Thermal Instability of a Micropolar Fluid Layer with Temperature-Dependent Viscosity
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RESEARCH ARTICLE
Thermal Instability of a Micropolar Fluid Layer with Temperature-Dependent Viscosity Joginder Singh Dhiman1 • Nivedita Sharma1
Received: 12 August 2015 / Revised: 12 September 2018 / Accepted: 22 December 2018 Ó The National Academy of Sciences, India 2019
Abstract In this paper, the effect of temperature-dependent viscosity on the onset of thermal convection in a micropolar fluid layer heated from below for each combination of rigid (the surfaces with non-slip condition) and dynamically free (the surfaces with stress-free condition) boundaries is investigated. It is shown here analytically that the principle of exchange of stabilities is valid for the problem, which means that instability sets in as stationary convection. The expressions for Rayleigh numbers for each combination of rigid and dynamically free boundary conditions are derived using Galerkin method. The effects of micropolar parameters and viscosity variation parameter on critical wave numbers and consequently on the critical Rayleigh numbers are computed numerically. Keywords Thermal convection Temperature-dependent viscosity Principle of exchange of stabilities Galerkin method Rayleigh number Microrotation
1 Introduction Microfluids exhibiting certain microscopic effects arising from the local structure and micromotions of the fluid elements were introduced and developed by Eringen [1]. These fluids support stress moments and body moments and are influenced by the spin inertia. Eringen’s theory has provided a good model to study a number of complicated fluids, & Joginder Singh Dhiman [email protected] 1
Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla 171005, India
including the flow of low-concentration suspensions, liquid crystal, blood and turbulent shear flows. However, Eringen [2] introduced a subclass of microfluids named micropolar fluid, which exhibits microrotational inertia. Physically, micropolar fluids may represent fluids consisting of rigid, randomly oriented particles suspended in a viscous medium, where the deformation of the particles is ignored. In this theory, the local fluid elements have the usual translatory degrees of freedom reckoned by the velocity vector and have in addition, degrees of freedom enabling the intrinsic rotatory motions described by the gyration vector. This constitutes a substantial generalization of the Navier–Stokes model since a new vector field, namely the angular velocity field or rotation of particles, is introduced. With the introduction of this new vector, one more vector equation is added in Navier–Stokes model which represents the conservation of angular momentum. Furthermore, four new viscosities are also introduced in the system of equations. If one of these viscosities, namely microrotation viscosity, becomes zero, the equation of conservation of the linear momentum becomes independent of the microstructure. Thus, the size of the microrotation viscosity coefficient allows us to measure the deviation of flows of micropolar fluids from t
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