A New Algorithm Based SLM to Find Critical Eigenvalue in the Rayleigh-Bénard-Brinkman Convection Problem with Boundary Conditions

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1 Advances in Computational Sciences and Technology ISSN Volume 10, Number 6 (017) pp Research India Publications A New Algorithm Based SLM to Find Critical Eigenvalue in the Rayleigh-Bénard-Brinkman Convection Problem with Boundary Conditions Dr. Ashoka S. B. MCA., Ph.D. Department o Computer Science, MSCWB, Bangalore , India. Abstract An eigen boundary value problem is solved or the critical eigenvalue in the case o a Rayleigh-Bénard-Brinkman convection problem bounded by permeable walls with Robin boundary condition on temperature and general boundary condition on velocity. Unknown initial values and guess eigenvalue required to initiate the successive linearization method are obtained using the single-term Rayleigh-Ritz method. The local thermal non-equilibrium between the solid and luid phases in the Newtonian luid-saturated porous medium results in coupled partial dierential equations. It is shown that ive steps o the method results in a convergent solution with an accuracy o 6 decimal digits. Keywords: porous medium, local thermal non-equilibrium, convection, linear stability, successive linearization method 1. INTRODUCTION Convection in porous medium usually known as the Horton-Rogers-Lapwood problem has been the ocus o interest o many researchers over the past several decades. The study initiated by Horton and Rogers [1] and Lapwood [] is a porous medium version o Rayleigh-Bénard problem in a clear luid layer and has wide variety o practical applications. These include oil reservoir modeling, geothermal engineering, building thermal insulation, grain storage and nuclear waste repository

2 1558 Dr. Ashoka S. B. and so on. A comprehensive study on the convection in porous media is well documented in the books Ingham and Pop [3], Nield and Bejan [4] and Vaai [5]. Most o the studies in literature concerning Horton-Rogers-Lapwood problem assume that there is no temperature dierence between the luid and solid phases, at any location o the porous medium system. This is called as local thermal equilibrium (LTE) model. In many practical situations LTE model ails to comprehend the heat transport in porous medium. Hence it is important to assume that the solid and luid phases are not in local thermal equilibrium by considering two-ield model o the energy equation and this model is popularly non as the local thermal non-equilibrium (LTNE) model in the literature. Heat pipes, cooling o computer chips, ood processing, microwave heating are ew applications where LTNE eects are predominant. Vowing to these applications there have been number o works in literature concerning the LTNE model. Banu and Rees [6] were the irst amongst others to study the onset o Darcy-Bénard convection using a LTNE model and obtained the critical Rayleigh number as a unction o local thermal non-equilibrium parameters. Postelnicu and Rees [7] extended the work o Banu and Rees [6] to Darcy-Brinkman-Bénard convection by considering a orm-drag term, using a thermal non-equilibrium with stress-ree boundaries. It was shown that the inclusion o quadratic drag term in the momentum equation has no eect on stability criteria since the basic state being motionless. Further, they recovered LTE results by perorming asymptotic analysis or large inter-phase heat transer coeicient H. Malashetty, et. al. [8] studied the onset o Lapwood-Brinkman convection using a LTNE model and reiterated the results obtained by Postelnicu and Rees [7]. Nouri-Borujerdi et. al. [9] studied the eect local thermal non-equilibrium on conduction in a luid saturated porous channel with a uniorm heat source and obtained conditions under which the LTNE can be assumed. Postelnicu [10] extended the earlier work by Postelnicu and Rees [7] to rigid boundaries and gave a quantitative comparison between the results o stress-ree and rigid boundaries. Above-cited works concern Newtonian liquid as a working medium. In recent past non-newtonian luids are being used in place o their Newtonian counterparts as most o the industrially important luids exhibit non- Newtonian characteristics. With this view point researchers have extended the use o LTNE eects in non-newtonian luid saturated porous medium. For example, Shivakumara et. al. [11] studied the onset o convection in a viscoelastic luidsaturated sparsely packed porous layer using a thermal non-equilibrium model. They showed that viscoelastic eects enhance stabilizing eect o the interphase heat transer coeicient. Recently, Malashetty et. al. [1] studied the eect o thermal nonequilibrium on the onset o convection in a couple stress luid saturated porous layer and showed that interphase heat transer coeicient increases the critical Rayleigh number or the onset o convection with the increasing values o couple stress parameter. In literature there are works addressing the LTNE on the double diusive convection in a luid saturated porous layer. Double diusive convection in a porous

