Non-Darcy unsteady mixed convection flow near the stagnation point on a heated vertical surface embedded in a porous medium with thermal radiation and variable viscosity

Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Non-Darcy unsteady mixed convection flow near the stagnation point on a heated vertical surface embedded in a porous medium with thermal radiation and variable viscosity I.A. Hassanien a,*, T.H. Al-arabi b a b

Department of Mathematics, Faculty of Science, Assiut University, 71516 Assiut, Egypt Department of Mathematics, Faculty of Girls, Al-Madina Al-monawra, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 30 May 2007 Received in revised form 26 October 2007 Accepted 29 February 2008 Available online 14 March 2008 PACS: 47.11.BC 47.55p 44.20.+b 44.35.+c 02.70.BF

a b s t r a c t The unsteady mixed convection boundary layer flow near the stagnation point on a heated vertical plate embedded in a fluid saturated porous medium is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature. Both the buoyancy assisting and the buoyancy opposing flow situations are considered with combined effects of the first and second order resistance of solid matrix of non-Darcy porous medium, variable viscosity and radiation. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. The features of the flow and the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The surface shear stress and the heat transfer of the present study are compared with the available results and a good agreement is found. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: Unsteady flow Mixed convection Porous media Non-similar solution Finite difference

1. Introduction The study of mixed convection flow finds applications in several industrial and technical processes such as solar central receivers exposed to wind current, electronic devices cooled by fans, heat exchangers, placed in low-velocity-environment, etc. In the history of fluid dynamics, considerable attentions have been given to study of 2-D stagnation point flows. Hiemenz [1] derived an exact solution of the steady flow and Eckert [2] considered the corresponding heat transfer problems. The combined forced and free convection in the region of two-dimensional stagnation point on a heated vertical non-isothermal flat plate has been studied by Hassanien and Gorle [3]. The above studies deal with steady flow. In several problems, the flow may be unsteady which might be caused by the change in the free stream velocity or in the surface temperature, or in both. When there is an impulsive change in the velocity field, the inviscid flow is developed instantaneously. However, the flow in the viscous layer near the wall is developed slowly and hence becomes fully developed steady flow after sometimes. During a small time the flow is dominated by * Corresponding author. E-mail address: [email protected] (I.A. Hassanien). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.02.011

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

1367

Nomenclature cf ebk f F1 g Grx k K Kkw m, n Nu Pr qr R Rex t, t* T T1 Ue u, v x, y

skin friction coefficient Plank’s function dimensionless stream function empirical constant gravitational acceleration Grashof number thermal conductivity permeability mean absorption constants Nusselt number Prandtl number irradiative heat flux radiation parameter Reynolds number dimensional and dimensionless times, respectively temperature free stream temperature velocity of the potential flow outside the boundary layer velocity components along x, y directions, respectively distance along and normal to the surface

Greek symbols a thermal diffusivity b coefficient of volumetric thermal expansion r viscosity parameter D second order resistance g pseudo-similarity variable c first order resistance f dimensionless time k bouncy parameter q fluid density l dynamic viscosity viscosity at the ambient fluid l1 Subscripts w condition at the wall 1 condition at infinity

the viscous forces and the unsteady acceleration, but for a large time it is dominated by the viscous forces, the pressure gradient and the convective acceleration. For a small period of time the flow is generally independent of the conditions far upstream and at the leading edge or at stagnation point. However, for a large period of time the flow depends on these conditions. The unsteady mixed convection flow in the stagnation region of a heated horizontal and vertical plates have been investigated by Amin and Riley [4] and Seshadri et al. [5], respectively. The boundary layer flow development of a viscous fluid on a semi-infinite flat plate due to the impulsive motion of the free stream has been investigated by Dennis [6] and Watkins [7]. The corresponding problem over a wedge has been studied by Williams and Rhyne [8]. In all the above mentioned studies, the viscosity of the fluid was assumed constant. However, it is known that these physical properties may change significantly with temperature. To accurately predict the flow behavior, it is necessary to take into account this variation of viscosity, (see, Hassanien et al. [9], Gary et al. [10], Mehta and Sood [11], Ling and Dybbs [12], Niled et al. [13], and Hossin and Munir [14]). Moreover, most of the existing studies for this problem neglected the radiation effects. If the radiation is taken into account in some industrial applications such as glass production and furnace design and in space technology applications (such as propulsion system, plasma physics, cosmical flight aerodynamics rocket and geophysics) then the governing equations become quite complicated and hold to be solved. However, Cogley et al. [15] showed that, in the optically thin limit, the fluid does not absorb its own emitted radiation, but the fluid does absorb radiation emitted by the boundaries. Cogley et al. [15] showed that, in the optically thin limit for a non-gray gas near equilibrium the flowing relation holds

