MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 33 (2006) 148–156 www.elsevier.com/locate/mechrescom
Eﬀect of variable viscosity on nonDarcy free or mixed convection ﬂow on a vertical surface in a ﬂuid saturated porous medium S. Jayanthi, M. Kumari
*
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India Available online 7 October 2005
Abstract This paper analyzes the variable viscosity eﬀects on nonDarcy free or mixed convection ﬂow on a vertical surface in a ﬂuid saturated porous medium. The viscosity of the ﬂuid is assumed to be a inverse linear function of temperature. Velocity and heat transfer are found to be signiﬁcantly aﬀected by the variable viscosity parameter, Ergun number, Peclet number or Rayleigh number. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Variable viscosity; Free or mixed convection ﬂows; NonDarcy; Vertical surface; Porous medium
1. Introduction Fluid ﬂow and heat transfer through porous medium have been of considerable interest, especially in the past decade. This is primarily because of numerous applications of ﬂow through porous media, such as storage of radioactive nuclear waste materials transfer, separation processes in chemical industries, ﬁltration, transpiration cooling, transport processes in aquifers, ground water pollution etc. The problem of natural convection heat transfer with constant viscosity from a vertical plate in a saturated porous medium are discussed in references (Cheng and Minkowycz, 1977; Kumari et al., 1985; Plumb and Huenefeld, 1981). Mixed convection ﬂow along vertical surfaces in a porous medium has been studied by several investigators (Cheng, 1977; Hsieh et al., 1993a,b; Nakayama and Shenoy, 1993; Wang et al., 1990). The fundamental analysis of convection through porous media with temperature dependent viscosity is driven by several contemporary engineering applications from cooling of electronic devices to porous journal bearings and is important for studying the variations in constitutive property. The eﬀect of variable viscosity for convective heat transfer through porous media are studied by several investigators (Bagai, 2004; Elbashbeshy, 2000; Gray et al., 1982; Horne and Sullivan, 1978; Kassoy and Zebib, 1975; Ling and Dybbs, 1992;
*
Corresponding author. Tel.: +91 80 2293 2314; fax: +91 80 2360 0146. Email address:
[email protected] (M. Kumari).
00936413/$  see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.09.001
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156
149
Pantokratoras, 2004; Strauss and Schubert, 1977). Lai and Kulacki (1990) considered the variable viscosity eﬀect for mixed convection ﬂow along a vertical plate embedded in saturated porous medium. The variable viscosity eﬀects on nonDarcy, free or mixed convection ﬂow on a horizontal surface in a saturated porous medium are studied by Kumari (2001). Considering the importance of inertia eﬀects for ﬂow in a porous medium, the problem of free or mixed convective heat transfer along a vertical surface in a ﬂuid saturated porous medium is studied in this paper. The viscosity of the ﬂuid is assumed to be an inverse linear function of temperature. The governing equations for the ﬂow are solved using an implicit ﬁnitediﬀerence scheme (Cebeci and Bradshaw, 1984). 2. Governing equations Consider the problem of steady, laminar, incompressible, twodimensional, nonDarcy, free or mixed convection ﬂow along a heated vertical plate embedded in a saturated porous medium. It is assumed that the ﬂuid and the solid matrix are everywhere in local thermal equilibrium, the thermophysical properties of the ﬂuid are homogeneous and isotropic. With these assumptions and the application of the Boussinesq and boundarylayer approximations, the governing system of conservation equations can be written as (Kumari et al., 1985; Plumb and Huenefeld, 1981) ux þ v y ¼ 0