3 A New Algorithm Based SLM to Find Critical Eigenvalue layer using a thermal non-equilibrium model was studied by Malashetty et. al. [13] and they showed that interphase heat transer coeicient has a signiicant say on the stationary, oscillatory and inite amplitude double diusive convection in a luid saturated porous medium. Malashetty and Heera [14] and Malashetty et. al. [15] perormed linear and non-linear stability analyses o the double diusive convection in a rotating porous layer (tightly/sparsely packed) using a thermal non-equilibrium model. In these studies it was shown that the rotation increases the stabilizing eect o interphase heat transer coeicient in all the three modes o double diusive convection, namely, stationary, oscillatory and inite amplitude. Works o Malashetty et. al. [16] and Shivakumara et. al. [17] takes into account o thermal non-equilibrium eects on convective instability in an anisotropic porous layer. In all the studies mentioned so ar concerning LTNE either stress-ree or rigid boundaries with isothermal conditions are made use. But, the interace conditions (slip conditions) and the thermal boundary conditions o the third type are the most general and practical boundary conditions. To the best o authors knowledge there are no works concerning Rayleigh-Bénard-Brinkman convection with LTNE and two boundary condition. Hence, we propose to investigate the same in the present paper.. Mathematical Formulation We consider a layer o luid saturated porous medium o depth d which is heated rom below and cooled rom above, as depicted in Figure 1. The upper surace is held at a temperature T 0 while the lower one is at T 0 T. It is assumed that both orm-drag and boundary eects are signiicant, that the porous medium is isotropic but that local thermal equilibrium does not apply. Thus the governing equations, i.e., the continuity, momentum and energy equations, subject to the Boussinesq approximation, take the orm q t q 0, (1) 1 q q p q q K e T c c q. T k T hts T, t s 1 c 1 ks Ts hts T s T t 0 g T T k ˆ, (), (3) (4)

4 1560 Dr. Ashoka S. B. Fig. 1. Schematic diagram o the physical coniguration. where, q u, v is the velocity vector, t is time, is the density, is the porosity o the porous, medium, p is the pressure, e is the eective viscosity, K is the permeability o the porous medium, is the coeicient o thermal expansion, g is acceleration due to gravity, c is speciic heat at constant pressure, T is the temperature, k is the thermal conductivity, h is the inter-phase heat transer coeicient. Further, the properties o luid phase are indicated with the subscript and those o solid phase are denoted by the subscript s. The ollowing velocity boundary conditions at the permeable walls are made use (see, Siddheshwar [18]): u 1 v 0, u at y 0, y K u v 0, u at y d. y K (5) Here, 1 and are dimensionless quantities depending on the material parameters which characterizes the structure o the permeable boundary membranes. We deine the ollowing dimensionless variables

5 A New Algorithm Based SLM to Find Critical Eigenvalue c d,,,, d k * * * x y x y t t k,,, c d c d * * * q u v u v q k * e k T T0 Ts T0 p p,,. c K T T (6) Using (6) in equations (1) - (4) and dropping asterisks or simplicity, we obtain Da Pr q 0, (7) q 1 ˆ q q p q q Ra j, (8) t q. H t M t, (9) H. (10) The non-dimensional numbers appearing in equations (7) - (10) are: Da e K d the Darcy number, Pr c k e the Prandtl number, gkdt Ra the Darcy-Rayleigh number, k hd H k the inter-phase heat transer parameter, k (1 ) k s the porosity-scaled conductivity ratio and M ( c) ( c) s k k s the diusivity ratio.

