Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

Chen YIN, Chunwu WANG, Shaowei WANG

doi:10.1007/s10483-020-2663-7

Applied Mathematics and Mechanics >

2020 , Vol. 41 >Issue 11: 1631 - 1650

DOI: https://doi.org/10.1007/s10483-020-2663-7

Articles

Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

Expand

1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. Department of Engineering Mechanics, School of Civil Engineering, Shandong University, Jinan 250061, China

Received date: 2020-06-09

Revised date: 2020-07-19

Online published: 2020-10-24

Supported by

Project supported by the National Natural Science Foundation of China (Nos. 11702135, 11271188, and 11672164), the Natural Science Foundation of Jiangsu Province of China (No. BK20170775), the China Postdoctoral Science Foundation (No. 2016M601798), and the Jiangsu Planned Project for Postdoctoral Research Funds of China (No. 1601169B)

Fold

Abstract

The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ-QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls (LRs) only depend on the depth ratio. With the existence of the shear flow, the effects of the depth ratio, the Reynolds number, the Prandtl number, the stress relaxation, and strain retardation times on the transverse rolls (TRs) are also studied. Additionally, the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system. In contrast to the case for Newtonian fluids, the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases.

Key words： viscoelastic fluid; thermal convection; Poiseuille flow; fluid-porous system

Cite this article

Chen YIN, Chunwu WANG, Shaowei WANG . Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow[J]. Applied Mathematics and Mechanics, 2020 , 41(11) : 1631 -1650 . DOI: 10.1007/s10483-020-2663-7