ð1Þ 2
ðluÞy þ K ðlu Þy ¼ Kq1 gbT y
ð2Þ
uT x þ vT y ¼ aT yy
ð3Þ
The boundary conditions are vðx; 0Þ ¼ 0; T ðx; 0Þ ¼ T w ðxÞ ¼ T 1 þ Axk T ðx; 1Þ ¼ T 1 ; uðx; 1Þ ¼ 0 ðfor free convection flowÞ uðx; 1Þ ¼ U 1
ð4Þ
ðfor mixed and forced convection flowÞ
Here x and y are the distances along and perpendicular to the surface, respectively; u and v are the velocities in the x and y directions, respectively; T is the temperature; g is the acceleration due to gravity; K is the permeability of the porous medium; K is the form drag coeﬃcient in the Ergun equation; a and b are, respectively, the thermal diﬀusivity and the coeﬃcient of thermal expansion; q1 and l are, respectively, the density and viscosity of the convecting ﬂuid; A is a constant and k is the parameter representing the variation of the wall temperature. The subscripts x and y denote partial derivatives with respect to x and y, respectively. The subscripts w and 1 denote the conditions at the wall and in the free stream region, respectively. It is assumed that the viscosity of the ﬂuid varies inversely as a linear function of temperature and can be written as (Kumari, 2001; Lai and Kulacki, 1990) 1=l ¼ 1=l1 ½1 þ RðT T 1 Þ ¼ aðT T e Þ
ð5Þ
where a = R/l1 and Te T1 = 1/R, a 5 0, R 5 0. Here a and Te are constants, and their values depend on the reference state and thermal property of the ﬂuid, i.e., R. The viscosity of a liquid usually decreases with increasing temperature and it increases for gases. In general for liquids a > 0 and for gases a < 0. On using Eq. (5), Eq. (2) can be rewritten as ðT T e Þuy þ 2K q1 aðT T e Þ2 uuy ¼ uT y þ Kq1 gbaðT T e Þ2 T y
ð6Þ
(a) Free convection ﬂow On applying the following transformations g ¼ ðy=xÞRax1=2 ; u ¼ ow=oy; v¼
wðx; yÞ ¼ aRax1=2 f ðn; gÞ;
v ¼ ow=ox;
ða=xÞRax1=2 ½ððk
hðn; gÞ ¼ ðT T 1 Þ=ðT w T 1 Þ; 0
u ¼ ða=xÞRax f ðn; gÞ
þ 1Þ=2Þf þ knfn þ ðg=2Þðk 1Þf 0
Tw > T1
150
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156
he ¼ ðT e T 1 Þ=ðT w T 1 Þ ¼ 1=ðRðT w T 1 ÞÞ h he ¼ ðT T e Þ=ðT w T 1 Þ;
n ¼ ðx=dÞ
Rax ¼ ðx=aÞ½KgbðT w T 1 Þ=m1 ;
k
Er ¼ K a=md;
Rad ¼ ðd=aÞ½Kgbðad k Þ=m1
ð7Þ
to Eqs. (1), (6), (3) and the boundary conditions (4), we ﬁnd that Eq. (1) is satisﬁed identically, Eqs. (6) and (3) are transformed to f 00 ððh he Þ=he Þn2ErRad f 0 f 00 ¼ ½ð1=ðh he ÞÞf 0 ððh he Þ=he Þh0 h00 þ ð1=2Þðk þ 1Þf h0 kf 0 h ¼ kn½f 0 hn h0 fn
ð8Þ ð9Þ
and the boundary conditions (4) become At g ¼ 0 : f ¼ 0;
h ¼ 1;
As g ! 1 : f 0 ! 0;
h!0
ð10Þ
Here n and g are the dimensionless variables, f, f 0 and h are the dimensionless stream function, dimensionless velocity and dimensionless temperature, respectively, he is a parameter which deﬁnes the eﬀect of variable viscosity of the ﬂuid, d is the pore diameter, Er and Rad are, respectively, the Ergun number and Rayleigh number based on the pore diameter. The value of the parameter he is determined by the viscosity of the ﬂuid in consideration with the operating temperature diﬀerence. The eﬀects of variable viscosity can be neglected for large values of he which implies either R or (Tw T1) are small. On the other hand, for a smaller value of he, either the ﬂuid viscosity changes considerably with temperature or the temperature diﬀerence is high. In either case, the eﬀects of variable viscosity is expected to become very important. It may be noted that for Er = k = 0 Eqs. (8) and (9) reduce to equations of free convection Darcy ﬂow over a vertical plate which are studied by Lai and Kulacki (1990). Furthermore, for k = 0, and he ! 1 we obtain the equations of free convection ﬂow over a vertical plate which are considered by Kumari et al. (1985). (b) Mixed convection ﬂow On using the following transformation g ¼ ðy=xÞPex1=2 ; u ¼ ow=oy;