1368

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

oqr ¼ 4ðT  T w ÞI; ð1Þ oy oebk  R1 where I ¼ 0 K kw oT w dk, qr is the radiative heat flux, Kkw is the mean absorption and is ebk the Plank’s function. Therefore, the aim of the present study is to analyze the combined effects of variable viscosity and radiation on the unsteady mixed convection flow at a two-dimensional stagnation point on a heated vertical plate embedded in a porous medium. The unsteadiness in the flow field is caused by impulsively creating motion in the free stream and at the same time suddenly raising the surface temperature above the surrounding. The problem is formulated in such a way that for t = 0 it is represented by the Raylaigh type equation and for t ? 1 it is represented by the Hremenz type equations. The partial differential equations governing the flow and the heat transfer have been solved numerically using the implicit finite difference Scheme. Particular cases of the present results are compared with those of Hassanien et al. [16]. 2. Mathematical analysis Let us consider a semi-infinite vertical plate embedded in a saturated porous medium with uniform ambient temperature T1. it is assumed that at time t = 0 the ambient fluid is impulsively moved with a velocity Ue and at the same time the surface temperature is suddenly raised. Fig. 1 shows the flow field over a heated vertical surface where the upper half of the field is assisted by the buoyancy force, but the lower part is opposed by the buoyancy force. The surface of the plate is assumed to have an arbitrary temperature. The fluid properties are assumed to be isotropic and constant, except for the fluid viscosity which is assumed to be a linear function of temperature (see Hassanien et al. [9])   T  T1 ; ð2Þ l ¼ l1 1 þ r Tw  T1 where l is the dynamic viscosity, l1, is the viscosity at the ambient fluid. Under the above assumptions along with Boussinesq approximation, the unsteady laminar boundary layer equations governing the mixed convection flow are given by ou ov þ ¼ 0; ox oy

ð3Þ

  ou ou ou oU e 1 o ou e F 1 e2 þ þu þv ¼ Ue l þ gbðT  T 1 Þ þ ðl U e  luÞ þ 1=2 ðU 2e  u2 Þ; ox ot ox oy q oy oy qK 1 K

ð4Þ

! oT oT oT o2 T 1 oqr :  þu þv ¼a ot ox oy oy2 k oy

ð5Þ

The initial conditions are uðx; yÞ ¼ vðx; yÞ ¼ 0;

Tðx; yÞ ¼ T 1 for t < 0:

ð6Þ

The boundary conditions for t P 0 are uðx; 0Þ ¼ vðx; 0Þ ¼ 0; Tðx; 1Þ ¼ T 1 ;

uðx; 1Þ ¼ U e ¼ ax; n

Tðx; 0Þ ¼ T w ðxÞ ¼ bx ;

a > 0;

b > 0;

n P 0;

Fig. 1. Physical model and coordinate system.

ð7Þ

1369

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

where u, v are the velocity components along x, y directions, respectively, T is the fluid temperature, g is the magnitude of the acceleration due to gravity, q is the fluid density, b is the coefficient of volumetric thermal expansion, k is the thermal conductivity, K is the permeability, e is the porosity and a is the thermal diffusivity. The index n = 0 for constant surface temperature and n = 1 for linear surface temperature. It is worth mentioning that we encounter certain difficulties in formulating the problem of the boundary layer development due to the impulsive motion (see Seshadri et al. [5] and Hassanien et al.  ¼ yðq=l1 tÞ1=2 ; t ¼ U e t=x, and for large time solu[16]). For small time solution (initial unsteady flow) we can use the scale y ; tÞ system, tion (final steady state flows) we can use the scale g = y(Ueq/l1x)1/2, t ¼ U e t=x. If the problem is formulated in ðy the short time solution first in properly, but the large time solution does not fit. This implies that we have to find a scaling of the y-coordinate which behaves like y(q/l1t)1/2 for small time and as y(Ueq/l1x)1/2 for large time. Further, it is convenient to choose a new time scale f so that the region of time integration may become finite. Such transformation have been given by  1=2 aq yf1=2 ; f ¼ 1  expðt Þ g¼ l1 0

t ¼ at;

a > 0; uðx; y; tÞ ¼ axf ;  1=2 al1 f1=2 f ðg; fÞ; vðx; y; tÞ ¼  q

ð8Þ

Tðx; y; tÞ ¼ T 1 þ ðT w  T 1 Þhðg; fÞ; Pr ¼

l1 ; qa

Rex ¼

k¼

Gr x Re2x

;

