Stability of Thermal Convection in a Fluid-Porous System Saturated with an Oldroyd-B Fluid Heated from Below
Tóm tắt
A linear stability analysis is conducted for thermal convection in a two-layer system composed of a fluid layer overlying a porous medium saturated with an Oldroyd-B fluid heated from below. It is found that the convection pattern in the system is controlled by the porous medium when the ratio of the depth of the fluid layer to that of the porous medium is small. However, the fluid layer takes a predominant role if the depth ratio exceeds a critical value. Compared with stationary convection, the switching point from a porous-dominated mode to a fluid-dominated mode for oscillatory convection is located at a lower depth ratio. The effects of different parameters on stationary convection and oscillatory convection are also investigated in detail.
Tài liệu tham khảo
Allen, M., Behie, A., Trangenstein, J.: Multiphase Flow in Porous Media: Mechanics, Mathematics, and Numerics. Speringer, Berlin (1988)
Avramenko, A.A., Kuznetsov, A.V.: Stability of a suspension of gyrotactic microorganisms in superimposed fluid and porous layers. Int. J. Heat Mass Transf. 31(8), 1057–1066 (2004)
Avramenko, A.A., Kuznetsov, A.V.: Linear instability analysis of a suspension of oxytactic bacteria in superimposed fluid and porous layers. Transp. Porous Media 61(2), 157–175 (2005)
Avramenko, A.A., Kuznetsov, A.V.: The onset of convection in a suspension of gyrotactic microorganisms in superimposed fluid and porous layers: effect of vertical throughflow. Transp. Porous Media 65(01), 159–176 (2006)
Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(01), 197–207 (1967)
Bénard, H.: Les tourbillons cellulaires dans une nappe liquide. Rev. Gen. Sci. Pures. Appl. 11, 1261–1268 (1900)
Carr, M.: Penetrative convection in a superposed porous-medium-fluid layer via internal heating. J. Fluid Mech. 509, 305–329 (2004)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)
Chang, M.H.: Stability of convection induced by selective absorption of radiation in a fluid overlying a porous layer. Phys. Fluids 16(10), 3690–3698 (2004)
Chang, M.H.: Thermal convection in superposed fluid and porous layers subjected to a horizontal plane Couette flow. Phys. Fluids 17(6), 064106 (2005)
Chang, M.H.: Thermal convection in superposed fluid and porous layers subjected to a plane Poiseuille flow. Phys. Fluids 18(3), 035104 (2006)
Chen, F., Chen, C.F.: Onset of finger convection in a horizontal porous layer underlying a fluid layer. J. Heat Transf. 110(2), 403–409 (1988)
Copley, S., Giamei, A., Johnson, A., Hornbecker, M.: The origin of freckles in unidirectionally solidified castings. Metall Trans. 1, 2193–2204 (1970)
Dongarra, J.J., Straughan, B., Walker, D.W.: Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22(4), 399–434 (1996)
Drazin, P., Reid, W.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)
Genc, G., Rees, D.A.S.: The onset of convection in horizontally partitioned porous layers. Phys. Fluids 23(6), 064107 (2011)
Hill, A.A., Straughan, B.: Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149 (2008)
Hill, A.A., Straughan, B.: Poiseuille flow in a fluid overlying a highly porous material. Adv. Water Res. 32(11), 1609–1614 (2009)
Horton, C., Rogers, F.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)
Jones, I.P.: Low reynolds number flow past a porous spherical shell. Proc. Camb. Philos. Soc. 73(01), 231–238 (1973)
Kim, M.C., Lee, S.B., Kim, S., Chung, B.J.: Thermal instability of viscoelastic fluids in porous media. Int. J. Heat Mass Transf. 46(26), 5065–5072 (2003)
Kolkka, R.W., Ierley, G.R.: On the convected linear stability of a viscoelastic Oldroyd B fluid heated from below. J. Non-Newtonian Fluid Mech. 25(2), 209–237 (1987)
Kumar, A., Bhadauria, B.: Double diffusive convection in a porous layer saturated with viscoelastic fluid using a thermal non-equilibrium model. Phys. Fluids 23, 054101 (2011)
Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Camb. Philos. Soc. 44(04), 508–521 (1948)
Malashetty, M., Shivakumara, I., Kulkarni, S., Swamy, M.: Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model. Transp. Porous Media 64, 123–139 (2006)
Malashetty, M.S., Tan, W., Swamy, M.: The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer. Phys. Fluids 21, 084101 (2009)
Moler, C., Stewart, G.: Algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10(2), 241–256 (1973)
Nield, D.A.: Onset of convection in a fluid layer overlying a layer of a porous medium. J. Fluid Mech. 81(03), 513–522 (1977)
Nield, D., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)
Rayleigh, L.: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag. 32, 529–546 (1916)
Sample, A., Hellawell, A.: The mechanisms of formation and prevention of channel segregation during alloy solidification. Metall. Trans. A 15, 2163–2173 (1984)
Sparrow, E.M., Goldstein, R.J., Jonsson, V.K.: Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18(04), 513–528 (1964)
Straughan, B.: Effect of property variation and modelling on convection in a fluid overlying a porous layer. Int. J. Numer. Anal. Methods 26(1), 75–97 (2002)
Straughan, B., Walker, D.W.: Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127(1), 128–141 (1996)
Tan, W., Masuoka, T.: Stokes’ first problem for an Oldroyd-B fluid in a porous half space. Phy. Fluids 17(2), 023101 (2005)
Vest, C.M., Arpaci, V.S.: Overstability of a viscoelastic fluid layer heated from below. J. Fluid Mech. 36(03), 613–623 (1969)
Zhang, Z., Fu, C., Tan, W.: Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below. Phys. Fluids 20(8), 084103 (2008)
Zhao, S.C., Liu, Q.S., Liu, R., Nguyen-Thi, H., Billia, B.: Thermal effects on Rayleigh-Marangoni-Bénard instability in a system of superposed fluid and porous layers. Int. J. Heat Mass Transf. 53(15–16), 2951–2954 (2010)