wðx; yÞ ¼ aPe1=2 x f ðn; gÞ;
v ¼ ow=ox;
hðn; gÞ ¼ ðT T 1 Þ=ðT w T 1 Þ;
u ¼ ða=xÞPex f ðn; gÞ ð11Þ
v ¼ ða=xÞPex1=2 ½ð1=2Þf þ knfn ðg=2Þf 0 Þ Pex ¼ U 1 x=a;
Tw > T1
0
Ped ¼ U 1 d=a;
n ¼ ðx=dÞ
k
Eqs. (6) and (3) reduce to f 00 ððh he Þ=he Þ2ErPed f 0 f 00 ¼ ½f 0 =ðh he Þ nðRad =Ped Þððh he Þ=he Þh0
ð12Þ
h00 þ ð1=2Þf h0 kf 0 h ¼ kn½f 0 hn h0 fn
ð13Þ
and the corresponding boundary conditions (4) become At g ¼ 0 : f ¼ 0;
h ¼ 1;
As g ! 1 : f 0 ! 1;
h!0
ð14Þ
Here Ped is the Peclet number based on the pore diameter. It may be noted that Eqs. (12) and (13) under boundary conditions (14) for k = Er = 0 reduce to the equations of mixed convection Darcy ﬂow over a vertical plate considered by Lai and Kulacki (1990). Furthermore, for he ! 1, Eqs. (12) and (13) reduce to the equations of mixed convection ﬂow which are studied by Hsieh et al. (1993b) for Er = 0 and the corresponding equations of Wang et al. (1990), are obtained by putting k = Er = 0. (c) Forced convection ﬂow The dimensionless equations for the forced convection ﬂow are obtained by putting Rad/Ped = 0 in Eqs. (12) and (13). These are given by
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156
151
f 00 ððh he Þ=he Þ2ErPed f 0 f 00 ¼ f 0 h0 =ðh he Þ h00 þ ð1=2Þf h0 kf 0 h ¼ kn½f 0 hn h0 fn
ð15Þ ð16Þ
and the corresponding boundary conditions are At g ¼ 0 : f ¼ 0;
As g ! 1 : f 0 ! 1;
h ¼ 1;
h!0
ð17Þ
For Er = k = 0 Eqs. (15) and (16) reduce to those of forced convection ﬂow over a vertical plate which is investigated by Lai and Kulacki (1990). It may noted that the equations of constant ﬂuid viscosity for free, mixed and forced convection ﬂows can be obtained by putting he ! 1 in the corresponding dimensionless equations. The heat transfer coeﬃcient in terms of the Nusselt number Nux is expressed as Nux Rax1=2 ¼ ½h0 ðn; 0Þnc Nux Pex1=2
for free convection
ð18Þ
¼ ½h ðn; 0Þmc
for mixed convection
ð19Þ
¼ ½h0 ðn; 0Þfc
for forced convection
ð20Þ
0
Here the subscripts nc, mc and fc correspond to free, mixed and forced convection respectively. The free convection asymptotes can be obtained by rewriting Eq. (18) as Nux Pex1=2 ¼ ðRad =Ped Þ
1=2
n½h0 ðn; 0Þnc
3. Results and discussion The dimensionless diﬀerential equations for free convection, mixed convection and forced convection are solved numerically using Kellerbox method. The details of the method is omitted as it is available in Cebeci and Bradshaw (1984). In order to check the accuracy of the method, the results for mixed convection ﬂow over a vertical plate with constant viscosity in a porous medium are compared with the results of Hsieh et al. (1993b) and Cheng (1977). The velocity and heat transfer at the wall (f 0 (n, 0), h 0 (n, 0)) for various values of ErRad for the case of free convection ﬂow are compared with those obtained by Plumb and Huenefeld (1981) and Kumari et al. (1985). These results are not presented, in order to conserve the space. For
10
1.5
x
Er = 0, n = 1 λ=0
f (ξ,0 )
5
Er = 0, n = 1, λ=0
Presents results x Lai & Kulacki (1990)
x
1
x x xx x x x
x