Grx ¼ gbðT w  T 1 Þ

x3 q2 ; l21

ax2 q : l1

Using (8) in Eqs. (4)–(7) we get 2

00

2

ð1 þ rhÞf 000 þ rh0 f 00 þ 21 gð1  fÞf 00 þ fff þ fð1  f 0 Þ þ khf þ cfð1  ð1 þ rhÞf 0 Þ þ Dfð1  f 0 Þ ¼ fð1  fÞ 0

pr1 h00  Rfðh  1Þ þ 21 gð1  fÞh0 þ fðf h0  nf hÞ ¼ fð1  fÞ

oh ; of

of 0 ; of

ð9Þ

ð10Þ

1e is the first order resistance parameter, D = [F1e4Rexl1/aKq]1/2 is the second order parameter, R = 4aI/ka is the where c ¼ laKq radiation parameter, k > 0 for the buoyancy assisting flow and k < 0 for the buoyancy opposing flow. The boundary conditions (7) reduce to

f ð0; fÞ ¼ f 0 ð0; fÞ ¼ 0;

f 0 ð1; fÞ ¼ 1;

hð1; fÞ ¼ 0;

hð0; fÞ ¼ 1:

ð11Þ

It may be noted that the buoyancy parameter k is the function of stream wise distance x unless the surface temperature (Tw  T1) varies linearly with x (i.e. n = 1). For n = 1, k is constant. Eqs. (9) and (10) are coupled non-linear parabolic partial differential equations, but for f ¼ 0ðt ¼ 0Þ (initial unsteady flow) and f ¼ 1ðt ! 1Þ (final steady stat flow) they reduce to ordinary differential equations. For f = 0, Eqs. (9) and (10) reduce to ð1 þ rhÞf 000 þ rh0 f 00 þ 21 gf 00 ¼ 0; 00

1

ð12Þ

0

h þ 2 prgh ¼ 0:

ð13Þ

For f = 1, Eqs. (9) and (10) reduce to 2

00

2

ð1 þ rhÞf 000 þ rh0 f 00 þ ff þ ð1  f 0 Þ þ cð1  ð1 þ rhÞf 0 Þ þ Dð1  f 0 Þ þ kh ¼ 0; 0

pr 1 h00  Rðh  1Þ þ ðf h0  nf hÞ ¼ 0:

ð14Þ ð15Þ

The boundary conditions for (12)–(15) are f ð0Þ ¼ f 0 ð0Þ ¼ 0;

f 0 ð1Þ ¼ 1;

hð1Þ ¼ 0;

hð0Þ ¼ 1:

ð16Þ

Eqs. (12) and (13) are coupled linear equations and (14) and (15) are coupled non-linear equations. At r = 0, Eqs. (12) and (13) under conditions (16) admit closed from solutions which are given by f ¼ gerfcðg=2Þ  ðpÞ1=2 ½1  expðg2 =4Þ;

h ¼ erfcðpr1=2 g=2Þ:

ð17Þ

Hence f 00 ð0Þ ¼ ðpÞ1=2 ;

h0 ð0Þ ¼ ðpr=pÞ1=2 :

ð18Þ

Eqs. (14) and (15) do not admit closed from solutions. Eqs. (9) and (10) under conditions (11) for f = 1, r = R = 0 (steady case) are identical to those of Ramachandran et al. [17]. Further (9) under conditions for k = 0 (forced convection flow), r = R = 0 is the same as that of Williams and Rhyne [8] if we put m = 1 in there equation. Furthermore, Eqs. (9) and (10) with r = R = 0 are the same as that of Hassanien et al. [16].

