual purpose, that of

providing an explicit solution to a problem that has physical relevance and as a means for testing the efficiency of complex numerical schemes for flows in complicated flow domains. The Stokes’ second problem describes the oscillatory flat plate in a semiinfinite flow domain with a specific frequency. Tanner (1962) has investigated the exact solution to this Stokes’ first problem for a Maxwell fluid. The impulsive motion of a flat plate in a viscoelastic fluid was analyzed by Taipel (1981). Exact

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K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

solution for unsteady flow of nonNewtonian fluid due to an oscillating wall was presented by Rajagopal (1982). Prezoisi and Joseph (1987) have discussed the Stokes first problem for viscoelastic fluids.

Erdogan (1995) has investigated the unsteady flow of viscous fluid due to an oscillating plane wall by using Laplace transform technique. Puri and Ky the (1998) have discussed an unsteady flow problem which deals with non-classical heat conduction effects and the structure of waves in Stokes’ second problem. Much work has been published on the flow of fluid over an oscillating plate for different constitutive models (Zeng and Weinbaum, 1995; Asghar et al., 2002; Ibrahem et al., 2006). The theory of couple stresses in fluids, developed by Stokes (1966), represents the simplest generalization of the classical theory and allows for polar effects such as the presence of couple stresses and body couples. Stokes’ first and second problems for an incompressible couple stress fluid under isothermal conditions were studied by Devakar and Iyengar (2008).

Reddappa et al. (2009) have investigated the non-classical heat conduction effects in Stokes’ second problem of a micropolar fluid under the influence of a magnetic field.

The mechanism of the flow in porous media involves many problems vital to science and industry. Notable examples are: to name a few, recovery of fuels from underground oil and gas reservoirs; mass and heat transfer operations in packed columns; solid catalyzed reactions; and fate of pollutants in soil. Many studies related to non-Newtonian fluids saturated in a porous medium have been carried out.

Dharmadhikari and Kale (1985) studied

experimentally the effect of non-Newtonian fluids in a porous medium. Chen and Chen (1988) investigated the free convection flow along a vertical plate embedded in a porous medium. Rees (1996) analyzed the effect of inertia on free convection over a horizontal surface embedded in a porous medium.

Nakayama (1991) investigated the effect of buoyancy-induced flow over a nonisothermal body of arbitrary shape in a fluidsaturated porous medium. A ray-tracing method for evaluating the radiative heat transfer in a porous medium was examined

by Argento and Bouvard (1996). The pulsatile flow of couple stress fluid through a porous medium with periodic body acceleration and magnetic field was investigated by Rathod and Tanveer (2009).

In view of these, we investigated the thermal effects on Stokes’ second problem for couple stress fluid through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied in detail with the help of graphs.

2. FORMULATION OF THE PROBLEM The equations of motion that characterize couple stress fluid flow are similar to the Navier- Stokes equations and are given by:

dρ + ρ div ( q ) = 0 dt

(1)

( )

1 (s)  2 curl ( ρ c ) + div τ dq ρ = ρ f + dt  + 1 curl ( div ( M ) )  2

µ

 − q − ρ 1 − α (θ − θ ∞ )  gδ i1  k 

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(2)

K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

where ρ is the density of the fluid, α - the co-efficient of thermal expansion, g - the (s)

acceleration due to gravity, τ is the symmetric part of the force stress diad, µ is the viscosity of the fluid, k is the permeability of the porous medium, M is the couple stress diad and f , c are the body force per unit mass and body couple per unit mass respectively.

The constitutive equations concerning the force stress tij , and the rate of deformation tensor dij are given by: 1 tij = − pδ ij + λ div ( q ) δ ij + 2 µ d ij − ε ijk  m, k + 4ηωk , rr + ρ ck  2

(3) The couple stress tensor mij that arises in the theory has the linear constitutive relation

1 mij = mδ ij + 4ηω j ,i + 4η 'ωi . j 3 1 In the above ω = curlq 2

(4) is the

spin vector, ωi , j is the spin tensor, m is the trace of couple stress tensor mij , p is the fluid pressure and ρ ck is the body couple vector. Comma in the suffixes denotes covenant differentiation and ωk ,rr stands for

ωk ,11 + ωk ,22 + ωk ,33 . The quantities λ and µ , are the viscosity coefficients and η , η ' are the couple stress viscosity coefficients.

These material constants are constrained by the inequalities µ ≥ 0,

3λ + 2 µ ≥ 0,

η ≥ 0,

η' ≤η

(5)

There

is

a

length

125

parameter

l = η / µ , which is a characteristic measure of the polarity of the fluid model and this parameter is identically zero in the case of non-polar fluids.