−θ (0 )
x x
x x x x x x x x
x x x x x x x x x
0
5 10
0.5
x
5
0
θe
5
10
x x
x x x x x x x x x x x x x x x x x
0 10
5
Fig. 1. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
Present results Forced convection Present results Free convection
x Lai & Kulacki (1990)
x x x x x x x x x x x x x x x x x x
0
θe
5
10
152
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156 3
30
Er = 0, Ped = 1, Rad/Ped = 1, n = 1 λ=0
Mixed convection
x
20 x x
f (ξ,0 )
10
x x x x x x

10
0
x x
x
x
x x xx 0.1 xx x xx 1 x x x x x x x x x x x x x x x x x
x
Lai & Kulacki (1990)
x
x x x x x x
x
0.1
x x xx x x x x xx x x x x x
10
x x x x x x x
x

1
1
1 x x x x x x x x x x x x x x x x x
x x x x x x x xxxx x x x x x x
5
0
θe
5
0 10
10
5
x
10 x x x x x x x
x x
1 x x x x x x x x x x x x x x x x x 0.1 xx
0.1
10 10
Present results x Lai & Kulacki (1990)
x
x x
x
2
10
x
Er = 0, Ped = 1, Rad/Ped = 1, n =1 λ=0
x
Rax/Pex
Present results
−θ (ξ,0)
Rax/Pex
Mixed convection
0
θe
5
10
Fig. 2. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
Er = k = 0 and n = 1 the velocity and heat transfer at the wall (f 0 (n, 0), h 0 (n, 0)) for the case of free, forced and mixed convection ﬂows compared with those of Lai and Kulacki (1990). These are presented in Figs. 1 and 2. In all the cases the results are found to be in good agreement. The velocity and heat transfer parameter asymptotically approaches to the case of constant viscosity case as he ! 1, as seen from Fig. 3. This implies either R or (Tw T1) are very small. This means the variation of ﬂuid viscosity is negligible. Figs. 4–6 depict the eﬀect of the Ergun number Er, Rayleigh number Rad and Peclet number Ped on the velocity and heat transfer at the wall (f 0 (n, 0), h 0 (n, 0)), with the variable viscosity parameter he for the case of free, forced and mixed convection respectively. It is observed that the velocity and heat transfer at the wall, decrease as the Ergun number Er increases. This is true for both liquids and gases. Also, for any particular value of the Ergun number Er, the velocity and heat transfer at the wall, for the case of mixed convection ﬂows are greater than that of forced and free convection ﬂows. It is also observed that the eﬀect of the Rayleigh 20
5
Er = 0.01, Ped (Rad) = 1, Rad/Ped = 10 λ = 0.5, ξ = 1
Constant viscosity Variable viscosity
4
Er = 0.01, Pedor Rad = 1 Rad/Ped = 10.0, λ = 0.5, ξ = 1.0.
Constant viscosity Variable viscosity
15

Mixed convection
Mixed convection
−θ (ξ,0 )
f (ξ ,0 )
Mixed convection
10
3
Mixed convection

2
Forced convection
Forced convection Forced convection
5
Free convection
Free convection
Forced convection
1
Free convection 0 10
5
0
θe
5
10
0 10
5
Fig. 3. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
Free convection 0
θe
5
10
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156 2
2.5
Rad/Ped = 10 λ = 0.5, ξ = 1 Er = 0.001 Er = 0.01 Er = 0.1
f (ξ ,0 )
2
10
5 Rad = 1
1
1

Er = 0.001 Er = 0.01 Er = 0.1
λ = 0.5, Rad = 1
1.5
Rad = 1
1.5
Rad = 10, ξ = 1
Free convection
−θ (ξ,0 )
Free convection

