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

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International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com

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 couple stress liquid. Shankar et al. [9] analyzed the combined effect of couple stresses and a uniform horizontal AC electric field on the stability of buoyancy-driven parallel shear flow of a vertical dielectric fluid. Many authors have studied the effect of periodic modulation on the onset of convection on a heated fluid layer. The gravitational modulation, which can be realized by vertically 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 generation 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 modulation 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. Since f varies sinusoidally with time, the only 1 ∂ 2 4 6 time independent term in the right-hand side of (33) is X2 = ∇ −∇ + C ∇ ; 2 Pr ∂t a R1β1 T0. Hence R1 = 0 and it follows that all the odd coefﬁcients R3, R5 and so on in (26) are zero. Expand- ∂ X = Ri 1 + 2M ing the right-hand side of (33) using Fourier series and by 3 ∂t inverting the operator L term by term we obtain T1 as ∞ ∂ ∂ e−iωt − 1 + 2M +∇2 . T = a2R β YRe sin (nπz) , (34) ∂t ∂t 1 0 1 L (ω, n) n=1 1 where SOLUTION L (ω, n) = −a2R Yβ − k4η γ − k4η Ri 1 0 n n n n n n ω2k2 2MRiω2k2 We assume the solution of (24) in the form − n + n Pr Pr = + 2 (T , R) (T0, R0) ε(T1, R1) k γ (26) + iω 2a2MRY + n n +ε2(T , R ) +··· 0 Pr 2 2 k2Ri where T0 and R0 are the eigenfunction and eigenvalue + n − k4η + 2MRik4η (35) Pr n n n n respectively for the unmodulated system and Ti and Ri , for i ≥ 1, are the correction to T0 and R0 in the presence 2 = 2 2 + 2 = + 2 of modulation. kn n π a ; ηn 1 Ckn; = + 2 =− 2 + 2 Substituting (26) in (24) and equating the corresponding βn 1 Mkn; γn kn 2Mω . terms, we obtain the following system of equations: (29) for T2 becomes LT = 0, (27) 2 0 LT2 = a R0(βn − i 2Mω) fT1 (36) +a2R Yβ T , =− + ∇2 2 1 0 LT1 (R1 R0f )X1 1 T0, (28) We use (36) to determine R2, the ﬁrst non-zero correction =− + ∇2 to R0. LT2 (R1 R0f )X1 1 T1 (29) 2 2 − + ∇2 a R0 Y (R2 R1f )X1 1 T0, R =− 2 4 ∞ ∗ (37) where [N(ω, n) + N (ω, n)] = ∇2 + |L (ω, n)|2 L R0 X1 1 X2 X3. (30) n=1 1 The marginal stability solution for (27) is where ∗ denotes a complex conjugate and = ∗ − T0 = sin (πz) exp [i(lx + my)] (31) N(ω, n) L1(ω, n)(βn i 2Mω) where l and m are the wave numbers in the x and y direc- 2 + 2 = 2 tions, respectively, such that l m a . Substituting RESULTS AND DISCUSSION (31) in (27), we obtain k6 η k4 η Ri R = 1 1 − 1 1 , (32) This paper presents an analytical study of the effects of 0 2 2 a Yβ1 a Yβ1 gravity modulation and internal heat generation on the

2691 International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 5 (2018) pp. 2688–2693 © Research India Publications, http://www.ripublication.com

FIGURE 2. Variation of R2c with ω for various values of FIGURE 3. Variation of R2c with ω for various values of Cattaneo number couple stress parameter onset of convection in a couple stress fluid layer employing Maxwell-Cattaneo law. The analysis presented is based on the assumptions that the amplitude of the gravity modulation is very small compared to the mean gravity and that the convective currents are weak so that nonlinear effects may be neglected. The violation of these assumptions would alter the results significantly when the modulating frequency, ω, is low. This is because the perturbation method imposes the condition that the amplitude of εT1 should not exceed that of T which in turn gives the condition ω>ε. 0 FIGURE 4. Variation of R2c with ω for various values of Thus the validity of the results depends on the value of the Prandtl number modulating frequency, ω. When ω 1, the period of modulation is large and it affects the entire volume of the fluid. For large frequencies the effect of modulation disappears the values of the other governing parameters. The effect of because the buoyancy force takes a mean value leading to modulation vanishes for large frequencies. the equilibrium state of the unmodulated case. Thus the The effect of couple stress parameter on the onset of effect of modulation is significant only for the small and convection is shown in Figure 3. It can be seen that the moderate values of ω. ([14]) effect of increasing the couple stress parameter is to make the system more stable. C is indicative of the concentra- Figures 2, 3, 4 and 5 show the variation of R2c with respect to ω for various governing parameters. We see from tion of the suspended particles and the effect of suspended particles is to increase the viscosity of the fluid and thereby these figures that R2c is positive over the entire range of values of ω indicating that the effects of gravity modula- having an stabilizing influence on the system. tion and internal heat generation is to stabilize the system Figure 4 depicts the variation of R2c with ω for different i.e., convection occurs at a later point compared to the values of Prandtl number, Pr. We find that the magnitude unmodulated system. of the correction Rayleigh number decreases with increase in Prandtl number. This indicates that Prreduces the stabi- Figure 2 shows the variation of R2c with the modulating frequency ω for different values of Cattaneo number, M. lizing effect. This effect is more pronounced for moderate Cattaneo number has stabilizing effects for small values frequencies. of the frequency. Cattaneo number is proportional to the Figure 5 shows the variation of R2c with ω for different relaxation time and the speed of the temperature propaga- values of internal Rayleigh number, Ri. We observe that tion will decrease for increasing relaxation time and hence R2c decreases with increase in Ri; thus destbilizing the delay the onset of convection. However this effect reverses system. Increase in Ri amounts to increase in energy supply and becomes destabilizing for moderate values because of to the system. This results in deviations from the basic state the decrease in the size of the convection cells. The value temperature distribution and thus in the enhancement of the of ω at which stabilizing effect reverses is dependent on thermal disturbances.

