Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

International Communications in Heat and Mass Transfer 38 (2011) 335–339

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet☆ V. Kumaran a,⁎, A.K. Banerjee a, A. Vanav Kumar a, I. Pop b a b

Department of Mathematics, National Institute of Technology, Tiruchirappalli - 620015, India Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e

i n f o

Available online 2 December 2010 Keywords: Unsteady boundary layer flow Stretching sheet Magnetic field Finite difference scheme Dissipation

a b s t r a c t In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ N 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Crane [4] gave an analytic solution for a steady flow past a linearly stretching sheet. In his paper he also addressed the heat transfer problem. Followed by this, many papers on steady stretching sheet problems are studied. It is worth mentioning that the momentum and heat transfer in an MHD Newtonian and non-Newtonian fluid flows over a stretching sheet have been studied extensively in the recent years because of its ever increasing usage in many practical applications, see Abel et al. [16], Fang and Zhang [17], Ray Mahapatra et al. [18], Noor et al. [19], Prasad et al. [20] etc. Few papers on unsteady stretching sheet problems are also published in the literature. In most of these papers unsteadiness is due to time dependent stretching rate as studied by Chen [5], Abel et al. [7], Ishak et al. [8], Andersson et al. [9] and Tsai et al. [11], or, impulsively started stretching as in Mehmood et al. [1], Wang et al. [2], Kumari and Nath [6] and Liao [3]. Recently, Kumaran et al. [13] considered a step change in the magnetic field with time over a boundary layer flow past a stretching sheet. In the present study we consider, a novel unsteady problem, namely the initial steady heat transfer in an MHD flow past a stretching sheet, becomes transient due to the consideration of viscous dissipation when time t′ N 0. Although the dissipation mechanism is in-built in the heat transfer process, in order to study the transient dissipation effect alone theoretically, the transient heat transfer due to dissipation is considered. However, the present

☆ Communicated by A.R. Balalerishnan ⁎ Corresponding author. E-mail address: [email protected] (V. Kumaran). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.11.011

solution when t′ → ∞ (the steady boundary layer flow and heat transfer past a stretching sheet with dissipation) is of practical importance. An important effect which bears great importance on heat transfer is the viscous dissipation. When the viscosity of the fluid and/or the velocity gradient is high, the dissipation term becomes more important although it disappears at infinity. Consequently, the effects of viscous dissipation are also included in the energy equation. For many cases, such as polymer processing, which is at very high temperature and flows appeared in glacier physics, viscous dissipation plays an important role (see Cortell [14]). On the other hand, numerical results confirm that, in the laminar forced convection in straight microchannels, both the temperature dependence of viscosity and viscous dissipation effects cannot be neglected in a wide range of real operative circumstances (see Celata et al. [15]).

2. Formulation 2.1. Initial steady state flow and heat transfer (t′ ≤ 0) — the solution of Char [10] Consider a laminar steady boundary layer flow of an incompressible electrically conducting Newtonian fluid past a stretching sheet. The sheet issues from a thin slit at x′=0,y′ = 0, where x′-axis is along the horizontal direction of flow, y′-axis is normal to the flow, u′ is the horizontal velocity component and v′ is the vertical velocity component. The stretching speed is proportional to the distance from the origin and the stretching rate is β(N 0). The fluid is under the influence of the magnetic field of strength B0, which acts in the direction normal to the stretching sheet. The sheet is assumed to be at temperature T′w, far away

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V. Kumaran et al. / International Communications in Heat and Mass Transfer 38 (2011) 335–339

Nomenclature B0 cp Ec M N Nux Nu Pr t′ T′ T′0 T′w T′∞ (u′, v′) (x′, y′) t T0 T (u, v) (x, y)

u

strength of the magnetic field, kg s− 2A− 1 specific heat at constant pressure, J/kg K Eckert number Magnetic parameter number of iterations local Nusselt number average Nusselt number Prandtl number time, s Transient temperature field, K Initial temperature field, K Temperature at the sheet, K Temperature of the ambient fluid, K velocity components along and normal to the sheet, m s− 1 coordinates along and normal to the sheet, m dimensionless time dimensionless initial temperature field, Eq. (4) dimensionless transient temperature field, Eq. (9) dimensionless velocity components dimensionless coordinates