After neglecting body forces and body couples, the equations governing the couple stress fluid dynamics as given by Stokes (1966) are (6) divq = 0  µ curl ( curl ( q ) ) −  ∂q  + ( q.∇q ) = −∇p −  curl curl curl ( curl ( q ) )  ∂t  

ρ

( (

  − µ q + α g (θ − θ∞ ) k

))

(7) We consider the unsteady flow of an incompressible, couple stress fluid through a porous medium which fills the half space y > 0 above a flat (solid) plate occupying xz-plane. Initially, we assume (has both fluid and plate are at rest. At time t = 0 + , whether we allow the plate to start with a constant velocity U along x-axis or oscillate with velocity U cos ωt the flow occurs only in x - direction. Therefore, the velocity is expected to be in the form q = u ( y, t ) , 0, 0 and it automatically

(

)

satisfies the continuity Eq. (6).

Under these assumptions the Eq. (7) becomes

ρ

∂u ∂ 2u ∂ 4u µ = µ 2 − η 4 − u + α g (θ − θ ∞ ) ∂t ∂y ∂y k (8)

The energy equation (MCF model) is given by (Ibrahem et al., 2006)

τθ tt + θt =

χ θ ρ c p yy

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(9)

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K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

where ωij is the vorticity, χ the thermal conductivity, θ the temperature, and τ the thermal relaxation time.

Introducing the non-dimensional variables

θ − θ0 u y U , , y = , t = t, θ = θw − θ0 U l l η ρUl l2 = , R = (10) u=

µ

µ

into Equations (8) and (9), we get (after dropping bars)

∂u ∂ 2u ∂ 4u 1 u + Gθ = − − ∂t ∂y 2 ∂y 4 Da λ pθtt + pRθt = θ zz vρ c p τ U 02 = Cp . ,λ= here p = χ v R

The non-dimensional conditions are θ ( y, t ) = eiωt at y=0

θ ( y, t ) → 0 u ( y, t ) = eiωt ∂u =0 ∂y 2 u ( y, t ) → 0

as

y→∞

at

y=0

at

y=0

as

y→∞

2

∂ 4U ∂ 2U  1  − 2 +  iRω +  U = GΘ 4 ∂y ∂y Da   Θ ''+ (λ pω 2 − iω pR)Θ = 0 The conditions are

corresponding

(15) (16)

boundary

Θ (0) = 1, Θ (∞ ) = 0, U (0) = 1, U ''(0) = 0, U (∞ ) = 0

(17) Solving the Equations (15) and (16) using the boundary conditions Eq. (17), we get Θ( y ) = e − my (18)

(11)

U ( y ) = c1e− m1 y + c2e− m2 y

(12)

where m =

(19)

−λ pω 2 + iω p ,

m2 2 m12 c1 = 2 , c2 = − 2 , m2 − m12 m2 − m12 boundary

m1 =

1− r 1+ r , m2 = 2 2 2

 1  and r = 1 − 4   − 4iRω .  Da  The solution of Equations (13) and (14) are given by θ ( y, t ) = e− ( my −iωt ) (20) (13)

3. SOLUTION To solve the nonlinear system (11) and (12) with the boundary conditions (13), we assume that

u ( y, t ) = U ( y )eiωt , θ ( y, t ) = Θ( y)eiωt (14) If we substitute by Eq. (14) in Equations (11) and (12) and the boundary conditions (13), we get

u ( y, t ) = c1e− ( m1 y −iωt ) + c2e− ( m2 y −iωt ) (21) The rate of heat transfer coefficient in terms of Nusselt number Nu at the wall of the plate is given by

Nu = −

∂θ ∂y

= meiωt

(22)

y =0

4. DISCUSSION OF THE RESULTS velocity

Fig. 1 depicts the variation of Reu with y for different values

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K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

of Darcy number Da with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , t = 0.1 and G = 5 . It is found that, the velocity Reu osscilates with y . Further it is found that, the velocity Reu first increases and then decreases with increasing Da .

u with y for different values of Darcy number Da with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , t = 0.1 and G = 5 is shown in Fig. 2. It is noted that, the velocity u The variation of velocity

increases with an increase in Da .

Fig. 3 depicts the variation of velocity Reu with y for different values of Grashof number G with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , Da = 0.1 and t = 0.1 . It is noted that, the Reu first decreases and then increases with increasing G. The variation of velocity u with y for different values of Grashof number G with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , Da = 0.1 and t = 0.1 is depicted in Fig. 4. It observed that, the absolute velocity u first decreases and then decreases with an increase in

G.

