On the Boussinesq approximation for fluid systems with deformable interfaces

Academic Article

Abstract

  • The assumptions underlying the Boussinesq approximation place restrictions on its applications to systems with free surfaces and interfaces. In this paper we reconsider the limits in which the Boussinesq approximation is valid and develop a generalization of the approximation which allows for self-consistent application to such systems. The Boussinesq limit is characterized two parameters, G = gL3/v2, and ε = βθ, where G is a dimensionless measure of gravitational acceleration or system size and ε is the product of the fluid's coefficient of thermal expansion with a characteristic temperature difference. The Boussinesq limit, in general, corresponds to G → ∞ and ε → 0 while the product Gε (equal to the familiar Grashof number) remains finite. We consider two problems involving deformable boundaries: the stability of a two-layer fluid system heated from above and below and the influence of buoyancy on long-wavelength Marangoni instability. In the first two examples, we examine the conditions required to consistently account for the effects of a deformable surface on thermal convection while simultaneously applying the Bousinesq approximation. In particular, the effect of the deformable surface can be included through a term proportional to the product of Gδ with the deflection, ζ, of the interface from planarity, but it is required to treat the density of the two fluids as equal in the equations of motion (i.e., δ → 0). In the second example, that of long-wavelength Marangoni instability, we find that to simultaneously consider the effects of thermal buoyancy together with a deformable surface we must either treat the surface as undeformable (G → ∞ and ε → 0) or consider finite G and take Gε and ε to be independent parameters in the equations of motion. This leads to a result which explicitly reveals the role of buoyancy as a destabilizing influence on long-wavelength instability.
  • Published In

    Digital Object Identifier (doi)

    Author List

  • Lyubimov DV; Lyubimova TP; Alexander JID; Lobov NI
  • Start Page

  • 1159
  • End Page

  • 1168
  • Volume

  • 22
  • Issue

  • 8