1370

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

The physical quantities of interest in this problem are the skin friction coefficient and the Nusselt number, which are defined by   2l ou ¼ 2f1=2 Re1=2 ð1 þ rhÞf 00 ðf; 0Þ; f > 0: ð19Þ cf ¼ x qU 2e oy y¼0 Similarly the heat transfer coefficient in terms of the Nusselt number can be written as   1 oT 0 Nu ¼ x ¼ f1=2 Re1=2 0 < f 6 1: x h ðf; 0Þ; ðT w  T 1 Þ oy y¼0

ð20Þ

We now discuss the local non-similarity method to solve Eqs. (9) and (10). Since it was already seen in the paper of Sparrow [18], that for the problem of coupled local non-similarity equations considerations of equations up to the second level of truncations gives almost accurate results comparable with the solutions from other methods. For the first level of truncation, derivatives in Eqs. (9) and (10) can be neglected. Thus, the governing equations for the first level of the truncation are 2

00

2

ð1 þ rhÞf 000 þ rh0 f 00 þ 21 gð1  fÞf 00 þ fff þ fð1  f 0 Þ þ khf þ cfð1  ð1 þ rhÞf 0 Þ þ Dfð1  f 0 Þ ¼ 0;

ð21Þ

0

pr1 h00  Rfðh  1Þ þ 21 gð1  fÞh0 þ fðf h0  nf hÞ ¼ 0: At the second level of truncation, we introduce new functions F ¼ level of traction thus, we have the following equations:

ð22Þ of ; of

G¼

oh of

and restore all the neglected terms in the first

Table 1 Values of the shear stress f00 (f, 0) and surface heat transfer h0 (f, 0) with time f when k = 1 and Pr = 0.7 c

0

D

0

r

0

R

0

Present result

Hassanien et al. [16]

1

0

0

0

Present result

Hassanien et al. [16]

1

1

0

0

n = 0.5

f

Present result

Hassanien et al. [16]

n = 0.0

n = 1.0

f00 (f, 0)

h0 (f, 0)

f00 (f, 0)

h0 (f, 0)

f00 (f, 0)

h0 (f, 0)