 2 2 ∂T0 ∂T 1 ∂ T0 ∂u + Ec +v 0 = Pr ∂y2 ∂x ∂y ∂y

ð4Þ

u = x; v = 0; T0 = 1 at y = 0 for x≥0;

ð5Þ

u→0; T0 →0 as y→∞ for x≥0:

When Ec = 0, Eqs. (2)–(5) admit a closed form solution and is given by, pffiffiffiffiffiffiffiffiffiffiffiffi − 1 + My

u = xe

pffiffiffiffiffiffiffiffiffiffiffiffi e− 1 + My −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ;v = 1+M

Pr −pffiffiffiffiffiffiffiffiffi y

T0 ð yÞ = e

F



1 + M

Pr Pr 1 + M;1 + M

F



ð6Þ

− + 1; 1 −Pr + Me

Pr Pr 1 + M;1 + M

pffiffiffiffiffiffiffiffiffiffiffiffi  1 + My

+ 1; 1 −Pr + M



;

ð7Þ

where F(a, b, z) is the Kummer's function (Char [10]). Substituting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y = η = 1 + M in Eq. (7) one can get the solution obtained by Char [10] and more recently obtained by Kumari and Nath [6]. 2.2. Transient unsteady-state heat transfer (t′ N 0) Assuming that the flow and temperature field when t ≤ 0, is given by Eqs. (6)–(7) and introducing the dimensionless quantities (along with Eq. (1))

Greek symbols α thermal diffusivity, m2s− 1 β(N 0) stretching rate, s− 1 Δt step size with respect to time t Δx step size with respect to x Δy step size with respect to y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η y 1+M μ dynamic viscosity, kg m− 1s− 1 ν kinematic viscosity, m2s− 1 ρ density, kg m− 3 σ electrical conductivity, kg− 1m3A2

  ′ −T ∞′ Þ t = βt ′ ; T ðx; y; t Þ = T ′ −T ∞′ Þ = T w

ð8Þ

the dimensionless transient temperature field when t N 0 is governed by,  2 ∂T ∂T ∂T 1 ∂2 T ∂u + Ec ;ðt N 0Þ +u +v = Pr ∂y2 ∂t ∂x ∂y ∂y

ð9Þ

t N 0 : T = 1 at y = 0; T→0 as y→∞; Subscripts w sheet condition ∞ free stream condition

t≤0 :

=

rffiffiffiffi rffiffiffiffi β β u′ v′ ; y = y′ ; u = pffiffiffiffiffiffi ; v = pffiffiffiffiffiffi ; T0 ν ν νβ νβ

T 0′ −T ∞′ σB0 ν νβ ; Pr = ; Ec = M= ′ −T ∞′ ′ −T ∞′ Þ α ρβ Tw cp ðT w

  1 ∂T : Nux = −x T ∂y y = 0

ð12Þ

2.3. Numerical solution

ð2Þ

2

∂u ∂u ∂ u −Mu; +v = u ∂x ∂y ∂y2

ð11Þ

ð1Þ

where ρ is the density of the fluid, cp is the specific heat, μ is the dynamic viscosity, ν is the kinematic viscosity, σ is the electrical conductivity and α is the thermal diffusivity of the fluid, the boundary layer equations governing the initial state of the present problem take the dimensionless form, ∂u ∂v + = 0; ∂x ∂y

T = T0 ð yÞ for x≥0; y≥0:

We assume that Eq. (6) governs the flow field for t N 0 also. The dimensionless form of the local Nusselt number is given by

the ambient fluid temperature is T′∞ and T′0 is the temperature of the fluid. Defining the dimensionless variables x = x′

ð10Þ

ð3Þ

The unsteady dimensionless temperature distribution is obtained as in Muthucumaraswamy and Ganesan [12], using the implicit finite difference method of Crank–Nicholson type. The local skin friction and local Nusselt number have been computed using the Newton Cotes formula. When Prandtl number Pr = 0.71, the step sizes Δx = 0.002, Δy = 0.0125 and Δt = 0.01 are chosen. The domain of computation assumed is 0 ≤ x ≤ 1 and 0 ≤ y ≤ 15. In the case of Pr = 7.0 the domain of computation is 0 ≤ x ≤ 1 and 0 ≤ y ≤ 6.25. The convergence criteria Table 1 Values of t the time to reach the steady state for various M and Pr values. M

