Linear Bifurcation of Diffusive Convection Micropolar Fluid in Porous Media
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RESEARCH PAPER
Linear Bifurcation of Diffusive Convection Micropolar Fluid in Porous Media Hossein Rahmaani Fazel1 • Behrouz Raesi1 Received: 30 October 2019 / Accepted: 10 July 2020 Ó Shiraz University 2020
Abstract In this paper, we consider diffusive micropolar equations in porous media and present linear stability regimes of parameters. We find all parametric eigenvalues and eigenvectors of the linear part of the Brinkman–Eringen equations. Then, we show that this linear part of the operator is sectorial. We also find two critical parameters that dictated Hopf and saddle-node bifurcations of the problem. Finally, we establish the principle of exchange of stabilities. Keywords Micropolar equations Porous media Diffusive convection Linear stability Bifurcation Mathematics Subject Classification Primary 37L15 Secondary 35B32
1 Introduction Diffusive convection arises when heat concentrations in the fluid effect on the layer density of the fluid. Variation in heat distribution causes the change in layer density of the fluid and is responsible for the instability of the rest state. Chun Hsia, Tian Ma, and Shouhong Wang determined linear and nonlinear regimes of stability of rest state in twoand three-dimensional usual double-diffusive convection in Hsia et al. (2008a, b). They investigated bifurcation attractors of the problems and found two critical Rayleigh values that characterize saddle-node and Hopf bifurcations of the problem. We have generalized their results on the linear part of the equation of micropolar fluid in a porous medium, and this problem is richer than its classical counterparts. And this problem is richer than its classical counterparts. For studying a dozen of linear and nonlinear bifurcation of Newtonian flows and Rayleigh–Benard threshold of instability, the readers are referred to Ma and Wang (2005, 2014). Undoubtedly one of the most important and applicable parts of fluid studies is the study of micropolar fluid which & Behrouz Raesi [email protected] Hossein Rahmaani Fazel [email protected] 1
Department of Mathematics, Shahed University, 18155/ 159, Tehran, Iran
has been increasing in recent decades. The nonlinear and attractor bifurcation of our problem will be arising in our next paper. The micropolar fluid is liquid with microstructure. It belongs to a class of fluid with nonsymmetric stress tensor that is called a polar fluid, which is generalized Navier– Stokes model of classical fluid. Micropolar fluid represents liquid consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is neglected (Lukaszewicz 1999). The equation of micropolar fluid raised in Migoun (1981), for the first time, by C.A. Eringen is worth studying as a very well balanced. It is a reasonable generalization of the classical Navier–Stokes equation, covering, both in applications and in theory, many more physical systems than the classical model. Flow in porous media is a topic
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