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Effect of Gravity Modulation and Internal Heat Generation on Rayleigh-Bénard Convection in Couple Stress Fluid with Maxwell-Cattaneo Law

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Effect of Gravity Modulation and Internal Heat Generation on Rayleigh-Bénard Convection in Couple Stress Fluid with Maxwell-Cattaneo Law International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com Effect of Gravity Modulation and Internal Heat Generation on Rayleigh-Bénard Convection in Couple Stress Fluid with Maxwell-Cattaneo Law Maria Thomas Research scholar, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, Karnataka-560029, India. Sangeetha George K. Department of Mathematics, CHRIST (Deemed to be University) Bengaluru, Karnataka-560029, India. Abstract: The linear stability of a horizontal layer of cou- where τ represents the time lag that is required for the heat ple stress fluid heated from below in the presence of time flux to reach a steady state following a perturbation to the periodic body force and internal heat generation is con- temperature gradient. Delayed response implies that the sidered. The classical Fourier heat law is replaced by the system has some thermodynamic inertia and the param- Maxwell-Cattaneo law to assimilate inertial effects. The eter τ is a measure of this inertia. Bissell [1, 2] studied regular perturbation method based on the small amplitude Rayleigh-Bénard convection problem using the hyperbolic of modulation is employed to compute the critical Rayleigh heat-flow model and developed a linear theory for ther- number and the corresponding wave number. It is shown mal instability by oscillatory modes. Stranges et al. [3] that the onset of convection can be advanced or delayed by examined the thermal convection for fluids possessing sig- proper regulation of various governing parameters. nificant thermal relaxation time and found that non-Fourier effects are important when the Cattaneo number is sig- Keywords: Couple Stress Fluid, Maxwell-Cattaneo Law, nificant. Pranesh and Ravi [4] and Shivakumara [5] have Gravity Modulation, Internal Heat Generation. respectively studied temperature modulation in a Newto- nian fluid and fluid-saturated Brinkman porous medium with Cattaneo effect. Understanding non-Newtonian fluid behaviour is very INTRODUCTION important for the better explanation of the behaviour of rheologically complex fluids such as liquid crystals, poly- meric suspensions that have long-chain molecules and Fourier’s law describes the phenomenon of heat conduc- lubrication. Couple stress fluid theory introduced by Stokes tion in most practical engineering applications. However, [6] takes into account the presence of couple stresses, some systems can exhibit time lag in response or wave-like body couples and non-symmetric stress tensor. This model behaviour. Since Fourier’s law fails to account such mech- results in equations that are similar to the Navier Stokes anisms, there were several theoretical developments to equations and thus facilitating a comparison with the results understand the nature of the deviations. Maxwell-Cattaneo for the classical non polar case. Pranesh and Sangeetha [7] law is given by studied the effect of temperature modulation on the onset ∂Q of convection in a dielectric couple stress fluid. Bhadau- Q + τ =−κ∇T , ∂t ria et al. [8] investigated the effect of temperature and International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com gravity modulation on double diffusive convection in a cou- ple stress liquid. Shankar et al. [9] analyzed the combined effect of couple stresses and a uniform horizontal AC elec- tric field on the stability of buoyancy-driven parallel shear flow of a vertical dielectric fluid. Many authors have studied the effect of periodic modu- lation on the onset of convection on a heated fluid layer. The gravitational modulation, which can be realized by verti- cally oscillating horizontal liquid layer, acts on the entire FIGURE 1. Physical configuration of the problem volume of the liquid and has a stabilizing or destabiliz- Under the Boussinesq approximation, the governing ing effect depending on the amplitude and frequency of the equations are given by ([13, 4]) modulation. Natural convection driven by internal heat gen- eration is observed in many physical phenomena in nature ∇.q = 0, (2) such as in the earth’s mantle. Understanding the effects of internal heat generation is also significant for reactor ∂q safety analysis, metal waste, spent nuclear fuel, fire and ρ0 + (q. ∇)q =−∇p + ρg(t) combustion studies and strength of radioactive materials. ∂t (3) + ∇2 − ∇4 Bhadauria