Thermorheological effect on thermal nonequilibrium porous convection with heat generation

International Journal of Engineering Science 74 (2014) 55–64

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Thermorheological effect on thermal nonequilibrium porous convection with heat generation S. Saravanan ⇑, V.P.M. Senthil Nayaki Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

a r t i c l e

i n f o

Article history: Received 25 April 2013 Received in revised form 31 July 2013 Accepted 16 August 2013 Available online 25 September 2013 Keywords: Porous medium Thermorheology Thermal nonequilibrium Brinkman’s equation

a b s t r a c t Thermorheological effect on the sufficient conditions for the onset of natural convection in a fluid saturated porous medium with uniformly distributed internal heat sources is studied. The flow through the medium is governed by Brinkman’s equation. The constituent phases of the medium are assumed lack thermal equilibrium. A nonlinear temperature dependent viscosity is considered. The resulting eigenvalue problem is solved using the Galerkin weighted residual method. Results indicate that the temperature dependence of viscosity and the presence of heat sources produce effects that are favourable for convection to set in. A possible application of this study in a practical situation is highlighted. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Problems in fluid mechanics involving the onset of convection in porous media are of continued interest in order to resolve the intricacies involved in several technological operations such as underground disposal of nuclear waste, petroleum reservoir operations, casting and welding in manufacturing process and chemical and food processing. In particular convection currents in porous media containing internally distributed heat sources are of great importance in geophysical applications, synthesis reactions of several chemicals, etc. Many studies on convective instabilities in porous media in the presence of internal heat sources with or without mass flux are available in the open literature (see for example Rudraiah, Veerappa, and Balachandra Rao (1982), Yoon, Kim, and Choi (1998), Nouri-Borujerdi, Noghrehabadi, and Rees (2007) and Saravanan (2009)). Most of the works addressing convective instabilities in porous media have invoked Boussinesq approximation. This is not true in some situations, especially when the operating temperature of the system rises beyond a certain level. In particular the viscosity of many liquids is a strong decreasing function of temperature. For example the viscosity of the liquid nitroethane decreases sharply from 1.354 mPa s to 0.337 mPa s when the temperature increases from 248.15 K to 373.15 K at atmospheric pressure. Hence the variation in viscosity was taken into account and its effect on convection in porous media was discussed as early as 1950s by Rogers and Morrison (1950). Over the past few decades a number of authors have considered this effect (see for example Kassoy and Zebib (1975), Patil and Vaidyanathan (1982), Richardson and Straughan (1993), Nield (1996), Saravanan and Kandaswamy (2004), Hooman and Gurgenci (2008a), Shivakumara, Mamatha, and Ravisha (2010), Vanishree and Siddheshwar (2010) and Rajagopal, Saccomandi, and Vergori (2010)). The works addressing variable viscosity in addition to internal heat generation are limited. Postelnicu, Grosan, and Pop (2001) have analyzed forced convective flow of a internally heat generating and variable viscosity fluid over a flat plate embedded in a porous medium. Bagai (2004) has analyzed the effect of the variable viscosity as well as the internal heat generation on the steady free convection boundary layers over a non-isothermal axisymmetric body embedded in a fluid ⇑ Corresponding author. E-mail address: [email protected] (S. Saravanan). 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.08.004