153
5
λ = 1.0 λ = 0.8
λ = 0.5, λ = 0.8 λ = 1.0 Rad = 1
5
10
0.5
λ = 0.0
10
5 λ = 0.0
10
0.5
0 10
5
0
θe
5
0 10
10
5
0
θe
5
10
Fig. 4. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
2
2.5
Rad/Ped = 10 λ = 0.5, ξ = 1
Forced convection Ped = 1 2 5
Er = 0.001 Er = 0.01 Er = 0.1
1.5
10
λ = 1.0
10
1.5
Er = 0.001 Er = 0.01 Er = 0.1
Ped = 1
−θ (ξ,0 )
f (ξ ,0 )
2
Forced convection


1
5
2 Ped = 1,2,5,10 λ = 1.0
λ = 0.5
λ = 0.5
1
λ = 0.0
λ = 0.0
0.5 0.5
0 10
Ped = 1,2,5,10
5
0
5
10
0 10
5
θe
0
5
10
θe Fig. 5. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
number Rad on velocity and heat transfer at the wall for free convection and the eﬀect of the Peclet number on the velocity and heat transfer at the wall in the case of mixed and forced convection is analogous to the eﬀect of the Ergun number Er on the velocity and heat transfer at the wall (f 0 (n, 0), h 0 (n, 0)). The eﬀect of the parameter k representing the variation of the wall temperature on the heat transfer at the wall h 0 (n, 0) with he is also shown in Figs. 4–6. It is observed that the heat transfer at the wall h 0 (n, 0) increases with the parameter k for free, forced and mixed convection ﬂows. It is also observed that the parameter k has no signiﬁcant eﬀect on the velocity at the wall f 0 (n, 0) and so it is not shown in the ﬁgures. The reason for such an eﬀect is that the parameter k does not explicitly occur in the equation of motion. The eﬀect of the parameter Rad/Ped on the velocity and heat transfer (f 0 (n, 0), h 0 (n, 0)) at the wall for mixed convection ﬂows with he is shown in Fig. 7. The velocity and heat transfer at the wall increase as Rad/Ped increases. This is true for all values of he.
154
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156 5
30
Rad/Ped = 10, ξ = 1
Mixed convection
Er = 0.001 Er = 0.01 Er = 0.1
4
20
−θ (ξ ,0 )
f (ξ ,0 )
λ = 1.0


Ped = 1 2 10
5 10
Ped = 1
2
5
Rad/Ped = 10, ξ = 1 Er = 0.001 Er = 0.01 Er = 0.1
Mixed convection
3
10
Ped = 1 5
.0
λ=1
2
10 5
10 λ = 0.5
2
5
0
5
1 10
10
Ped = 1
λ = 0.5
λ = 0.0 0 10
2
λ = 0.0 5
0
5
10
θe
θe Fig. 6. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
4
20
Er = 0.01, Ped = 10 λ = 0.5, ξ = 1.0.
Mixed convection
Mixed convection
Er = 0.01, Ped = 10.0, λ = 0.5, ξ = 1
3
Rad/Ped = 10

10
−θ (ξ ,0 )
f (ξ ,0 )
Rad/Ped = 10 Rad/Ped = 10
Rad/Ped = 10

5.0 2
5.0
5.0 5.0 1.0 0 10
1.0
1.0
1.0
1
0.1
0.1 0.1
0.1 5
0
θe
5
10
10
5
0
5
10
θe Fig. 7. Eﬀect of he on f 0 (n, 0) and h 0 (n, 0).
The variation of the velocity and heat transfer at the wall (f 0 (n, 0), h 0 (n, 0)) with n for diﬀerent values of he for free, forced and mixed convection ﬂows is displayed in Fig. 8. The velocity and heat transfer at the wall decrease as he increases. Also for any value of he the velocity and heat transfer at the wall are less aﬀected with n for the case of free and forced convection ﬂows. For mixed convection ﬂows, for any ﬁxed value of he, the velocity and heat transfer at the wall increase with n. The heattransfer at the wall (h 0 (n, 0)) as a function of the mixed convection parameter Rad/Ped is presented in Fig. 9. This ﬁgure also shows the limiting cases of free and forced convection ﬂows. It is observed that the heat transfer is more for variable viscosity case, than the constant viscosity case for liquids (he < 0) and it has the opposite trend for gases (he > 0). Similar behaviour has also been observed for the case of vertical plate by Lai and Kulacki (1990) and for the case of a horizontal plate by Kumari (2001).
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156 35
2
Er = 0.01, Rad or Ped = 10, Rad/Ped = 0 or 10 f’(ξ,0) λ=0.5,ξ = 1 −θ (ξ,0)
Er = 0.01, Ped = 10, Rad/Ped = 10 λ = 0.5