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Pr = 0.71 Pr = 7.0

3.16 1.95

3.29 2.00

3.39 2.05

3.45 2.09

3.49 2.13

3.51 2.17

3.52 2.20

3.52 2.23

3.50 2.26

3.48 2.29

3.46 2.32

V. Kumaran et al. / International Communications in Heat and Mass Transfer 38 (2011) 335–339

a

b

0.5

0.14

Pr = 7.0, M = 0.0, Ec = 1.0

Pr = 0.71, M = 0.0, Ec = 1.0

0.12

337

0.4

(T - T0)

(T - T0)

0.1

t = 0.01, 0.02, 0.1, 0.2, 1.0, 2.0, 3.16

0.08 0.06

0.3

t = 0.01, 0.02, 0.1, 0.2, 1.0, 1.95 0.2

0.04 0.1 0.02 0

0

1

2

3

4

5

0 0

6

0.5

1

y

c

d

0.4

t = 0.01, 0.02, 0.1, 0.2, 1.0, 2.0, 2.17

t = 0.01, 0.02, 0.1, 0.2, 1.0, 2.0, 3.51 (T - T0)

(T - T0)

Pr = 7.0, M = 1.0, Ec = 1.0

Pr = 0.71, M = 1.0, Ec = 1.0

0.1

2

0.5

0.14 0.12

1.5

y

0.08 0.06

0.3

0.2

0.04 0.1 0.02 0

0

1

2

3

4

5

0

6

0

0.5

1

1.5

2

y

y

Fig. 1. Transient profiles of excess temperature due to dissipation at x = 1, for Ec = 1.0.

a

1

b

Pr = 0.71, M = 0.0

3

Pr = 7.0, M = 0.0 2

0

-Ty⏐x=1.0, y=0

-Ty⏐x=1.0, y=0

0.5

Ec = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0

-0.5

-1 0

1

Ec = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0

0 -1

0.5

1

1.5

2

2.5

3

3.5

-2

4

0

0.5

1

1.5

c

1

2

2.5

3

3.5

3

3.5

t

t

d

Pr = 0.71, M = 1.0

3

Pr = 7.0, M = 1.0 2

-Ty⏐x=1.0, y=0

-Ty⏐x=1.0, y=0

0.5

0

-0.5

Ec = 0, 0.2, 0.4, 0.6, 0.8, 1.0

1 0 -1

Ec = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 -2 -1 0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

t

1.5

2

t Fig. 2. Profiles of the transient surface heat flux, Nux at x = 1.0.

2.5

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V. Kumaran et al. / International Communications in Heat and Mass Transfer 38 (2011) 335–339

a

b

0.2

0.5

Pr = 7.0, M = 0.0

Pr = 0.71, M = 0.0 0.4

(T - T0)

(T - T0)

0.15

0.1

Ec = 0.2, 0.4, 0.6, 0.8, 1.0 0.05

0

0.3

Ec = 0.2, 0.4, 0.6, 0.8, 1.0 0.2

0.1

0

2

4

6

8

10

12

0

14

0

0.5

1

1.5

y

c

d

0.2

2.5

3

0.5

Pr = 7.0, M = 1.0

Pr = 0.71, M = 1.0 0.4

0.1

(T - T0)

0.15

(T - T0)

2

y

Ec = 0.2, 0.4, 0.6, 0.8, 1.0

0.3

Ec = 0.2, 0.4, 0.6, 0.8, 1.0 0.2

0.05 0.1

0

0

2

4

6

8

10

12

0

14

0

0.5

1

1.5

2

2.5

3

y

y

Fig. 3. Profiles of the final steady state excess temperature due to dissipation at x = 1.