56

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

saturated porous medium. He concluded that the viscosity variation contributes a pronounced effect on the velocity profile and the heat transferred is more for a less viscous fluid. Saravanan and Kandaswamy (2004) have studied the stability of convective motion of a variable viscosity fluid contained in a vertical layer generated purely by uniformly distributed internal heat sources in the presence of a transverse magnetic field. It was found that thermal running waves are the most unstable modes and dominate the shear ones when the viscosity decreases. Studies available on convection in porous media are usually based on thermal equilibrium between the solid and the fluid phases. Nevertheless this assumption cannot be used when their temperatures deviate significantly in situations such as the one involving sudden or high speed convective flows. In such situations the hot fluid stream can penetrate into the porous bed and hence in a representative elementary volume its temperature becomes sufficiently higher than that of the adjacent solid phase. This necessitates to consider the heat transfer characteristics of the fluid and solid phases separately. This Local thermal nonequilibrium (LTNE) situation was discussed in the pioneering work of Combarnous (1972). It was then discussed elaborately by the Rees and his coworkers (see Banu and Rees (2002) and Rees and Pop (2005)) using the two equation model proposed by Nield and Bejan (1999). It was concluded that a thermal shock wave is formed within the fluid phase when the velocity of the fluid is sufficiently large. The presence of heat transfer between the phases caused the strength of the thermal shock to degrade with time. Following this several investigations have been done employing the LTNE model (see for example, Shivakumara et al. (2010), Nield and Kuznetsov (2010) and Saravanan and Jegajothi (2010) and the references cited therein). The objective of the present paper is to report the onset of natural convection in a porous medium heated from below that is saturated with a heat generating fluid exhibiting temperature sensitive viscosity. Most of the earlier studies have used the classical Darcy’s law. It does not capture the flow structure near the bounding rigid surfaces where close packing of the porous fillings is not possible. It is silent about the boundary effects that could become increasingly significant when the porous medium is made up of coarser fillings. Hence we shall employ Brinkman’s correction to the Darcy’s law which incorporates a Laplacian term analogous to that appearing in the Navier–Stokes equations. According to Rajagopal (2007) the equations due to Darcy and Brinkman can be obtained systematically by making severe approximations on a system of more general constitutive equations which are based on the mixture theory. Brinkman’s correction can take care of the boundary effects and has been successfully used in recent studies dealing with convection in complex fluids (see Nield and Kuznetsov (2010) and Sunil, Sharma, and Mahajan (2011)). In addition the porous medium is assumed to be in LTNE state. We also consider the porous medium to be confined between two rigid boundaries, a more realistic situation. We believe that the results of this study can provide a theoretical support for understanding various processes taking place in packed bed reactors, heat exchangers, etc. (see Levenspiel (1999)). In particular packed bed reactors, which are used in chemical and petroleum industries, contain basically a tube filled with randomly arranged catalytic pellets through which fuels are fed with high speed at various temperatures. Coarser pellets are normally used in order to have high surface area which could speed up the reaction rate. Several exothermic oxidation and hydrogenation processes are usually conducted in these reactors. For example the organic compound isopropanol is synthesized in these reactors by hydrogenating acetone. In this process a stream of acetone which remains in the liquid state enters the tube and releases heat during hydrogenation. One should notice that the viscosity of acetone drops to less than one half its value for an increase in its temperature by 75 K (see Table 2). This specific example corrresponds to a situation in which all the effects to be discussed in this work, viz., Brinkman’s extension, LTNE, internal heating and viscosity variation are simultaneously present. 2. Mathematical formulation We consider a horizontal fluid saturated porous layer of height d which is heated from below and cooled from above. The lower and the upper surfaces of the layer are held at temperatures T l and T u ð< T l Þ respectively. The porous medium is isotropic and homogeneous and is of infinite horizontal extent. It exhibits LTNE and hence we shall employ a two-field model for the energy exchange. However it is assumed that at the bounding surfaces the solid and fluid phases have identical temperatures. Both the two phases of the layer generate heat at a uniform rate q000 . We shall employ Brinkman’s correction which has a Laplacian term analogous to that appearing in the Navier–Stokes equations. To describe the geometry we choose a Cartesian coordinate system with its origin at the lower surface and z axis pointing vertically upwards. The viscosity l of the fluid is assumed to depend on temperature. A nonlinear approximation characterizing as a function of temperature yields the thermorheological equation of state in the form

h

l ¼ l0 1  M1 ðT f  T u Þ  M2 ðT f  T u Þ2

i

ð1Þ

where M 1 and M 2 are empirical constants. The thermal conductivity is treated as a constant as it so for almost all liquids. The complete system of dimensional conservation equations relevant to the above situation is !

rq ¼0 !

ð2Þ !

!

l q þKðrp  q g Þ  K le r2 q ¼ 0

ðqcÞf

! @T f þ ðqcÞf ðq rÞT f ¼ kf r2 T f þ hðT s  T f Þ þ q000 f @t

ð3Þ ð4Þ

57

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

@T s ¼ ð1  Þks r2 T s  hðT s  T f Þ þ ð1  Þq000 s @t   q ¼ q0 1  bðT f  T u Þ ð1  ÞðqcÞs

!