References

[1] WORSTER, M. G. Instabilities of the liquid and mushy regions during solidification of alloys. Journal of Fluid Mechanics, 237(1), 649-669(1992)
[2] CHEN, Q. S., PRASAD, V., and CHATTERJEE, A. Modeling of fluid flow and heat transfer in a hydrothermal crystal growth system:use of fluid-superposed porous layer theory. Journal of Heat Transfer, 121(4), 1049-1058(1999)
[3] MATTHEWS, P. C. A model for the onset of penetrative convection. Journal of Fluid Mechanics, 188(1), 571-583(1988)
[4] EWING, R. E. Multidisciplinary interactions in energy and environmental modeling. Journal of Computational and Applied Mathematics, 74(1-2), 193-215(1996)
[5] NIELD, D. A. Onset of convection in a fluid layer overlying a layer of a porous medium. Journal of Fluid Mechanics, 81(3), 513-522(1977)
[6] CHEN, F. and CHEN, C. F. Onset of finger convection in a horizontal porous layer underlying a fluid layer. Journal of Heat Transfer, 110(2), 403-409(1988)
[7] STRAUGHAN, B. Effect of property variation and modelling on convection in a fluid overlying a porous layer. International Journal for Numerical and Analytical Methods in Geomechanics, 26(1), 75-97(2002)
[8] LIU, R., LIU, Q. S., and ZHAO, S. C. Influence of Rayleigh effect combined with Marangoni effect on the onset of convection in a liquid layer overlying a porous layer. International Journal of Heat and Mass Transfer, 51(25-26), 6328-6331(2008)
[9] ZHAO, S. C., LIU, Q. S., LIU, R., NGUYEN-THI, H., and BILLIA, B. Thermal effects on Rayleigh-Marangoni-Bénard instability in a system of superposed fluid and porous layers. International Journal of Heat and Mass Transfer, 53(15-16), 2951-2954(2010)
[10] HILL, A. A. and STRAUGHAN, B. Global stability for thermal convection in a fluid overlying a highly porous material. Proceedings of the Royal Society A:Mathematical Physical and Engineering Sciences, 465(2101), 207-217(2009)
[11] BAGCHI, A. and KULACKI, F. A. Natural convection in fluid-superposed porous layers heated locally from below. International Journal of Heat and Mass Transfer, 54(15-16), 3672-3682(2011)
[12] BAGCHI, A. and KULACKI, F. A. Natural Convection in Superposed Fluid-Porous Layers, Springer, New York (2014)
[13] DIXON, J. M. and KULACKI, F. A. Mixed Convection in Fluid Superposed Porous Layers, Springer International Publishing, Berlin (2017)
[14] ZHANG, Y., SHAN, L., and HOU, Y. Well-posedness and finite element approximation for the convection model in superposed fluid and porous layers. SIAM Journal on Numerical Analysis, 58(1), 541564(2020)
[15] CHANG, M. H., CHEN, F., and STRAUGHAN, B. Instability of Poiseuille flow in a fluid overlying a porous layer. Journal of Fluid Mechanics, 564, 287-303(2006)
[16] HILL, A. A. and STRAUGHAN, B. Poiseuille flow in a fluid overlying a porous medium. Journal of Fluid Mechanics, 603, 137-149(2008)
[17] HILL, A. A. and STRAUGHAN, B. Poiseuille flow in a fluid overlying a highly porous material. Advances in Water Resources, 32(11), 1609-1614(2009)
[18] CHANG, M. H. Thermal convection in superposed fluid and porous layers subjected to a horizontal plane Couette flow. Physics of Fluids, 17(6), 064106(2005)
[19] CHANG, M. H. Thermal convection in superposed fluid and porous layers subjected to a plane Poiseuille flow. Physics of Fluids, 18(3), 035104(2006)
[20] ZEESHAN, A., SHEHZAD, N., ELLAHI, R., and ALAMRI, S. Z. Convective Poiseuille flow of Al₂O₃-EG nanofluid in a porous wavy channel with thermal radiation. Neural Computing and Applications, 30(11), 3371-3382(2018)
[21] ELLAHI, R., ZEESHAN, A., SHEHZAD, N., and ALAMRI, S. Z. Structural impact of kerosene-Al₂O₃ nanoliquid on MHD Poiseuille flow with variable thermal conductivity:application of cooling process. Journal of Molecular Liquids, 264, 607-615(2018)
[22] HASSAN, M., MARIN, M., ALSHARIF, A., and ELLAHI, R. Convective heat transfer flow of nanofluid in a porous medium over wavy surface. Physics Letters A, 382(38), 2749-2753(2018)
[23] AVILA, R. Linear instability analysis of the onset of thermal convection in an Ekman-Couette flow. International Journal of Heat and Mass Transfer, 154, 119635(2020)
[24] ZHANG, Z., FU, C., and TAN, W. Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below. Physics of Fluids, 20(8), 084103(2008)
[25] MALASHETTY, M. S., TAN, W., and SWAMY, M. The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer. Physics of Fluids, 21, 084101(2009)
[26] YIN, C., FU, C., and TAN, W. Onset of thermal convection in a Maxwell fluid-saturated porous medium:the effects of hydrodynamic boundary and constant flux heating conditions. Transport in Porous Media, 91(3), 777-790(2011)
[27] BHATTI, M. M., ZEESHAN, A., TRIPATHI, D., and ELLAHI, R. Thermally developed peristaltic propulsion of magnetic solid particles in biorheological fluids. Indian Journal of Physics, 92(4), 423-430(2017)
[28] SHEHZAD, N., ZEESHAN, A., and ELLAHI, R. Electroosmotic flow of MHD power law Al₂O₃-PVC nanouid in a horizontal channel:Couette-Poiseuille flow model. Communications in Theoretical Physics, 69(6), 655-666(2018)
[29] SUN, Q., WANG, S., ZHAO, M., YIN, C., and ZHANG, Q. Weak nonlinear analysis of Darcy-Brinkman convection in Oldroyd-B fluid saturated porous media under temperature modulation. International Journal of Heat and Mass Transfer, 138, 244256(2019)
[30] HEMANTHKUMAR, C., RAGHUNATHA, K. R., and SHIVAKUMARA, I. S. Nonlinear convection in an elasticoviscous fluid saturated anisotropic porous layer using a local thermal nonequilibrium model. Heat Transfer, 49(4), 16911712(2020)
[31] YIN, C., FU, C., and TAN, W. Stability of thermal convection in a fluid-porous system saturated with an Oldroyd-B fluid heated from below. Transport in Porous Media, 99(2), 327-347(2013)
[32] YIN, C., NIU, J., FU, C., and TAN, W. Thermal convection of a viscoelastic fluid in a fluid porous system subjected to a horizontal plane Couette flow. International Journal of Heat and Fluid Flow, 44, 711-718(2013)
[33] LEBON, G., PARMENTIER, P., TELLER, O., and DAUBY, P. C. Bénard-Marangoni instability in a viscoelastic Jeffreys' fluid layer. Rheologica Acta, 33(4), 257-266(1994)
[34] MARTINEZ-MARDONES, J. and PEREZ-GARCIA, C. Linear instability in viscoelastic fluid convection. Journal of Physics:Condensed Matter, 2(5), 1281(1990)
[35] TAN, W. and MASUOKA, T. Stokes' first problem for an Oldroyd-B fluid in a porous half space. Physics of Fluids, 17(2), 023101(2005)
[36] BEAVERS, G. S. and JOSEPH, D. D. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30(01), 197-207(1967)
[37] JONES, I. P. Low Reynolds number flow past a porous spherical shell. Mathematical Proceedings of the Cambridge Philosophical Society, 73(01), 231-238(1973)
[38] DONGARRA, J. J., STRAUGHAN, B., and WALKER, D. W. Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Applied Numerical Mathematics, 22(4), 399-434(1996)
[39] MOLER, C. B. and STEWART, G. W. Algorithm for generalized matrix eigenvalue problems. SIAM Journal on Numerical Analysis, 10(2), 241-256(1973)

Options

Outlines

模态框（Modal）标题

Abstract

Cite this article

References