0 0.5 1 0 0.5 1

0.56419 1.23018 1.82292 0.56419 1.22960 1.82292

0.47204 0.39895 0.34962 0.47204 0.40053 0.34962

0.56419 1.22232 1.76760 0.56419 1.21450 1.76760

0.47204 0.46347 0.53697 0.47204 0.47852 0.53697

0.56419 1.20390 1.70632 0.56419 1.19286 1.70632

0.47204 0.57850 0.76406 0.47204 0.59387 0.76406

0 0.5 1 0 0.5 1

0.56419 0.96358 1.42876 0.56419 0.96301 1.41875

0.47204 0.38895 0.29892 0.47204 0.39053 0.29912

0.56419 0.96001 1.37650 0.56419 0.95219 1.37350

0.47204 0.46890 0.46346 0.47204 0.45285 0.46344

0.56419 0.94688 1.32408 0.56419 0.93582 1.32207

0.47204 0.53043 0.67502 0.47204 0.5486 0.66505

0 0.5 1 0 0.5 1

0.56419 0.87013 1.30316 0.56419 0.86955 1.29315

0.47204 0.38480 0.27372 0.47204 0.38594 0.27707

0.56419 0.86835 1.26011 0.56419 0.86054 1.25701

0.47204 0.45381 0.43310 0.47204 0.44077 0.43296

0.56419 0.85766 1.20698 0.56419 0.8467 1.21477

0.47204 0.51902 0.63640 0.47204 0.52718 0.62679

0

0

.25

0

Present result

0 0.5 1

0.48697 1.06667 1.59389

0.47205 0.39903 0.34269

0.48697 1.05772 1.53603

0.47205 0.45812 0.52414

0.48697 1.03787 1.47276

0.47205 0.56577 0.7427

1

0

.25

0

Present result

0 0.5 1

0.48697 1.21967 1.77638

0.47205 0.40022 0.34327

0.48697 1.21137 1.7369

0.47205 0.46271 0.52992

0.48697 1.19487 1.69249

0.47205 0.57615 0.75886

1

1

.25

0

Present result

0 0.5 1

0.48697 1.42539 2.05594

0.47205 0.40071 0.3489

0.48697 1.41681 2.02304

0.47205 0.46906 0.54288

0.48697 1.40088 1.98517

0.47205 0.59125 0.78392

0

0

.25

0.1

Present result

0 0.5 1

0.48697 1.06649 1.60827

0.47205 0.39587 0.31274

0.48697 1.05848 1.54496

0.47205 0.45352 0.50664

0.48697 1.03913 1.47757

0.47205 0.56103 0.7336

1

0

.25

0.1

Present result

0 0.5 1

0.48697 1.21991 1.7853

0.47205 0.39715 0.31247

0.48697 1.21213 1.74236

0.47205 0.45819 0.51159

0.48697 1.19585 1.6954

0.47205 0.57141 0.74917

1

1

.25

0.1

Present result

0 0.5 1

0.48697 1.42578 2.06276

0.47205 0.39757 0.31739

0.48697 1.41756 2.02718

0.47205 0.46451 0.52402

0.48697 1.40175 1.98735

0.47205 0.58649 0.77395

1371

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376 2

00

2

ð1 þ rhÞf 000 þ rh0 f 00 þ 21 gð1  fÞf 00 þ fff þ fð1  f 0 Þ þ khf þ cfð1  ð1 þ rhÞf 0 Þ þ Dfð1  f 0 Þ ¼ fð1  fÞF 0 ;

ð23Þ

0

pr1 h00  Rfðh  1Þ þ 21 gð1  fÞh0 þ fðf h0  nf hÞ ¼ fð1  fÞG:

ð24Þ

Subject to the boundary conditions Fð0; fÞ ¼ F 0 ð0; fÞ ¼ F 0 ð1; fÞ ¼ 0;

ð25Þ

Gð1; fÞ ¼ Gð0; fÞ ¼ 0:

The introduce of the two new dependent variables F and G in the problem requires two equations with appropriate boundary 2 2 conditions. This can be obtained by differentiating (23) and (24) with respect to f and neglecting the terms oof2F and oof2h which leads to 000

00

00

2

00

ð1 þ rhÞF 000 þ rðGf þ h0 F 00 þ G0 f 00 Þ þ 21 gðF 00  f 00  fF 00 Þ þ ff þ fðfF þ Ff  2f 0 F 0 þ cF 0 þ 2DF 0 f 0 Þ þ ð1  f 0 Þð1  DÞ þ cð1  f 0 Þ þ kðh þ fGÞ ¼ ð1  2fÞF 0 ;

ð26Þ 0

0

Pr1 G00  R½ðh  1Þ þ fG þ 21 g½ð1  fÞG0  h0  þ f h0  nf h þ f½fG þ h0 F  nðf 0 G þ hF 0 Þ ¼ ð1  2fÞG:

ð27Þ

Subject to the boundary conditions Fð0; fÞ ¼ F 0 ð0; fÞ ¼ F 0 ð1; fÞ ¼ 0;

ð28Þ

Gð1; fÞ ¼ Gð0; fÞ ¼ 0:

Table 2 Values of the shear stress f00 (f, 0) and surface heat transfer h0 (f, 0) with time f when k = 1 and Pr = 0.7 c

0

D

0

r

0

R

0

Present result

Hassanien et al. [16]

1

0

0

0

Present result

Hassanien et al. [16]

1

1

0

0

n = 0.5

f

Present result

Hassanien et al. [16]

n = 0.0

n = 1.0

f00 (f, 0)

h0 (f, 0)

f00 (f, 0)

h0 (f, 0)

f00 (f, 0)

h0 (f, 0)