6 156 Dr. Ashoka S. B. The usual Rayleigh number, which is based on the mean properties o the porous medium, is given by 1 * Ra R. We now introduce the steam unction xy, that takes care o continuity equation (7) as ollows u, v. y x (11) We substitute equation (11) into equations (8) - (10) and eliminating pressure rom the momentum equation we get the ollowing equations: Da, 4 Da Pr t x, y Ra, x, H t x, y M t (1), (13) H. (14) The basic state is assumed to be quiescent and the heat transport takes place through conduction alone and hence we have 0, 1 y, 1 y. (15) b b b We now superimpose a small perturbation on the basic state as ollows, 1 y, 1 y. (16)

7 A New Algorithm Based SLM to Find Critical Eigenvalue Substituting equation (16) in equations (1) - (14) and linearizing we obtain Da Pr t t Ra, (17) x Da 4 x H M t, (18) H. (19) It can be easily proved that equations (17) - (19) obey the principal o exchange o stabilities and hence the time derivative terms in those equations can be neglected to get 0 Da Ra x y x y x, (0) x H 0 x y y x H 0, (1). () The boundary conditions or the perturbed variables are assumed in the ollowing orm 0, Ds 0, 0 L BiL y y y y * and BiL 0 at y 0 0, Ds 0, 0 U BiU y y y y * and BiU 0 at y 1, (3). (4) Here, DsL 1d K is the slip Darcy number at the lower surace, DsU d K is the slip Darcy number at the upper surace, Bi, Bi are the Biot numbers o the luid L U

8 1564 Dr. Ashoka S. B. * * phase at lower and upper suraces respectively, Bi, Bi are the Biot numbers o the solid phase at lower and upper suraces respectively. Equations (0) () admit solutions in the orm L U x, y cos kx y x, y sin kx y x, y sin kx y (5) where k is the horizontal wave number. By substituting equation (5) into equations (0) - (4) the ollowing set o equations is obtained: Da D k 1 D k Rak 0, (6) D k k H 0, (7) D k H 0, (8) 0, D DsLD 0, D BiL 0 * and D BiL 0 at y 0, (9) 0, D DsUD 0, D Bi U 0 * and U 0 at 1 D Bi y. (30) Equations (6) - (30) are solved using successive linearization method by treating Ra as an extra variable. The details o the method are discussed in the paper by Narayana et al. [19]. The initial solutions or the successive linearization method were obtained through single-term Galerkin method in the ollowing orm: 4 3 y y a1 y a y a3 y, y y a4y a5, y y a6y a7, (31) where, 1 5DsL 3DsU DsLDsU a1 1 4 Ds Ds Ds Ds L U L U,

9 A New Algorithm Based SLM to Find Critical Eigenvalue DsL 6 DsU a 1 4 Ds Ds Ds Ds L U L U U 6Ds a3 1 4 Ds Ds Ds Ds L U L U,, a a U 4 Bi Bi a, a L 4 5 BiU BiL BiLBiU BiL * * BiL BiU a6 6, a * * * * 7 * BiU BiL BiLBiU BiL,. It should be noted here that DsL, DsU 0represents stress-ree boundaries and DsL, DsU represents rigid boundaries. Similarly, the case BiL, BiU 0 represents adiabatic type temperature boundary condition while BiL, BiU represents isothermal type boundary condition. Other combinations o boundary conditions are obtained or some particular choice o values o these parameters. 3. RESULTS AND DISCUSSION A semi-analytical study is made o the linear stability analysis or the Rayleigh- Bénard-Brinkman convective instability problem that assumes local thermal nonequilibrium between the luid and solid phases and that considers general boundary conditions on velocity and temperatures. The governing equations result in a twopoint eigen boundary value problem with third type boundary conditions. It is the boundary conditions that pose obstacles in obtaining an analytical expression or the eigen value Rac (critical Rayleigh number). The most general ormulation o the problem acilitates the handling o limiting case problems which at the present time are being tackled as -independent problems. The integrated approach to the convective instability problem gives a new direction to the investigation o such problems. The main diiculty in solving such EBVP' s with generalized boundary conditions concerns convergence o the schemes, iterative or otherwise. In this paper we propose the successive linearization method (SLM). The method when used in the original orm does pose problems in the case o general boundary conditions. The usual problems encountered are the ollowing: 1. Sensitivity o the convergence o the solution to the choice o initial unknown guess values and eigen values.. Extreme values o the parameters many a time slow down convergence and warrant use o acceleration procedures.

