In this paper, we investigate the combined effect of internal heating and time periodic gravity modulation in a viscoelastic fluid saturated porous medium by reducing the problem into a complex non-autonomous Ginzgburg-Landau equation. Weak nonlinear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number is obtained in terms of the amplitude for oscillatory mode of convection. The influence of viscoelastic parameters on heat transfer has been discussed. Gravity modulation is found to have a destabilizing effect at low frequencies and a stabilizing effect at high frequencies. Finally, it is found that overstability advances the onset of convection, more with internal heating. The conditions for which the complex Ginzgburg-Landau equation undergoes Hopf bifurcation and the amplitude equation undergoes supercritical pitchfork bifurcation are studied.
[1] Ingham D.B. and Pop I.eds. (2005): Transport Phenomena in Porous Media. – Vol.3 1st Edn. Oxford: Elsevier.
[2] Nield D.A. and Bejan A. (2013): Convection in Porous Media. – 3rd edn. New York: Springer:
[3] Vafai K.ed. (2000): Handbook of Porous Media. – New York: Marcel Dekker.
[4] Gresho P.M. and Sani R. (1970): The effects of gravity modulation on the stability of a heated fluid layer. – J. Fluid Mech. vol.40 pp.783-806.
[5] Bhadauria B.S. and Bhatia P.K. Lokenath D. (2005): Convection in Hele-Shaw cell with parametric excitation. – Int. J. Non-Linear Mech. vol.40 No.4 pp.475-484.
[6] Malashetty M.S. and Padmavathi V (1997): Effect of gravity modulation on the onset of convection in a fluid and porous layer. – Int. J. Engg. Sci. vol.35 pp.829-839.
[7] Das R. and Pop I. (2000): The effect of G-jitter on vertical free convection boundary-layer flow in porous media. -Int Comm Heat Mass Transf vol.27 No.3 pp.415-424.
[8] Govender S. (2005a): Weak non-linear analysis of convection in a gravity modulated porous layer. – Transp. Porous Med. vol.60 pp.33-42.
[9] Siddhavaram V.K. and Homsy G.M. (2006): The effects of gravity modulation on fluid mixing Part 1. Harmonic modulation. – J. Fluid Mech. vol.562 pp.445-475.
[10] Saravanan S. and Sivakumar T. (2011): Thermovibrational instability in a fluid saturated anisotropic porous medium. – ASME J. Heat Transf. 133:051601.1-051601.9.
[11] Malashetty M.S. and Swamy M. (2011): Effect of gravity modulation on the onset of thermal convection in rotating fluid and porous layer. – Phys. Fluids vol.23 No.6 064108.
[12] Bhadauria B.S. Srivastava A. K. Sacheti N.C. and Chandran P. (2012): Gravity modulation of thermal instability in a viscoelastic fluid-saturated-anisotropic porous medium. – Z. Naturforch vol.67a pp.1-9.
[13] Bhadauria B.S. Siddheshwar P.G. Kumar J. and Suthar O. P. (2012): Non-linear stability analysis of temperature / gravity modulated Rayleigh-Benard convection in a porous medium. – Transp. Porous Med. vol.92 pp.633-647.
[14] Siddheshwar P.G. Bhadauria B.S. Mishra P. and Srivastava A.K. (2012): Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg-Landau model. – Int. J. Nonlinear Mech. vol.47 pp.418-425.
[15] Siddheshwar P.G. Bhadauria B.S. and Srivastava A. (2012): An analytical study of nonlinear double diffusive convection in a porous medium with temperature modulation / gravity modulation. – Transp. Porous Med. vol.91 pp.585-604.
[16] Bhadauria B.S. Siddheshwar P.G. and Suthar O. P. (2012): Non-linear thermal instability in a rotating viscous fluid layer under temperature/gravity modulation. – ASME J. Heat Trans. vol.134 No.10102502 doi.10.1115/1.4006868.
[17] Green T. III. (1968): Oscillating convection in an elasticoviscous liquid. – Phys. Fluids vol.11 1410. doi.10.1063/1.1692123
[18] Vest C.M. and Arpaci V.S. (1969): Overstability of a viscoelastic fluid layer heated from below. – J. Fluid Mech. vol.36 pp.13-623.
[19] Bhatia P.K. and Steiner J.M. (1972): Convective instability in a rotating viscoelastic fluid layer. – ZAMM vol.52 pp.321-327.
[20] Kim M.C. Lee S.B. Kim S. and Chung B.J. (2003): Thermal instability of viscoelastic fluids in porous media. – Int. J. Heat Mass Transfer vol.46 pp.5065-5072.
[21] Malashetty M.S. and Kulkarni S. (2009): The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model. – J. Non-Newton Fluid Mech. vol.162 No.3 pp.29-37.
[22] Wang S. and Tan W. (2011): Stability analysis of Soret-driven double-diffusive convection of Maxwell fluid in a porous medium. – Int. J. Heat Fluid Flow vol.32 No.1 pp.88-94.