ð5Þ ð6Þ !

where q is the velocity vector, p the pressure, q the density,  the porosity, j the diffusivity, g the gravitational acceleration, l the fluid viscosity, T the fluid temperature, h the inter-phase heat transfer coefficient, c the specific heat, K the permeability of the porous matter, le the effective viscosity and the subscript f and s refer to fluid and solid phases respectively. We eliminate the pressure from the momentum equation and non-dimensionalize the resulting pressure free equation and the energy equations for both phases using the following transformations, ðx; y; zÞ ¼ dðx ; y ; z Þ; ðu; v ; wÞ ¼

kf ðu ; v  ; w Þ=ðqcÞf d; p ¼ kf lp =ðqcÞf K; t ¼ ðqcÞf d2 t =kf ; T s ¼ ðT l  T u ÞT s þ T u ; T f

¼ ðT l  T u ÞT f þ T u . Hence we obtain, after

omitting the asterisks,

h

M1 T f þ M 2 T 2f

" # i @2T f @2T f @w @T f @w @T f @w @T f @u @T f @ v @T f 2 2  1 r w þ ðM1 þ 2M 2 T f Þ u v þ2 þ wr1 T f þ 2   @z @z @x @x @y @y @z @x @z @y @x@z @y@z

þ Rr21 T f þ Dar4 w ¼ 0

ð7Þ

! @T f þ ðq rÞT f  r2 T f  HðT s  T f Þ  Q f ¼ 0 @t

a

ð8Þ

@T s  r2 T s þ cHðT s  T f Þ  Q s ¼ 0 @t

ð9Þ 2

where M 1 ¼ M1 ðT l  T u Þ and M 2 ¼ M 2 ðT l  T u Þ2 are the linear and nonlinear viscosity variation parameters, Da ¼ K le =l0 d the Darcy number, R ¼ K q0 gbðT l  T u Þd=l0 jf the Rayleigh number based on fluid properties, Rc=ð1 þ cÞ ¼ 2

K q0 gbðT l  T u ÞdðqcÞf =½kf þ ð1  Þks l0 the Rayleigh number based on porous media properties, H ¼ hd =kf the nondimen2

2

sional inter-phase heat transfer coefficient, Q f ¼ qf d =kf ðT l  T u Þ the fluid heat generation parameter, Q s ¼ qs d =ks ðT l  T u Þ the solid heat generation parameter, a ¼ kf ðqcÞs =ks ðqcÞf the diffusivity ratio and c ¼ kf =ð1  Þks the porosity-modified conductivity ratio. The basic state is assumed to be of the form ! qb

!

¼ 0;

T f ¼ T fb ðzÞ;

T s ¼ T sb ðzÞ

In this case the temperatures of the fluid and solid phases have to satisfy 2

d T fb 2

þ H½T sb  T fb  þ Q f ¼ 0

ð10Þ

2

 cH½T sb  T fb  þ Q s ¼ 0

ð11Þ

dz 2 d T sb dz

with the boundary conditions

T fb ¼ T sb ¼ 1 at z ¼ 0 T fb ¼ T sb ¼ 0 at z ¼ 1 Since the fluid and the solid phase are not in thermal equilibrium, the use of appropriate thermal boundary conditions may pose a difficulty. However, the assumption that the solid and fluid phases have equal temperatures at the bounding surfaces made at the beginning of this section helps in overcoming this difficulty. The temperature distribution at the basic state across the layer is given by

T mb ¼ 

  Qm 2 Qm z þ 1 zþ1 2 2

ð12Þ

where the subscript m refers either the fluid or solid phase. It is to be noted that Eq. (12) is obtained with H ¼ 0. The basic state is perturbed and the quantities in the perturbed state are given by

ðu; v ; wÞ ¼ ðu0 ; v 0 ; w0 Þ;

T f ¼ T fb þ h;

T s ¼ T sb þ /

where the perturbations are very small and unsteady quantities. Substituting the above equations in Eqs. (7)–(9) and using the basic state solutions, we obtain the following linearized equations for the perturbed quantities, after neglecting the primes

58

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

    h i Qf @w þ Rr21 h þ Dar4 w ¼ 0 M 1 T fb þ M 2 T 2fb  1 r2 w þ ðM 1 þ 2M 2 T fb Þ Q f z þ 1 @z 2

ð13Þ

   Qf @h þ w Q f z þ  1  r2 h  Hð/  hÞ ¼ 0 @t 2

ð14Þ

a

@/  r2 / þ cHð/  hÞ ¼ 0 @t

ð15Þ

Now Eqs. (13)–(15) have to be solved with suitable boundary conditions. We shall now consider the boundaries to be rigid that are more appropriate when the boundary resistance is taken care of. Hence we have

w and

dw dz

¼ 0 at z ¼ 0; 1

ð16Þ

h ¼ / ¼ 0 at z ¼ 0; 1 In order to find the possible instability mode, the normal mode expansions are taken in the form