10 1566 Dr. Ashoka S. B. 3. Greater accuracy i required or the eigen values would mean greater the intensity o the above two problems. 4. Choice o basis unctions also makes contributions to all the three computational problems noted above. In the present problem we have circumvented all the above hurdles by adopting the ollowing scientiic procedure: The elegance in the SLM method can be appreciated only when it is implemented in an automated way in the chosen computer program. Since MATLAB implements algorithms mostly through matrices, the SLM based procedure that leans heavily on matrix based concepts is ideally suited or Chebyshev-based spectral SLM method. SciLab is another such package (open source) that provides natural support or implementation o SLM procedure STRESS-FREE BOUNDARIES RIGID BOUNDARIES Da = 0, H = 10 -, = 1 35 Ra c Bi L, Bi U Fig.. Critical Rayleigh number as a unction o Biot numbers o luid phase. We now move on to discuss some o the important results obtained through the computation. It is observed that the critical eigen value o permeable surace lies in between those o the stress-ree and rigid boundaries, i.e., (Rac)Free (Rac)Permeable (Rac)Rigid. These results are in line with those reported by Siddheshwar [18]. Also, the critical Raleigh number corresponding to the Robin type temperature boundary conditions lies in between those o isothermal and adiabatic type o temperature boundary conditions, i.e., (Rac)Adiabatic (Rac)Robin (Rac)Isothermal.

11 A New Algorithm Based SLM to Find Critical Eigenvalue Figure airms the above statement or both stress-ree and rigid boundaries. As mentioned earlier, the case BiL, BiU 0 represents adiabatic type temperature boundary condition, while BiL, BiU represents isothermal type boundary condition and the intermediate values o BiL, BiU represent Robin type boundary condition. It is evident rom Figure that or intermediate values o BiL, BiU the critical Rayleigh number is sandwiched between those o adiabatic and isothermal cases. Fig.3: Error plots or dierent values o Da, with CF = 0.005, = 1, Re = 10 and initial solutions. Figures 3 show the error plots or various values o Reynolds number or initial velocities we see that the Reynolds number appears in the nonlinear term and hence it would require one or two additional iterations to achieve convergence than it usually require. In the present problem, the non-boundary parameters that aect the critical Rayleigh number are the Darcy number Da, the inter-phase heat transer parameter H and the porosity-scaled conductivity ratio γ. We now discuss eects o these parameters on the onset o convection with upper and lower stress-ree-isothermal (FIFI) boundaries and upper and lower rigid-isothermal (RIRI) boundaries. Table 1 gives the critical Rayleigh number and the critical wave number or dierent values o Da, γ and LogH. It is clear that increasing values o LogH results in increasing the critical Rayleigh number indicating that the inter-phase heat transer parameter has a stabilizing eect on the onset o convection in the case o stress-ree-isothermal (FIFI) and rigid-isothermal (RIRI) boundaries. It is observed that the critical Rayleigh number is a decreasing unction o the parameter γ which mean that

12 1568 Dr. Ashoka S. B. convection sets in much earlier than expected or higher values γ vowing to its destabilizing eect in both FIFI and RIRI cases. Further, the Darcy number also increases the critical Rayleigh number in FIFI and RIRI case. Hence it has a stabilizing eect on the system considered. Results o the other mixed combinations o boundary conditions like rigid-stress-ree, isothermal-adiabatic are obtained as special cases o the general boundary conditions considered in the paper. We omit the discussion o the same or reasons o space. Table 1. Critical Rayleigh number and critical wave number or dierent values o Da, γ and LogH Da γ LogH FIFI RIRI kc Rac kc Rac