30
f (ξ ,0 ),−θ (ξ ,0)
(f (ξ ,0 ), −θ’ (ξ ,0 ))
Forced convection Free convection
−
θe = −2
1
2 2 2

−
25
20
θe = −2
15

2
0.5
2
10
Free convection Forced convection
0
f (ξ,0) −θ (ξ,0) 
θe = −2
1.5
0
155
1
2
5
0
3
2 2 0
1
ξ
ξ
2
3
Fig. 8. Eﬀect of n on f 0 (n, 0) and h 0 (n, 0).
10
Er = 0.01, Ped =1 λ = 0.5, ξ = 1 Mixed convection Forced convection asympotote Free convection asympotote
−θ (ξ ,0 )
5

θe = −2
∝
2
10
3
10
2
10
1
0
10
1
10
2
10
Rad/Ped Fig. 9. Heat transfer results as a function of Rad/Ped.
4. Conclusions Velocity and heat transfer are signiﬁcantly aﬀected by the variable viscosity parameter, Ergun number, Peclet number or Rayleigh number. In the case of variable viscosity it is seen that for liquids, the heat transfer is more than the constant viscosity, whereas for gases, it has the opposite trend. Acknowledgement One of the authors (MK) is thankful to the University Grants Commission, India, for the ﬁnancial support under the Research Scientist Scheme.
156
S. Jayanthi, M. Kumari / Mechanics Research Communications 33 (2006) 148–156
References Bagai, S., 2004. Acta Mech. 169, 187. Cebeci, T., Bradshaw, P., 1984. Physical and Computational Aspects of Convective Heat Transfer. Springer, New York. Cheng, P., 1977. Int. J. Heat Mass Transfer 20, 807. Cheng, P., Minkowycz, W.J., 1977. J. Geophys. Res. 82, 2044. Elbashbeshy, E.M.A., 2000. Int. J. Eng. Sci. 38, 207. Gray, J., Kassoy, D.R., Tadjeran, H., Zebib, A., 1982. J. Fluid Mech. 117, 233. Horne, R.N., Sullivan, OÕ.M.J., 1978. Trans. ASME J. Heat Transfer 100, 448. Hsieh, J.C., Chen, T.S., Armaly, B.F., 1993a. Int. J. Heat Mass Transfer 36, 1486. Hsieh, J.C., Chen, T.S., Armaly, B.F., 1993b. Int. J. Heat Mass Transfer 36, 1819. Kassoy, D.R., Zebib, A., 1975. Phys. Fluids 18, 1649. Kumari, M., 2001. Int. Commun. Heat Mass Transfer 28, 723. Kumari, M., Pop, I., Nath, G., 1985. Int. Commun. Heat Mass Transfer 12, 337. Lai, F.C., Kulacki, F.A., 1990. Int. J. Heat Mass Transfer 33, 1028. Ling, J.X., Dybbs, A., 1992. Trans. ASME J. Heat Transfer 114, 1063. Nakayama, A., Shenoy, A.V., 1993. Appl. Sci. Res. 50, 83. Pantokratoras, A., 2004. Int. J. Eng. Sci. 42, 1891. Plumb, O.A., Huenefeld, J.C., 1981. Int. J. Heat Mass Transfer 24, 765. Strauss, J.M., Schubert, G., 1977. J. Geophys. Res. 82, 325. Wang, C., Tu, C., Zhang, X., 1990. Acta Mech. Sinica 6, 214.