and Kiran [10] analyzed double diffusive con- µ q µ q, vection in an electrically conducting viscoelastic fluid layer heated from below and found that modulated gravity field ∂T + (q. ∇)T =−∇.Q + Q (T − T ), (4) can be used either to delay or enhance the heat and mass ∂t 1 0 transfer in the system. Sameena and Pranesh [11] studied the effects of fluctuating gravity in a weak electrically con- ∂Q ducting couple stress fluid with a saturated porous layer. τ +ω × Q =−Q − κ∇T , (5) ∂t 1 Vasudha et al. [12] studied the effect of gravity modula- tion on the onset of convection in a micropolar fluid with = − − internal heat generation using linear stability analysis. ρ ρ0[1 α(T T0)], (6) We study in the present paper the combined effect of where q is the velocity, T is the temperature, p is the pres- gravity modulation and internal heat generation on the sure, ρ is the density, ρ0 is the density at T = T0, Q is the onset of convection in a couple-stress fluid layer. Since heat flux vector, µ is the dynamic viscosity, µ is the couple the amplitude and frequency of the modulation are exter- stress viscosity, τ is the relaxation time, κ is the thermal nally controlled parameters, the onset of convection can conductivity, α is the coefficient of thermal expansion, Q1 be delayed or advanced by the proper tuning of these 1 is the internal heat source and ω = ∇×q. parameters. 1 2 The thermal boundary conditions considered are T = T0 + T at z = 0 MATHEMATICAL FORMULATION (7) and T = T0 at z = d. We consider a layer of couple stress fluid confined between The basic state of the fluid is quiescent and is given by two horizontal wall as shown in Figure 1.A Cartesian coor- q (z) = 0, ρ = ρ (z), p = p (z), dinate system (x, y, z) is chosen such that the origin is at b b b (8) the lower wall and z-axis is vertically upward. The sys- T = Tb(z), Q = Qb(z). tem is under the influence of a periodically varying vertical Substituting (8) in (2)-(6), the pressure pb, heat flux gravity field given by Qb, temperature Tb, and density ρb satisfy the following ˆ g(t) =−g0 (1 + ε cos ωt)k (1) equations: where ε is the amplitude of the modulation, g0 is the mean ∂pb =−ρbg0 (1 + ε cos ωt), (9) gravity and ω is the frequency of the modulation. ∂z 2689 International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com dQb = − We eliminate p from (17) by operating curl twice and Q1(Tb T0), (10) dz eliminate Q between (18) and (19). These equations are then non-dimensionalized using: ∂T =− b Qb κ , (11) ∗ ∗ ∗ x y z q ∂z (x , y , z ) = , , ; q∗ = ; d d d κ d 2 dQb d Tb ∗ t ∗ T ∗ ω =−κ , (12) t = ; T = ; ω = ; (21) dz dz2 d2 T κ 2 κ d = − − ρb ρ0[1 α(Tb T0)]. (13) to obtain, ignoring the asterisks: (12) is solved for Tb(z) subject to the boundary conditions 1 ∂ (∇2W) =∇4W − C∇6W + R (1 + εf ) ∇2T , (7) and we obtain Pr ∂t 1 T √ z (22) T = T + √ sin Ri(1 − ) (14) b 0 d sin ( Ri) √ ⎫ √ ⎪ 2 Ri ∂ 2 ⎪ Q1 d √ cos Ri(1 − z) 1 + 2M − M ∇ W ⎬ where Ri = is the internal Rayleigh number. sin ( Ri) ∂t κ ∂ ∂ ∂ ⎪ Let the basic state be disturbed by an infinitesimal ther- + Ri 1 + 2M − 1 + 2M +∇2 T = 0. ⎭ ∂t ∂t ∂t mal perturbation. We assume a solution for q, T , p, ρ and (23) Q in the form 2 2 2 + = + = + 2 ∂ ∂ 2 2 ∂ q =qb q , ρ ρb ρ , p pb p , where ∇ = + , ∇ =∇ + , (15) 1 2 2 1 2 ∂x ∂y ∂z T = Tb + T , Q = Qb + Q = µ Pr (Prandtl number), where q , ρ , p , T , Q represents the perturbed quantities ρ0 κ which are assumed to be small. Substituting (15) in (2)-(6) ρ αgd3 T R = 0 (Rayleigh number), and using the basic state equations we obtain the linearized µκ equations for the infinitesimal perturbations as: τκ M = (Cattaneo number), 2 d2 ∇ = .q 0, (16) µ C = (Couple stress parameter). µd2 ∂q 2 4 ρ0 =−∇p + µ∇ q − µ ∇ q (22) and (23) are solved for stress-free, isothermal, vanish- ∂t (17) ing couple- stress boundary conditions ([14]) and hence we − + ˆ ρ g0 (1 ε cos (ωt))k, have 2 √ ∂ W ∂T W T Ri √ z W = = T = 0 − √ cos Ri(1 − ) ∂z2 ∂t d sin ( Ri) d (18) at z = 0 and z = 1. =−∇.Q + Q1T , Eliminating W from (22) and (23), we obtain √ ∂ κτ T Ri √ z [R (1 + εf ) X ∇2 + X X ]T = 0 (24) 1 + τ Q =− √ cos Ri(1 − ) 1 1 2 3 ∂t 2 d sin ( Ri) d with the boundary conditions ∂q × −∇W − κ∇T , (19) ∂z ∂2T ∂4T ∂6T T = = = = 0 ∂z2 ∂z4 ∂z6 (25) ρ =−ρ0αT (20) at z = 0, 1. 2690 International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com 2 = 2 + 2 = + 2 = + 2 where where k1 π a , η1 1 Ck1 , β1 (1 Mk1 ). 2 −4 π Then (28) for T1 becomes Y = ; 2 Ri − 4 π 2 LT1 = a (R1 + R0f )Yβ1T0 (33) ∂ The solubility condition requires that the time independent X = Y 1 + 2M − M∇2 ; 1 ∂t part of the right-hand side of (33) should be orthogonal to T0.
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