a

b

0.5

Pr = 0.71, M = 0.0

2

Pr = 7.0, M = 0.0

1.5

0.4

1

0.3

Nux

Nux

0.5 0.2 0.1

Ec = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0

0

Ec = 0, 0.2, 0.4, 0.6, 0.8, 1.0

-1

-0.1 -0.2

0 -0.5

-1.5 0

0.2

0.4

0.6

0.8

-2

1

0

0.2

0.4

c

0.6

0.8

1

x

x

d

0.5

Pr = 0.71, M = 1.0

2

Pr = 7.0, M = 1.0

1.5

0.4

1

0.3

Nux

Nux

0.5 0.2 0.1

Ec = 0, 0.2, 0.4, 0.6, 0.8, 1.0

-0.5

0

-1

Ec = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 -0.1 -0.2 0

0

-1.5 0.2

0.4

0.6

0.8

1

-2

0

0.2

x

0.4

0.6

x Fig. 4. Profiles of the final steady state local Nusselt number.

0.8

1

V. Kumaran et al. / International Communications in Heat and Mass Transfer 38 (2011) 335–339 Table 2 Values of the final steady local Nusselt number (Nux) at x = 1.0. Ec

0.0 0.2 0.4 0.6 0.8 1.0 a

Pr = 0.71

a

Pr = 7.0

M = 0.0

M = 1.0

M = 2.0

M = 0.0

M = 1.0

M = 2.0

0.4586 0.4083 0.3581 0.3078 0.2576 0.2073

0.3841 0.3046 0.2251 0.1456 0.0661 − 0.0134

0.3378 0.2353 0.1329 0.0304 − 0.072 − 0.1745

1.8954 1.6208 1.3461 1.0715 0.7968 0.5222

1.8048 1.3233 0.8418 0.3602 − 0.1213 − 0.6028

1.7352 1.0756 0.4159 − 0.2437 − 0.9034 − 1.5630

Ec = 0: Char [10], Kumari and Nath [6].

used is the absolute difference between the values of (n + 1)th and (n) th time step of the nondimensional temperature less than 5 × 10− 5. 3. Results and discussion It is observed that t , the time to reach steady state is independent of Ec. The computed results reveal that T(x, y, t) = T0(y) + Ec x2h(y, t). From Table 1, a local maximum of steady time is seen around M = 1.3 for Pr = 0.71 and at a higher value of M for Pr = 7. The transient profiles of excess temperature due to dissipation at x = 1 when Ec = 1.0 (Fig. 1) show a remarkably increased value of excess temperature near the sheet for Pr = 7.0 than for Pr = 0.71. The maximum temperature is attained within the boundary layer very near to the sheet for Pr = 7.0. On the other hand the boundary layer thickness is more for Pr = 0.71 than Pr = 7.0. These effects are more pronounced for increasing the M values and also for increasing the Ec values (Fig. 3). The profiles of transient Nusselt number at x = 1 are shown in Fig. 2. It is seen that at x = 1, Nux decreases with the increasing time t and Ec. The rate of decrease in Nux is more for small times. Nux attains negative values also for increasing the Ec and M values. All these effects are more pronounced for increasing the Pr and M values. From Figs. 4(a)–(d) for the final steady state local Nusselt number Nux, it is seen that an increase in the Ec value leads to the decrease in the Nux value. The increase in Pr leads to an increase in Nux. There is a decrease in the value of Nux for an increase in M. In the downstream, Nux changes sign, for large Ec. Thus, the heat energy transfer is from the fluid to the sheet in the downstream when Ec is large. It is instructive to see that Nux is linear in x for Ec = 0 whereas Nux varies with Ec x2 under dissipative effect. These effects are also confirmed from Table 2 for the steady Nux at x = 1. 4. Conclusion Transient temperature field due to a step change in Ec (from Ec = 0 to Ec ≠ 0) given at time t = 0 in a steady MHD flow and heat transfer over a stretching sheet is obtained. Some of the important observations are: • T(x, y, t) is of the form: T0(y) + Ec x2h(y, t). • Under dissipative effect, the temperature attains the maximum near the sheet (not at the sheet) within the boundary layer.

339

• Under dissipative effect, the local Nusselt number increases up to some x, then decreases in the downstream and becomes negative also. • The time to reach the steady state remains constant with respect to Ec.

Acknowledgement The authors wish to thank the referee for helpful comments and suggestions that improved the content of the paper.

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