0 0.5 1 0 0.5 1

0.56419 0.5857 0.52289 0.56419 0.57650 0.52289

0.47207 0.41417 0.28051 0.47207 0.39402 0.28051

0.56419 0.58637 0.60713 0.56419 0.58664 0.60713

0.47207 0.45271 0.43624 0.47207 0.45323 0.43624

0.56419 0.59619 0.69166 0.56419 0.60428 0.69166

0.47207 0.53259 0.63325 0.47207 0.54548 0.63325

0 0.5 1 0 0.5 1

0.56419 0.35507 0.19821 0.56419 0.35476 0.18921

0.47207 0.39107 0.22119 0.47207 0.38719 0.20835

0.56419 0.84947 1.09595 0.56419 0.36098 0.25446

0.47207 0.45593 0.47719 0.47207 0.42976 0.33148

0.56419 0.37473 0.33322 0.56419 0.37262 0.32355

0.47207 0.48614 0.49303 0.47207 0.49903 0.49303

0 0.5 1 0 0.5 1

0.56419 0.28792 0.15188 0.56419 0.28160 0.15099

0.47207 0.38628 0.19840 0.47207 0.38340 0.18988

0.56419 1.13611 1.5505 0.56419 0.28701 0.20541

0.47207 0.46248 0.50831 0.47207 0.41832 0.30378

0.56419 0.30086 0.26613 0.56419 0.29965 0.26436

0.47207 0.46409 0.46331 0.47207 0.47698 0.45411

0

0

.25

0

Present result

0 0.5 1

0.48697 0.51721 0.47274

0.47207 0.41426 0.27599

0.48697 0.51633 0.54099

0.47207 0.44944 0.42741

0.48697 0.52182 0.60764

0.47207 0.52382 0.61762

1

0

.25

0

Present result

0 0.5 1

0.48697 0.7271 0.88883

0.47207 0.40965 0.29723

0.48697 0.72906 0.93316

0.47207 0.45195 0.46245

0.48697 0.73757 0.98002

0.47207 0.53898 0.67042

1

1

.25

0

Present result

0 0.5 1

0.48697 0.97832 1.30724

0.47207 0.40598 0.31695

0.48697 0.98167 1.33721

0.47207 0.45793 0.49518

0.48697 0.99039 1.36961

0.47207 0.55998 0.72003

0

0

.25

0.1

Present result

0 0.5 1

0.48697 0.51802 0.46264

0.47207 0.4133 0.26082

0.48697 0.51662 0.53512

0.47207 0.4477 0.41717

0.48697 0.52162 0.60464

0.47207 0.52159 0.61146

1

0

.25

0.1

Present result

0 0.5 1

0.48697 0.72742 0.88297

0.47207 0.4086 0.28193

0.48697 0.72901 0.92955

0.47207 0.45009 0.45246

0.48697 0.73721 0.9781

0.47207 0.53665 0.66462

1

1

.25

0.1

Present result

0 0.5 1

0.48697 0.97837 1.30362

0.47207 0.40477 0.30149

0.48697 0.98149 1.33501

0.47207 0.45591 0.48536

0.48697 0.99007 1.36846

0.47207 0.5576 0.71451

1372

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

Eqs. (21)–(24), and (26), (27) subject to the boundary conditions (25) and (28) were solved numerically using finite difference method as described in [17]. To conserve space, the details of the solution procedure are not represented here. 3. Results and discussion In order to validate our results, we have compared the skin friction coefficient and the surface heat transfer for the prescribed surface temperature for some values of f(0 6 f 6 1), r = 0 and R = 0 with those of Hassanien et al. [16] as shown in Tables 1, 2. The results are found to be in excellent agreement. Therefore, it can be concluded that the developed code can be used with great confidence to study the problem discussed in this paper. It is clearly seen that surface shear stress increases as viscosity, first and second order resistance increases. The opposite effects are observed for surface heat transfer and this is in accordance with the results presented in Tables 1 and 2. Farther more, for f > 0, the buoyancy assisting flow have higher values of skin friction coefficient and the surface heat transfer.

3.5

___ λ = 1 − − λ = −1

3

Pr=0.7 7 20

2–1cf √ Rex

2.5

2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

ζ Fig. 2. Variation of the skin friction coefficient with time f for n = 1, c = 0.6, D = 0.5, r = 0.2 and R = 0.01.

3

___ λ = 1 − − λ = −1

2.5

20

− θ ′ (ζ , 0 )

2 7

1.5

1 Pr=0.7

0.5

0 0

0.2

0.4

0.6

0.8

1

ζ Fig. 3. Variation of the surface heat transfer (h0 (f, 0)) with time f for n = 1, c = 0.6, D = 0.5, r = 0.2 and R = 0.01.

1373

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

The effects of first resistance parameter c, second resistance parameter D, variable viscosity parameter r, Prandtl number Pr and radiation parameter R on the skin friction coefficient and with the surface heat transfer time f(0 6 f 6 1) for the buoyancy assisting and opposing flows (k = ±1) for n = 1 (the non-isothermal surface) are shown in Figs. 2–9. Figs. 2 and 3 illustrate the effects of Pr on the skin friction coefficient and surface heat transfer with time f for the buoyancy assisting and opposing flows for n = 1. It is clearly seen that at the start of motion (f = 0), the buoyancy force parameter k and the Prandtl number Pr have no effect on the surface shear stress f00 (f, 0) and these effects become pronounced with increasing f. The steady state is reached at f = 1(t* ? 1). The surface heat transfer changes little with f except when Pr is large, Pr = 20. For a fixed time f, the surface heat transfer increases significantly with Pr, because the higher Prandtl number fluid has a lower thermal conductivity which results in thinner thermal buoyancy layer and hence a higher heat transfer rate at the surface. For f > 0, heat transfer for the buoyancy opposing flow is less than that of the buoyancy assisting flow. Figs. 4 and 5 show the effects of variable viscosity parameter r and first resistance parameter c on skin friction coefficient and surface heat transfer with f for the buoyancy assisting and opposing flows (k = ±1). It is clearly seen that surface sheer stress increases with increasing the viscosity, and first resistance for the two cases k = ±1, but surface heat transfer decreases