½wðzÞ; hðzÞ; /ðzÞ ¼ ½WðzÞ; HðzÞ; UðzÞ eiðlxþmyÞþrt where a ¼ becomes

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l þ m2 is the horizontal wave number and r is the complex frequency of disturbance. Then Eqs. (13)–(15)

h 0000 i h i  Da W ðzÞ  2a2 W 00 ðzÞ þ a4 WðzÞ þ M 1 T fb þ M2 T 2fb  1 W 00 ðzÞ  a2 WðzÞ    Qf þ ðM 1 þ 2M2 T fb Þ Q f z þ  1 W 0 ðzÞ  Ra2 HðzÞ ¼ 0 2 

H00 ðzÞ  ðr þ a2 þ HÞHðzÞ  Q f z þ



ð17Þ

 Qf  1 WðzÞ þ HUðzÞ ¼ 0 2

ð18Þ

U00 ðzÞ  ðar þ a2 þ cHÞUðzÞ þ cHHðzÞ ¼ 0

ð19Þ

The eigenvalue problem defined by Eqs. (17)–(19) together with the boundary conditions is a two-point boundary value problem with variable co-efficients and can be solved to obtain a relation between parameters.

3. Method of solution The boundary value problem is solved using the Galerkin method which is superior than other methods of its kind (Finlayson, 1972). This method expands W; H and U in terms of trial functions with unknown coefficients. The coefficients are then found by imposing the condition that the trial function based weighted residual is zero in the domain of interest. In

Table 1 Comparison of present results with Postelnicu (2008).

c

H

Present study

Postelnicu (2008)

ac

Rc

ac (1 term)

Rc (1 term)

ac (10 term)

Rc (10 term)

1 105

3.160 4.770

1852.243 7.22  106

3.171 4.765

1885.93 7.31  106

3.173 4.704

1840.523 7.034  106

1

1 103 105

3.161 3.124 3.116

1847.985 3491.830 3524.489

3.167 3.130 3.120

1881.67 3556.54 3590.99

3.168 3.133 3.120

1836.336 3470.010 3504.071

10

1 105

3.136 3.108

1822.346 1938.422

3.143 3.11982

1855.92 1975.24

3.144 3.120

1811.060 1927.428

1 105

3.37 10.00

48.664 12940.71

3.377 10.271

49.5312 13091.9

3.291 10.032

45.259 12481.318

1

1 103 105

3.36 3.31 3.30

48.565 91.949 92.863

3.370 3.316 3.299

49.4323 93.618 94.5782

3.285 3.230 3.215

45.163 85.441 86.291

10

1 105

3.32 3.28

47.954 51.082

3.335 3.29891

48.8195 52.0233

3.249 3.214

44.573 47.464

Da = 1 0

Da = 0.001 0

59

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64 Table 2 Viscosity and Viscosity variation parameters values for different liquids. Liquids

Tri chloromethane

Hydrazine

Mercury

Acetone

Methyl propionate

0.988 (248) 0.706 (273)

0.876 (298) 0.628 (323)

1.526 (298) 1.402 (323)

0.54 (248) 0.395 (273)

0.581 (273) 0.431 (298)

0.537 (298) 0.427 (323)

0.48 (348) 0.384 (373)

1.312 (348) 1.245 (373)

0.306 (298) 0.247 (323)

0.333 (323) 0.266 (348)

2.2677 1.4475

1.9334 0.9915

1.2440 0.5148

2.2162 1.2016

1.9919 0.9904

l in mPa s (T in K)

(see Haynes and Lide (2010))

M1 M2

choosing the trial functions one should ensure that they satisfy the boundary conditions and form a complete set. The number of trial functions has then to be fixed such that it does not affect the results significantly. We have made one such analysis in a similar problem based on the LTNE model (see Saravanan (2009)). In order to avoid elaborate numerical computations we shall find an approximate solution using a one-term Galerkin method. Accordingly we multiply Eqs. (17)–(19) by WðzÞ; HðzÞ and UðzÞ respectively and integrate them across the layer. We then replace WðzÞ by A1 W 1 ðzÞ; HðzÞ by B1 H1 ðzÞ and UðzÞ by C 1 U1 ðzÞ, where W 1 ðzÞ; H1 ðzÞ and U1 ðzÞ are trial functions which will be fixed later. This procedure leads to

h D E

A1 Da D2 W 1 ; D2 W 1 þ 2a2 hDW 1 ; DW 1 i þ a4 hW 1 ; W 1 i  ½M 1 T fb þ M 2 T 2fb  1½hDW 1 ; DW 1 i þ a2 hW 1 ; W 1 i   þðM 1 þ 2M 2 T fb Þ½Q f z þ Q f =2  1 hDW 1 ; W 1 i  B1 Ra2 hH1 ; W 1 i ¼ 0