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[2] Bissell, J. J., 2016, “Thermal convection in a magnetized conducting fluid with the Cattaneo–Christov heat-flow model,” Proc. R. Soc. A., 472(2195), pp. 20160649. [3] Stranges, D. E., Khayat, R. E., and Debruyn, J., 2016, “Finite thermal convection of non-Fourier fluids,” Int J Therm Sci., 104, pp. 437–447. [4] Pranesh, S., and Kiran, R. V., 2015, 2015, “Rayleigh- Bénard Chandrasekhar convection in an electrically conducting fluid using Maxwell-Cattaneo law with temperature modulation of the boundaries,” Int J Eng Research & Tech, 4(10), pp. 174–184. [5] Shivakumara, I. S., Ravisha, M., Ng CO, and Varun, V. FIGURE 5. Variation of R with ω for various values of 2c L., 2015, “A thermal non-equilibrium model with Cat- internal Rayleigh number Ri taneo effect for convection in a Brinkman porous layer,” Int J Non Linear Mech, 71, pp. 39–47. CONCLUSION [6] Stokes, V. K., 1966, “Couple stresses in fluids,” Phys Fluids, 9(9), pp. 1709–1715. [7] Pranesh, S., and Sangeetha George, 2014, “Effect of The effect of gravity modulation and internal heat genera- imposed time periodic boundary temperature on the tion on the onset of convection in a couple stress fluid with onset of Rayleigh-Bènard convection in a dielectric cou- Maxwell-Cattaneo law is studied using a linear stability ple stress fluid,” Int. J. Appl. Math. Comput., 5(4), pp. analysis and the following conclusions are drawn: 1–13. [8] Bhadauria, B. S., Siddheshwar, P. G., Singh, A. K., and • Cattaneo number is stabilizing for small frequencies Vinod K. Gupta, 2016, “A local nonlinear stability anal- and destabilizing for moderate frequencies. ysis of modulated double diffusive stationary convection • Couple stress parameter stabilizes the system. in a couple stress liquid,” J. Appl. Fluid Mech., 9(3), pp. • Prandtl number enhances the destabilizing effect of 1255–1264. modulation. [9] Shankar, B. M., Jai Kumar, and Shivakumara, I. S., 2016, • Internal Rayleigh number destabilizes the system. “Stability of natural convection in a vertical dielectric couple stress fluid layer in the presence of a horizontal Low frequency gravity modulation and internal heat gener- ac electric field,”Appl Math Modelling, 40(9), pp. 5462– ation have a significant influence on stability of the system. 5481. [10] Bhadauria, B. S., and Palle Kiran, 2015, “Chaotic and The study is significant in areas of low temperature fluids, oscillatory magneto-convection in a binary viscoelastic granular flows and liquid metals. The problem gives insight fluid under G-jitter Int J Heat Mass Transfer,” 84, pp. into external means of controlling the onset convection. 610–624. [11] Sameena Tarannum,and Pranesh, S., 2016, “Effect of gravity modulation on the onset of Rayleigh-Bènard Convection in a weak electrically conducting couple ACKNOWLEDGEMENT stress fluid with saturated porous layer,” Int J Eng Res, 5, pp. 914–928. The authors sincerely thank the management of CHRIST [12] VasudhaYekasi, Pranesh, S., and Shahnaz Bathul, 2016, “Effects of gravity modulation and internal heat gen- (Deemed to be University) for their incessant support eration on the onset of Rayleigh-Bènard convection and Dr. S Pranesh for his encouragement and insightful in a micropolar fluid,” J Advances Math, 12(6), pp. comments. 6270–6285. [13] Siddheshwar, P. G., and Pranesh, S., 2004, “An Ana- lytical Study of Linear and Non-linear Convection in Boussinesq–Stokes Suspensions,” Int J Non-Linear REFERENCES Mech, 39(1), pp. 165–172. [14] Malashetty, M. S., and Swamy, M. S., 2011, “Effect of [1] Bissell, J. J., 2015, “On oscillatory convection with gravity modulation on the onset of thermal convection the Cattaneo-Christov hyperbolic heat-flow model,” in rotating fluid and porous layer,” Phys. Fluids, 23, pp. Proc. R. Soc. A Math. Phys. Eng. Sci., 471(2175), 064108. pp. 20140845.

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