4

___ λ = 1 − − λ = −1

3.5

γ =σ =1

γ = 1, σ = 0

2–1cf √ Rex

3

γ = 0, σ = 1

2.5

γ =σ = 0

2

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

ζ Fig. 4. Variation of the skin friction coefficient with time f for n = 1, Pr = 0.7, D = 0 and R = 0.01.

1

___ λ = 1

− θ ′ (ζ , 0 )

0.8

γ =σ = 0

− − λ = −1

γ = 0, σ = 1

0.6

γ =σ =1

0.4

γ = 1, σ = 0 0.2

0 0

0.2

0.4

ζ

0.6

0.8

1

Fig. 5. Variation of the surface heat transfer (h0 (f, 0)) with time f for n = 1, Pr = 0.7, D = 0 and R = 0.01.

1374

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376 5

___ λ = 1

R = 3, Δ = 1

− − λ = −1

2–1cf √ Rex

4

R = 1, Δ = 1

R = 3, Δ = 0

3

R = 1, Δ = 0 2

1

0 0

0.2

0.4

0.6

0.8

1

ζ Fig. 6. Variation of the skin friction coefficient with time f for n = 1, Pr = 0.7, c = 0 and r = 0.1.

0.6

0.5

R = 3, Δ = 1

R = 3, Δ = 0

R = 1, Δ = 1

R = 1, Δ = 0

− θ ′ (ζ , 0 )

0.4

0.3

0.2

___ λ = 1

0.1

− − λ = −1 0 0

0.2

0.4

0.6

0.8

1

ζ Fig. 7. Variation of the surface heat transfer (h0 (f, 0)) with time f for n = 1, Pr = 0.7, c = 1 and r = 0.1.

with these parameters. The viscosity and first resistance parameters have no effect on both skin friction coefficient and the surface heat transfer at start of motion (f = 0), and the steady state is reached at f = 1. Figs. 6 and 7 display the effects of radiation and second resistance parameter on the skin friction coefficient and the surface heat transfer with time f when Pr = 0.7, r = 0.1, k = ±1 and n = 1. As mentioned in Figs. 4 and 5, we see that there is no effect on surface heat transfer at f = 0. It can be also seen that the skin friction coefficient increases as D and R increase with f for assisting flow (k = 1) but the opposite trend is observed for the buoyancy opposing flow (k = 1) for increasing R only. Also, we see that the surface heat transfer decreases with increasing D and R for k = ±1. The variation of the surface shear stress and the surface heat transfer with time f when the buoyancy parameter k = 0, 5, 10, Pr = 0.7 and n = 0.5, 0, 1 are shown in Figs. 8 and 9. It is seen that the shear stress and the surface heat transfer increase with k because positive buoyancy force acts like favorable pressure gradient which accelerates the motion and reduce both momentum and thermal boundary layers. For n = 0 (isothermal case) the heat transfer is less than n = 1 (non-isothermal case). For n = 0.5, the time variation of heat transfer is very small. The reason for this trend is that for n = 0, the surface temperature difference (Tw  T1) is less than that of n = 1. This results in lower heat transfer for n = 0 as compared to n = 1. Also, for gases (l1aT) reduction in the surface temperature causes thinner boundary layer which in turn increases the surface shear stress.

1375

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376 10

λ = 10

n=1 8

2–1cf √ Rex

5 6

4

0

2

0 0

0.2

0.4

ζ

0.6

0.8

1

Fig. 8. Variation of the skin friction coefficient with time f for Pr = 0.7, c = D = 0.5, r = 0.1 and R = 1.

1.2

λ = 0, n= –0.5,0,1 λ = 5, n= –0.5,0,1

0.9

− θ ′ (ζ , 0 )

λ = 10, n= –0.5,0,1

0.6

0.3

0 0

0.2

0.4

0.6

0.8

1

ζ Fig. 9. Variation of the surface heat transfer (h0 (f, 0)) with time f for Pr = 0.7, c = D = 0.5, r = 0.1 and R = 1.