ð20Þ

     B1 ðr þ a2 þ HÞhH1 ; H1 i þ hDH1 ; DH1 i þ A1 Q f z þ Q f =2  1 hW 1 ; H1 i  HC 1 hU1 ; H1 i ¼ 0

ð21Þ

  C 1 ðar þ a2 þ cHÞhU1 ; U1 i þ hDU1 ; DU1 i  cHB1 hH1 ; U1 i ¼ 0

ð22Þ

where hf ðzÞ; gðzÞi ¼

R1 0

f ðzÞgðzÞdz. It can be shown that the above system of equations yields a nontrivial solution only when

D1 ðG1  I1 J 1 Þ R¼    2 a J 1 Q f z þ Q f =2  1 hW 1 ; H1 i2

ð23Þ

Now one can identify two different cases at the marginal state. If r ¼ 0 then there is an exchange of stabilities and hence the stationary mode prevails at the onset of instability. Hence Eq. (23) becomes

106

105

104 2

10 104

1

Rc

Da = 5 Da = 10- 4 M 2 = − 0.2

103

γ = 0.1 Qf = 10

104 102

102 H=1 101

100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

M1 Fig. 1. Rc against M1 for different H.

0.7

0.8

0.9

1.0

60

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

M 2 = - 0.4, - 0.2, 0

104

Da = 5 Da = 10- 4 H=1 γ = 0.1 Qf = 10

103

Rc 102

- 0.4 - 0.2 M2 = 0

1

10

100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M1 Fig. 2. Rc against M 1 for different M 2 and small H.

RS ¼

D1 ðG1  E1 F 1 Þ    a2 F 1 Q f z þ Q f =2  1 hW 1 ; H1 i2

ð24Þ

On the other hand if r ¼ ix, where x – 0 is real then the oscillatory mode determines the onset condition. In this case one can arrive at, from Eq. (23),

h i D1 F 1 G1  E1 F 21  E1 a2 x2 hU1 ; U1 i2 i RO ¼ h   a2 F 22 þ a2 x2 hU1 ; U1 i2 Q f z þ Q f =2  1 hW 1 ; H1 i2

ð25Þ

and

x2 ¼

h i  F 21 hH1 ; H1 i þ G1 ahU1 ; U1 i

ð26Þ

a2 hH1 ; H1 ihU1 ; U1 i2

D E h i where D1 ¼ DaD11  D12 þ D13 , D11 ¼ D2 W 1 ; D2 W 1 þ 2a2 hDW 1 ; DW 1 i þ a4 hW 1 ; W 1 i, D12 ¼ M 1 T fb þ M 2 T 2fb  1      hDW 1 ; DW 1 i þ a2 hW 1 ; W 1 i D13 ¼ ðM 1 þ 2M 2 T fb Þ Q f z þ Q f =2  1 hDW 1 ; W 1 i; E1 ¼ ða2 þ HÞhH1 ; H1 i þ hDH1 ; DH1 i; F 1 ¼ 105

γ = 0.01, 0.1, 1

104

Da = 5 Da = 10- 4 H=1 M 2 = − 0.2

103

Rc

Qf = 10 102

γ = 0.01, 0.1, 1

101

100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M1 Fig. 3. Rc against M1 for different c and small H.

0.8

0.9

1.0

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

61

107 106

0.01

105

Rc

0.1 1

104 103

0.01 0.1

102 101

γ=1

H = 104 M 2 = − 0.2

Da = 5 Da = 10- 4

Qf = 10 100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M1 Fig. 4. Rc against M 1 for different c and large H.