13 A New Algorithm Based SLM to Find Critical Eigenvalue CONCLUSIONS In this paper we presented a semi-analytical study o the linear stability analysis or the Rayleigh-Bénard-Brinkman convective instability problem that assumes local thermal non-equilibrium between the luid and solid phases and that considers general boundary conditions on velocity and temperatures. The eigen boundary value problem was solved using the successive linearization method with the help o singleterm Galerkin method or initial solutions. The analysis predicts the critical Rayleigh number corresponding to the third kind temperature boundaries to be between those o adiabatic and isothermal cases. Further, the Darcy number and the inter-phase heat transer parameter have stabilizing eect on the system while the opposite is true in the case o porosity-scaled conductivity ratio. REFERENCES [1] C.W. Horton and F.T. Rogers, Convection currents in a porous medium, Journal o Applied Physics 16, (1945). [] E.R. Lapwood, Convection o a luid in a porous medium, Proceeding o the Cambridge Philosophical Society 44, (1948). [3] D.B. Ingham and I. Pop, Transport phenomena in porous media (Pergamon, Oxord, 1998). [4] D.A. Nield and A. Bejan, Convection in porous media (Springer-Verlag, New York, 1999). [5] K. Vaai, Handbook o porous media (Marcel Dekker, New York, 000). [6] N. Banu and D.A.S. Rees, Onset o Darcy-Bénard convection using a thermal nonequilibrium model, International Journal o Heat and Mass Transer 45, 1-8 (00). [7] A. Postelnicu and D.A.S. Rees, The onset o a Darcy-Brinkman convection using a thermal nonequilibrium model-part I: stress-ree boundaries, International Journal o Energy Research 7, (003). [8] M.S. Malashetty, I.S. Shivakumara and S. Kulkarni, The onset o Lapwood- Brinkman convection using a thermal non-equilibrium model, International Journal o Heat and Mass Transer 48, (005). [9] A. Nouri-Borujerdi, A.R. Noghrehabadi and D.A.S. Rees, The eect o local thermal non-equilibrium on conduction in porous channels with a uniorm heat source, Transport in Porous Media 69, (007). [10] A. Postelnicu, The onset o a Darcy Brinkman convection using a thermal nonequilibrium model. Part II, International Journal o Thermal Sciences 47,

14 1570 Dr. Ashoka S. B (008). [11] M.S. Malashetty, I.S. Shivakumara and S. Kulkarni, The onset o convection in a couple stress luid saturated porous layer using a thermal non-equilibrium model, Physics Letters A 373, (009). [1] I.S. Shivakumara, M.S. Malashetty and K.B. Chavaraddi, Onset o convection in a viscoelastic luid-saturated sparsely packed porous layer using a thermal nonequilibrium model, Candian Journal o Physics 84, (006). [13] M.S. Malashetty, M. Swamy, R. Heera, Double diusive convection in a porous layer using a thermal non-equilibrium model, International Journal o Thermal Sciences 47, (008). [14] M.S. Malashetty and R. Heera, Linear and non-linear double diusive convection in a rotating porous layer using a thermal non-equilibrium model, International Journal o Non-Linear Mechanics 43, (008). [15] M.S. Malashetty, I. Pop and R. Heera, Linear and nonlinear double diusive convection in a rotating sparsely packed porous layer using a thermal nonequilibrium model, Continuum Mechanics and Thermodynics 1, (009). [16] M.S. Malashetty, I.S. Shivakumara and S. Kulkarni, The onset o convection in an anisotropic porous layer using a thermal non-equilibrium model, Transport in Porous Media 60, (005). [17] I.S. Shivakumara, J. Lee, A.L. Mamatha and M. Ravisha, Boundary and thermal non-equilibrium eects on convective instability in an anisotropic porous layer, Journal o Mechanical Science and Technology 5, (011). [18] P.G. Siddheshwar, Convective instability o erromagnetic luids bounded by luid-permeable, magnetic boundaries, Journal o Magnetism and Magnetic Materials 149, (1995). [19] M. Narayana, S.S. Motsa and P. Sibanda, On a piecewise linearization solution o an eigen BVP and chaotic system due to magneto-convection in a two dimensional rectangular box, to be communicated (014).