4. Conclusion The non-Darcy unsteady mixed convection in the stagnation flow on a heated vertical surface embedded in a saturated porous medium has been studied taking into account the effects of variable viscosity and thermal radiation. Both the buoyancy assisting and opposing flow are considered. Numerical results for skin friction coefficient and heat transfer are presented graphically and in tabular from for various parameter conditions. It is found that the skin friction and heat transfer, in general increase with time and there is a smooth transition from small time solution to the large time solution for both cases of constant and variable viscosity. The skin friction as well as the coefficient increases as the viscosity parameter first and second order resistance parameters increase. This increment in the skin friction and heat transfer are found to be more for the buoyancy assisting flow than these of the buoyancy opposing flow.The surface heat transfer increases with Prandtl number and decreases with viscosity and radiation parameters. The surface heat transfer can considerably be reduced by using a lower Prandtl number fluid. It can also be reduced by imposing the buoyancy force in the opposite direction to that of the forced flow or by maintaining uniform temperature.

1376

I.A. Hassanien, T.H. Al-arabi / Commun Nonlinear Sci Numer Simulat 14 (2009) 1366–1376

References [1] [2] [3] [4] [5]

Hiemenz K. Die Grenzschicht an einem in den gleich formigen flussig keitsstrom eingetacuhten geraden krebzylinder. Dingl Polytech J 1911;32:321–4. Eckart ERG. Die Berechnung des warmeubergangs in der laminaren Grenzschicht umstromter koper. VDI Forsch 1942:416. Hassanien IA, Goral. Non-similar solutions for natural convection in micropolar fluids on a vertical plate. Int J Fluid Mech Res 2003;30:381–94. Amin N, Riley N. Mixed convection at a stagnation point. Quart J Mech Appl Math 1995;48:111–21. Seshadri R, Sreeshylan N, Nath G. Unsteady mixed convection flow in the stagnation region of e heated vertical plate due to impulsive motion. Int J Heat Mass Transfer 2002;45:1345–52. [6] Dennis SCR. The motion of a viscous fluid past an impulsively stared semi-infinite flat plate. J Inst Math Appl 1972;10:105–17. [7] Watkins CB. Heat transfer in the boundary layer over an impulsively stared flat plate. J Heat Transfer 1975;97:482–4. [8] Williams JC, Rhyne TH. Boundary layer development of a wedge impulsively set into motion. SIAM J Appl Math 1980;38:215–24. [9] Hassanien IA, Essawy AH, Moursy NM. Variable viscosity and thermal conductivity effects on combined heat and mass transfer in mixed convection over a UHFnUMF wedge in porous media the entire regime. Appl Math Comput 2003;145:667–82. [10] Gary J, Kassory DR, Tadjeran H, Zebib A. The effect of significant viscosity variation on convective heat transport in water-saturated porous media. J Fluid Mech 1982;117:233–49. [11] Mehta KN, Sood S. Transient free convection flow with temperature dependent viscosity in a fluid saturated porous medium. Int J Eng Sci 1992;30:1083–7. [12] Ling JX, Dybbs A. The effect of variable viscosity on forced convection over a flat plate submersed in a porous medium. ASME J Heat Transfer 1992;14:1063–5. [13] Niled DA, Porneala DC, Lage JL. A theoretical study with experimental verification of the viscosity effect on the forced convection through a porous medium channel. ASME J Heat Transfer 1999;121:500–3. [14] Hossin MA, Munir MdS, Rees DAS. Flow of viscous incompressible fluid with temperature-dependent viscosity and thermal conductivity past a permeable wedge with uniform surface heat flux. Int J Therm Sci 2000;39:635–44. [15] Cogley AC, Vincenty WG, Gilles SE. Differential approximation for radiative transfer in a non-gray-gas near equilibrium. AIAA J 1968;6:551–3. [16] Hassanien IA, Ibrahem FS, Omer GhM. Non-Darcian effects on unsteady mixed convection flow in the stagnation region of e heated vertical plate embedded in porous medium, submitted for publication. [17] Ramachandran N, Chen TS, Aramly BF. Mixed convection in stagnation flows adjacent to vertical surface. J Heat Transfer 1988;110:173–7. [18] Sparrow EM, Quack H, Boerner J. Local non-similarity boundary solutions. AIAA 1970;8:1936–42.