On the Oberbeck-Boussinesq Approximation

Abstract

This paper deals with a derivation (using a perturbation technique) of an approximation, due to Oberbeck8,9 and Boussinesq,1 to describe the thermal response of linearly viscous fluids that are mechanically incompressible but thermally compressible. The present approach uses a nondimensionalization suggested by Chandrasekhar2 and utilizing the ratio of two characteristic velocities as a measure of smallness, systematically derives the Oberbeck-Boussinesq approximation as a third-order perturbation. In the present approach, the material is subjected to the constraint that the volume change is determined solely by the temperature change in the body and uses a novel approach in deriving the thermodynamical restrictions. Consequently, it is free from the additional assumptions usually added on in earlier works in order to obtain the correct equations.

Penetrative ferroconvection via internal heating in a saturated porous layer with constant heat flux at the lower boundary

“…The flow in the porous medium is described by the Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity and the Oberbeck-Boussinesq approximation is assumed to be valid. Rajagopal et al [38] have presented a frame work within which the status of the Oberbeck-Boussinesq approximation can be clearly delineated, within the context of a full thermodynamical theory for the Navier-Stokes fluid.…”

Effect of Internal Heat Generation on the Onset of Brinkman-Benard Thermomagnetic Convection in a Saturated Porous Layer

“…In most of the studies, the Oberbeck-Boussinesq approximation has been quite mistreated, and the justifications that have been given for this approximation are largely incorrect. Rajagopal et al 19 have provided a systematic basis for this approximation and discussed some of the errors in the previous approaches. It is clear that there exists the following solution for the basic state:…”

On an Iterative Method for Approximate Solutions of a Generalized Boussinesq Model

“…Boussinesq type systems of hydrodynamic equations arise as zero order approximations to the coupling between the Navier-Stokes equation and the thermodynamic equation (see Joseph [15], Chandrasekhar [2], Feireisl [6], Rajagopal, Ruzicka and Srinivasa [25]). In the derivations of such systems, it is usual to assume that the fluid viscosity and thermal conductivity are positive constants; however, there are several important physical situations where such hypotheses are not adequate, and one must consider the possibility that such viscosity and thermal conductivity may be temperature dependent.…”