ða2 þ cHÞhU1 ; U1 i þ hDU1 ; DU1 i; G1 ¼ cH2 hH1 ; U1 i2 , I1 ¼ ðr þ a2 þ HÞhH1 ; H1 i þ hDH1 ; DH1 i, J 1 ¼ ðar þ a2 þ cHÞhU1 ; U1 iþ hDU1 ; DU1 i.We notice from Eq.(26) that x2 is always negative for a > 0 and c > 0. Hence r cannot be complex and it follows that the transition from stability to instability must occur via a stationary mode convection. The critical Rayleigh number Rc representing the onset of convection is then obtained by minimizing RS with respect to a. As we have proposed to follow a one-term expansion the trial functions we select need to be efficient enough. Here we make the choices W 1 ¼ z2 ð1  zÞ2 and H1 ¼ U1 ¼ zð1  zÞð1 þ z  z2 Þ. Gershuni and Zhukhovitskii (1976) first used these trial functions in the Rayleigh - Benard problem and found that they are good approximations to determine the convection threshold. We have also successfully used these functions in our recent work (see Saravanan and Sivakumar (2010)) concerning a similar problem in porous media. The results found using the above procedure for M1 ¼ M 2 ¼ Q f ¼ 0 and different values of the control parameters were compared with the existing results of Postelnicu (2008) who considered rigid boundaries (see Table 1). It can be seen that the value of Rc of the present study lies between those of 1 term and 10 term Galerkin method using a different set of trial functions. Hence we hope that these trial functions give better accurate results. Moreover in order to have an estimate of the accuracy of the one term Galerkin method, the result for the limiting clear fluid case was

8

7

Da=10-4, γ =0.1, M2=0, Qf=10, H=1 2 Da=10-4, γ =0.1, M2= -0.2, Qf=10, H=10 Da=10-4, γ =0.1, M2= -0.2, Qf=10, H=1 Da=10-4, γ =0.1, M 2= -0.2, Q f=0, H=1 Da=5, γ =0.1, M 2=-0.2, Q f =10, H=1

6

ac 5

4

3

2 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M1 Fig. 5. ac against M 1 for different parameters.

0.8

0.9

1.0

62

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

800

M 1 = 0.9 M 2 = −0.2

700

Da = 10- 2 600

10- 2

10- 3

Qf = 10 Qf = 0

Rc

500 10- 1

400 300 200 1

γ = 10

100 0

0

1

2

3

4

5

6

log10H Fig. 6. Rc against log 10 H for different c.

compared with the well known exact value Rc ¼ 1707:76 of Chandrasekhar (1961). It was found that the deviation of the present result was well below 2.5% and hence we can believe the results within an admissible error. 4. Results and discussion We consider the onset of natural convection in a fluid saturated porous medium with temperature dependent viscosity and internally distributed heat sources. The medium is bounded by rigid boundaries that are physically more sensible. Brinkman’s equation is used. We shall discuss the results for both Darcian and non-Darcian regimes using the LTNE model. The effect of M 1 for different values of H is displayed in Fig. 1. It decreases Rc and this effect is clearly seen when Da is small. Thus M1 enhances the onset of instability and its effect is significant in the Darcian regime. An increase in M 1 results in a decrease in viscosity which in turn contributes to the destabilizing effect causing convection. In fact Rc decreases against M1 monotonically and becomes zero at some finite value of M 1 . This shows that the heat generation alone can induce the instability for sufficiently large values of M 1 . At this stage one should note that in a similar study carried out by Shivakumara, Mamatha, and Ravisha (2010) it was found that the critical Rayleigh number increases against M 1 . This entirely opposite trend arises due to the fact that their Rayleigh number was based on the hot wall temperature in contrast to R of this study 50 10 1

40

γ Rc /(γ +1)

10- 1 30

10- 2

10 1

M 1 = 0.9

10- 1

20

γ = 10- 2

Qf = 10

10

0

M 2 = −0.2 Da = 10- 2 Qf = 0

0

1

2

3

4

log10H Fig. 7. cRc /(1 + c) against log 10 H for different c.

5

6

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

63

6.0 M 1 = 0.9 M 2 = −0.2

5.5

Qf = 10 Qf = 0

5.0

ac

10- 3

Da = 10-2

10- 2

4.5

10- 1

4.0 1 3.5

3.0

γ = 10 0

1

2

3

4

log10H Fig. 8. ac against log 10 H for different c.