Fig. 5 illustrates the variation of velocity Reu with y for different values of couple stress Reynolds number R with λ = 0.005 , p = 1 , ω = 10 , G = 5 and t = 0.1 . As R increases, it is seen that the velocity Reu first increases and then decreases with increasing R .

127

The variation of velocity u with y for different values of couple stress Reynolds number R with λ = 0.005 , p = 1 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1 is illustrated in Fig. 6. It is noted that, the absolute velocity u decreases with increasing R .

Fig. 7 shows the variation of velocity Reu with y for different values of p with λ = 0.005 , R = 0.5 , ω = 10 , G = 5 , Da = 0.1 and t = 0.1 . It is observed that, the the velocity Reu first decreases and then increases with an increase in p .

u with y for different values of p with λ = 0.005 , R = 0.5 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1 is shown in Fig. 8. It is found that, the absolute velocity u decreases with increasing p .

The variation of velocity

Fig. 9 depicts the variation of velocity Reu with y for different values of λ with p = 1 , R = 0.5 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1 . It is noted that, the velocity Reu fisrt increases and then decreases with an increase in λ .

The variation of velocity u with y for different values of λ with p = 1 , R = 0.5 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1 is depicted in Fig. 10. It is observed that, the absolute velocity u increases with increasing λ .

Fig. 11 illustrates the variation of temperature Reθ with y for different

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K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

values of p with λ = 0.005 and ω = 10 .

It is found that, the temperature Reθ initially increases and then decreases with increase in p .

The variation of temperature

θ

with y for different values of p with λ = 0.005 and ω = 10 is shown in Fig. 12. It is noted that, the absolute temperature θ decreases with increasing p .

Table-1 shows the effect of p on Nusselt number Re Nu with t = 0.1 , λ = 0.005 and ω = 1 . It is found that, the Re Nu increases with increasing p .

Table-2 depicts the effect of ω on Nusselt number Re Nu with t = 0.1 , λ = 0.005 and p = 1 . It is noted that, the Re Nu increases with increasing ω .

Reu

y for different values of Darcy number Da with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , Fig. 1 The variation of velocity

t=0.1 and G = 5 .

with

p on Nusselt number t = 0.1 , λ = 0.005 and

Table-1: Effect of

Re Nu

ω = 1.

with

p

Nu

0.2 0.5 0.7 1

0.2822 0.4462 0.5280 0.6310

Table-2: Effect of ω on Nusselt number Re Nu with t = 0.1 , λ = 0.005 and p = 1.

ω

Nu

0 1 2 3

0 0.6310 0.7755 0.7966

u

y for different values of Darcy number Da with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , t=0.1 and G = 5 .

Fig. 2 The variation of velocity

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with

K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

Reu

y for different values of Grshof number G with λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , Da = 0.1 and t=0.1.

Fig. 3 The variation of velocity

Reu

with

The variation of velocity

u

with

different values of Grshof number

G

y

for with

λ = 0.005 , p = 1 , ω = 10 , R = 0.5 , Da = 0.1 and t=0.1.

u

Da = 0.5 and t=0.1.

y for different values of Reynolds number R with λ = 0.005 , p = 1 , ω = 10 , G = 5 , Da = 0.5 and t=0.1.

Reu with y for different values of p with λ = 0.005 , R= 0.5, ω = 10 , Da = 0.1 , G = 5 and t = 0.1 .

u with y for different values of p with λ = 0.005 , R = 0.5 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1

Fig. 5 The variation of velocity

with

y

Fig. 4

129

for different values of Reynolds number R with λ = 0.005 , p = 1 , ω = 10 , G = 5 ,

Fig. 7 The variation of velocity

Fig. 6 The variation of velocity

with

Fig. 8 The variation of velocity

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K. Srinivasa Rao, et al., J. Comp. & Math. Sci. Vol.5 (2), 123-131 (2014)

Reu with y Fig. 10 The variation of velocity u with y for different values of λ with p = 1 , R = 0.5 , for different values of λ with p = 1 , R = 0.5 , ω = 10 , Da = 0.1 , G = 5 and t = 0.1 . ω = 10 , Da = 0.1 , G = 5 and t = 0.1 .

Fig. 9 The variation of velocity

Reθ p with λ = 0.005

Fig. 11 The variation of temperature for different values of and

ω = 10 .

with

y

Fig. 12 The variation of temperature for different values of and

ω = 10 .

θ

with

p with λ = 0.005

y

5. CONCLUSIONS

REFERENCES

In this chapter, we investigated the thermal effects on Stokes’ second problem for couple stress fluid through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. It is observed that, the Re u and u decreases with increasing G, p and

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