Large prandtl number finite-amplitude thermal convection with maxwell viscoelasticity

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Abstract
Nonlinear interactions of deviatoric stress components and the velocity field occur in all dynamic flows where convected elasticity is accounted for. By incorporating a linear Maxwellian constitutive relation (Oldroyd ‘B’ type) into a finite-amplitude convection model we quantify the magnitude of some of the effects of these nonlinear interactions. For viscoelastic flows the relevant nondimensional parameter is the ratio of viscoelastic constitutive relaxation time constant, λ1, to the basic flow process time. The Rayleigh number, Ra, and the nondimensional ratio of λ1 to thermal conduction time, τc, are part of the parameter space investigated. However, shorter basic flow time scales than that for thermal equilibration are of interest since most viscoelastic fluids have relatively small values of λ1 The ratio of λ1 to buoyant time [bcirc], or λ1/[bcirc], is, therefore, a pertinent parameter. Using both lithospheric and aesthenospheric values for λ1, the ratio appropriate to mantle convection is roughly bounded by O(1)[bcirc]>λ1/[bcirc]>O(10−6). Employing these bounds and computing low Rayleigh number time-dependent convective flows in a two-dimensional box, it is demonstrated that viscoelasticity has a negligible influence on quasi-steady heat transport even for λ1/[bcirc]∼O(1) For any time-dependent behavior with time scales as short, or shorter than, the buoyant time, [bcirc], viscoelasticity might be important to the local exchange of mechanical energy. The recoverable strain energy in the descending portion of the lithosphere is comparable to the local viscous dissipation. The magnitude of this recoverable component of shear is proportional to λ1/[bcirc].

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