[23] Kumar A. and Bhadauria B.S. (2011): Non-linear two dimensional double diffusive convection in a rotating porous layer saturated by a viscoelastic fluid. – Transp. Porous Med vol.87 pp.229-250
[24] Kumar A. and Bhadauria B.S. (2011): Double diffusive convection in a porous layer saturated with viscoelastic fluid using a thermal non-equilibrium model. – Phys. Fluids vol.223 pp. 967–983.
[25] Kumar A. and Bhadauria B.S. (2011): Thermal instability in a rotating anisotropic porous medium saturated with viscoelastic fluid. – Int. J. Nonlinear Mech. vol.46 pp.47-56.
[26] Bhadauria B.S. and Kiran P. (2014): Weak non-linear oscillatory convection in a viscoelastic fluid saturated porous medium under gravity modulation. – Transp. Porous Med. vol.104 pp.451-467.
[27] Bhadauria B.S. and Kiran P. (2014): Weak non-linear oscillatory convection in a viscoelastic fluid layer under gravity modulation. – Int. J. Nonlinear Mech. vol.65 pp.133-140.
[28] Bhadauria B.S. and Kiran P. (2014a): Weakly non-linear oscillatory convection in a viscoelastic fluid saturating porous medium under temperature modulation. – Int. J. Heat Mass Transfer vol.77 pp.843-851.
[29] Bhadauria B.S. and Kiran P. (2015): Chaotic and oscillatory magneto-convection in a binary viscoelastic fluid under G-jitter. – Int. J. Heat Mass Transfer vol.84 pp.610-614.
[30] Haajizadeh M. Ozguc A.F. and Tien C.L. (1984): Natural convection in a vertical porous enclosure with internal heat generation. – Int. J. Heat Mass Transfer vol.27 pp.1893-1902.
[31] Bhattacharya S.P. and Jena S.K. (1984): Thermal instability of a horizontal layer of micropolar fluid with heat source. – Proc. Indian Acad. Sci. (Math Sci) vol.93 No.1 pp.13-26.
[32] Parthiban C. and Patil P.R. (1997): Thermal instability in an anisotropic porous medium with internal heat source and inclined temperature gradient. – Int. Comm. Heat Mass Transfer vol.24 No.7 pp.104-1058.
[33] Magyari E. Pop I. and Postelnicu A. (2007): Effect of the source term on steady free convection boundary layer flows over a vertical plate in a porous medium Part I. – Transp. Porous Med. vol.67 pp.49-67.
[34] Bhadauria B.S. Kumar A. Kumar J. Sacheti N.C. and Chandran P. (2011): Natural convection in a rotating anisotropic porous layer with internal heat generation. – Transp. Porous Med. vol.90 No.2 pp.687-705.
[35] Bhadauria B.S. (2012): Double diffusive convection in a saturated anisotropic porous layer with internal heat source. – Transp. Porous Med. vol.92 pp.299-320.
[36] Bhadauria B.S. Hashim I. and Siddheshwar P.G. (2013): Effect of internal-heating on weakly non-linear stability analysis of Rayleigh-Benard convection under G-jitter. – Int. J. Nonlinear Mech. vol.54 pp.35-42.
[37] Bhadauria B.S Hashim I. and Siddheshwar P.G. (2013): Study of heat transport in a porous medium under G-jitter and internal heating effects. – Transp. Porous Med. vol.96 pp.21-37.
[38] Bhadauria B.S. Hashim I. and Siddheshwar P.G. (2013): Effects of time-periodic thermal boundary conditions and internal heating on heat transport in a porous medium. – Transp. Porous Med. vol.97 pp.185-200.
[39] Srivastava A. Bhadauria B.S. Siddheshwar P.G. and Hashim I. (2013): Heat transport in an anisotropic porous medium saturated with variable viscosity liquid under g-jitter and internal heating effects. – Transp. Porous Med. vol.99 pp.359-376.
[40] Altawallbeh A.A. Bhadauria B.S. and Hashim I. (2013): Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source. – Int. J. Heat Mass Transf. vol.59 pp.103-111.
[41] Horton C.W. and Rogers F.T. (1945): Convection currents in a porous medium. – J. Appl. Phys. vol.16 pp.367-370.
[42] Strogatz S.H. (2007): Non-Linear Dynamics and Chaos. – Levant Books Edition-I Kolkata India.
[43] Kuznetsov Y A. (1998): Topological Equivalence Bifurcations and Structural Stability of Dynamical Systems. – In Elements of Applied Bifurcation Theory. Edition-II Springer.
[44] Malkus W.V.R. and Veronis G. (1958): Finite amplitude cellular convection. – J. Fluid Mech. vol.4 pp.225-260.
[45] Venezian G. (1969) Effect of modulation on the onset of thermal convection. – J. Fluid Mech. vol.35 pp.243-254.