which is based on the cold wall temperature. This issue has already been addressed and discussed in detail by Nield (1996). We also notice from Fig. 1 that an increase in H homogenizes the temperature field across the constituent phases of the porous medium which would it turn postpones the possible instability. We fitted the data available for viscosity of different liquids (Haynes & Lide, 2010) using a second degree polynomial specified by Eq. (1) with T u = 100 K and T l = 400 K and calculated the possible values for the viscosity variation parameters (see Table 2). Based on this we assumed M 2 to be negative in this study. Fig. 2 displays a plot of Rc against M 1 for different values of M2 when H is small. It is seen that the effect of M 2 is destabilizing and is clearly observed only when Da is small. Moreover its effect is significant for larger M 1 and hence one should take care of this when the viscosity dependence on temperature is steep. Similar effects of M 2 was observed for large H. Figs. 3 and 4 show the effect of c for small and large H respectively. It is to be noticed that small value of H corresponds to a situation in which the two phases remain almost without any interaction and hence only the fluid phase takes part in convection. On the other extreme, for large values of H both the phases interact well and behave as a single phase. We note that an increase in c may be due to an increase in either the porosity or the relative fluid–solid conductivity ratio. When H is small the heat exchange between the two phases is also less and hence c leaves no effect. However when H is large the effect of c is naturally seen and is found to advance the onset of instability. It is clear from Figs. 1–4 that Rc decreases drastically as Da ! 0. This is due to a reduction in the viscous effects near the solid boundaries which in turn causes the fluid to move more easily near the boundaries. Fig. 5 displays the effect of M 1 on ac for all physical parameters. It is observed that ac remains constant either when the porous medium is away from the Darcy regime or when the fluid is not significantly heat generating. On the other hand it increases against M 1 and varies sharply whenever Rc approaches zero indicating the emergence of tall cells at the onset of convection.

and ac for different values of c and Q f when Figs. 6–8 illustrate the effects of H on Rc ; Rc 1þc c

M1 ¼ 0:9; M 2 ¼ 0:2; Da ¼ 0:01. One should note that Rc 1þc c is the critical Rayleigh number based on the mean properties of the porous medium which is used when using the LTE assumption. We observe that the onset of convection is advanced

when Q f takes higher values as it leads to higher buoyancy. It is obvious that for a fixed value of c both Rc and Rc 1þc c in

crease against H when it takes intermediate values. We also observed that Rc 1þc c approaches the known LTE limit corresponding to a clear fluid when Da ! 0 and H ! 1. Both the critical Rayleigh numbers are independent of c in the small and large H limits. The corresponding ac remains unaffected in these limits. However it attains its maximum value for intermediate values of H. The appearance of tall cells at the onset for intermediate values of H is a typical feature of this type of problem. 5. Conclusion Convective instability in a horizontal fluid saturated porous medium heated from below is examined when the solid and fluid phases are in LTNE state. The fluid saturating the porous medium is assumed to have temperature sensitive viscosity and the fluid saturated porous medium is having uniformly distributed internal heat sources. It is found that the viscosity

