Instability thresholds for thermal convection in a Kelvin\u2013Voigt fluid of variable order

B Straughan

doi:10.1007/s12215-020-00588-1

Abstract

We present numerical techniques for calculating instability thresholds in a model for thermal convection in a complex viscoelastic fluid of Kelvin–Voigt type. The theory presented is valid for various orders of an exponential fading memory term, and the strategy for obtaining the neutral curves and instability thresholds is discussed in the general case. Specific numerical results are presented for a fluid of order zero, also known as a Navier–Stokes–Voigt fluid, and fluids of order 1 and 2. For the latter cases it is shown that oscillatory convection may occur, and the nature of the stationary and oscillatory convection branches is investigated in detail, including where the transition from one to the other takes place.

Highlights

The subject of hydrodynamics is one with immense application
We present a computational procedure for calculating the neutral curves for instability associated with thermal convection
In the following we firstly describe the equations for thermal convection in a Kelvin–Voigt fluid of order 0, 1, 2, ... , L, and we outline the computational procedure in the general case

Summary

Introduction

The subject of hydrodynamics is one with immense application. The famous Navier–Stokes equations to describe flow of a linearly viscous, incompressible fluid are employed in many branches of applied mathematics and engineering. Straughan the stress may need to incorporate dependence on second gradients of the velocity field to describe such physical effects as the flattening of the parabolic profile in Poiseuille flow, cf Straughan [18] We stress that this does not preclude complex solution behaviour in a Newtonian fluid where extremely complicated solutions are known for the Navier–Stokes equations and approximations arising therefrom such as the primitive equations, see e.g. Gargano et al [19], Kukavica et al [20]. We present a computational procedure for calculating the neutral curves for instability associated with thermal convection This is a non-trivial process for Kelvin–Voigt fluids of variable order since it transpires that to calculate the critical Rayleigh number thresholds of instability one has to first calculate the eigenvalues of the variable order theory and employ these in the minimization process over all wave numbers. Practical applications of Kelvin–Voigt fluids are mentioned at the end of Sect. 2

The Kelvin–Voigt equations for thermal convection

Thermal convection in a fluid of variable order

Stability for thermal convection in a Navier–Stokes–Voigt fluid

Instability for a Kelvin–Voigt fluid of order one

Instability for a Kelvin–Voigt fluid of order two

Conclusions