Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

Abstract

a b s t r a c tThis article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly dependent on temperature at infinite Prandtl number. Although we verify the proposed techniques solely for viscosities that depend exponentially on temperature, the methods are extensible to other dependence laws. The set-up is a 2D domain with periodic boundary conditions along the horizontal coordinate which introduces a symmetry in the problem. This is the O(2) symmetry, which is particularly well described by spectral methods and motivates the use of these methods in this context. We examine the scope of our techniques by exploring transitions from stationary regimes towards time dependent regimes. At a given aspect ratio, stable stationary solutions become unstable through a Hopf bifurcation, after which the time-dependent regime is solved by the spectral techniques proposed in this article.

Confined rotating convection with large Prandtl number: Centrifugal effects on wall modes

“…However, it is not stable for stronger viscosity dependencies on temperature or infinite Prandtl number (see Ref. [35] for further details).…”

Symmetry and plate-like convection in fluids with temperature-dependent viscosity

“…The sign in the real part of the eigenvalue λ determines the stability of the solution: if it is negative, the perturbation decays and the stationary solution is stable, while if it is positive the perturbation grows over time and the stationary solution is unstable. For each unknown field expression, (5) is introduced into the system (1)-(3) and the equations are linearized inỹ, which are assumed to be small (see [9,8] for details). Together with their boundary conditions, the equations define a generalized eigenvalue problem.…”

“…The impact of the symmetry on the solutions displayed in convection problems with temperature-dependent viscosity has been addressed in [9,8], where a 2D physical set-up similar to ours is analyzed. The viscosity law considered in this work is similar to the one studied in [8], the main difference being that the viscosity change in our current setting is achieved within a narrower temperature gap.…”

Unsteady Convective Heat Transfer of Power-Law Fluid with Variable Fluid Properties in a Concentric Annulus Originating from a Polymer Flooding Process

“…where we consider the K À T and k À T relations for the thermodependent dynamic viscosity and thermal conductivity [15][16][17][18][19]:…”

“…However, these physical properties, especially fluid viscosity and thermal conductivity, are affected by temperature [16][17][18][19][20]. In the power-law rheological model, the power-law exponent also varies with temperature and this was investigated in experimental research [21], the results indicating that power-law index increases with increase in the temperature; however, this issue is rarely studied in numerical simulation.…”