64

S. Saravanan, V.P.M. Senthil Nayaki / International Journal of Engineering Science 74 (2014) 55–64

variation parameters and fluid heat generation parameter produce destabilizing effect on the motionless state. The effect of nonlinear viscosity variation parameter becomes significant only when either the porous medium is of Darcy type or l-T dependence is strong enough. The LTE results are recovered in the large H limit. Acknowledgements The authors thank UGC, India for its support through DRS SAP in Fluid Dynamics. One of the authors (V.P.M.S) would like to acknowledge UGC, India for its financial support through Junior Research Fellowship of BSR RFSMS programme. References Bagai, S. (2004). Effect of variable viscosity on free convection over a non-isothermal axisymmetric body in a porous medium with internal heat generation. Acta Mechanica, 169, 187–194. Banu, N., & Rees, D. A. S. (2002). The onset of Darcy–Benard convection using a thermal nonequilibrium model. International Journal of Heat and Mass Transfer, 45, 2221–2228. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Amen House, London: Oxford University Press. Combarnous, M. (1972). Description du transfert de chaleur par convection naturelle dans une couche poreuse horizontale a laide dun coefficient de transfert solidefluide. Comptes Rendus Mathématique. Académie des Sciences. Paris II, Serie A, 275, 1375–1378. Finlayson, B. A. (1972). Method of weighted residuals and variational principles. New York: Academic Press. Gershuni, G. Z., & Zhukhovitskii, E. M. (1976). Convective stability of incompressible fluids. Jerusalem: Keter press. Haynes, W. M., & Lide, D. R. (2010). Handbook of chemistry and physics (91st ed.). CRC Press, Taylor & Francis Group, pp. 6.229–6.233. Hooman, K., & Gurgenci, H. (2008a). Effects of temperature-dependent viscosity on Benard convection in a porous medium using a non-Darcy model. International Journal of Heat and Mass Transfer, 51, 1139–1149. Kassoy, D. R., & Zebib, A. (1975). Variable viscosity effects on the onset of convection in porous media. Physics of Fluids, 18, 1649–1651. Levenspiel, O. (1999). Chemical reaction engineering. New York: John Wiley. Nield, D. A. (1996). The effect of temperature dependent viscosity on the onset of convection in a saturated porous medium. Jounral of Heat Transfer, 118, 803–805. Nield, D. A., & Bejan, A. (1999). Convection in porous media. New York: Springer. Nield, D. A., & Kuznetsov, A. V. (2010). The effect of local thermal nonequilibrium on the onset of convection in a nanofluid. Journal of Heat Transfer, 132, 052405-1–052405-7. Nield, D. A., & Kuznetsov, A. V. (2010). Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transport in Porous Media, 81, 409–422. Nouri-Borujerdi, A., Noghrehabadi, A. R., & Rees, D. A. S. (2007). Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model. Transport in Porous Media, 69, 343–357. Patil, P. R., & Vaidyanathan, G. (1982). Effect of variable viscosity on thermohaline convection in a porous medium. Journal of Hydrology, 57, 147–161. Postelnicu, A. (2008). The onset of Darcy–Brinkman convection using a thermal non-equilibrium model. Part II. International Journal of Thermal Sciences, 47, 1587–1594. Postelnicu, A., Grosan, T., & Pop, I. (2001). The effect of variable viscosity on forced convection flow past a horizontal flat plate in a porous medium with internal heat generation. Mechanics Research Communications, 28(3), 331–337. Rajagopal, K. R. (2007). On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Mathematical Models & Methods in Applied Sciences, 17, 215–252. Rajagopal, K. R., Saccomandi, G., & Vergori, L. (2010). Stability analysis of Rayleigh–Bénard convection in a porous medium. Zeitschrift fur Angewandte Mathematik und Physik, 62, 149–160. Rees, D. A. S., & Pop, I. (2005). Local thermal nonequilibrium in porous medium convection. In D. B. Ingham & I. Pop (Eds.). Transport phenomena in porous media (Vol. III). New York: Elsevier. Richardson, L., & Straughan, L. (1993). convection with temperature dependent viscosity in a porous medium: Nonlinear stability and Brinkman effect. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 4(3), 223–230. Rogers, F. T., & Morrison, H. L. (1950). Convection currents in porous media. III. Extended theory of the critical gradient. Journal of Applied Physics, 21, 1177–1180. Rudraiah, N., Veerappa, B., & Balachandra Rao, S. (1982). Convection in a fluid saturated porous layer with nonuniform temperature gradient. International Journal of Heat and Mass Transfer, 25, 1147–1156. Saravanan, S. (2009). Thermal non-equilibrium porous convection with heat generation and density maximum. Transport in Porous Media, 76, 35–43. Saravanan, S., & Jegajothi, R. (2010). Stationary fingering instability in a non-equilibrium porous medium with coupled molecular diffusion. Transport in Porous Media, 84, 755–771. Saravanan, S., & Kandaswamy, P. (2004). Hydromagnetic stability of convective flow of variable viscosity fluids generated by internal heat sources. Zeitschrift fur Angewandte Mathematik und Physik, 55, 451–467. Saravanan, S., & Sivakumar, T. (2010). Onset of filtration convection in a vibrating medium: The Brinkman model. Physics of Fluids, 22, 034104. Shivakumara, I. S., Mamatha, A. L., & Ravisha, M. (2010). Effects of variable viscosity and density maximum on the onset of Darcy-Brinkman convection using a thermal nonequilibrium model. Journal of Porous Media, 13(7), 613–622. Shivakumara, I. S., Mamatha, A. L., & Ravisha, M. (2010). Boundary and thermal non-equilibrium effects on the onset of Darcy–Brinkman convection in a porous medium. Journal of Engineering Mathematics, 67, 317–328. Sunil Sharma, P., & Mahajan, A. (2011). Onset of Darcy–Brinkman ferroconvection in a rotating porous layer using a thermal non-equilibrium model: A nonlinear stability analysis. Transport in Porous Media, 88, 421–439. Vanishree, R. K., & Siddheshwar, P. G. (2010). Effect of rotation on thermal convection in an anisotropic porous medium with temperature-dependent viscosity. Transport in Porous Media, 81, 73–87. Yoon, D. Y., Kim, D. S., & Choi, C. K. (1998). Convective instability in packed beds with internal heat sources and throughflow. Korean Journal of Chemical Engineering, 15(3), 341–344.