摘要:

This paper studies the asymptotic structure of convection in an infinite Prandtl number fluid with strongly temperature‐dependent viscosity, in the limit where the dimensionless activation energy 1/ε is large, and the Rayleigh numberR, defined (essentially) with the basal viscosity and the prescribed temperature drop, is also large. We find that the Nusselt numberNis given byN~CεR1/5, whereCdepends on the aspect ratioa. The relative error in this result isO(R−1/10ε−1/4, ε1/2,R−2/5ε−2,R−2/20ε−1/24), so that we cannot hope to find accurate confirmation of this result at moderate Rayleigh numbers, though it should serve as a useful indicator of the relative importance ofRandε. For the above result to be valid, we requireR≳ 1/ε5≫1. More important, however, is the asymptotic structure of the flow: there is a cold (hence rigid) lid with sloping base, beneath which a rapid, essentially isoviscous, convection takes place. This convection is driven by plumes at the sides, which generate vorticity due to thermal buoyancy, as in the constant viscosity case (Roberts, 1979). However, the slope of the lid base is sufficient to cause a large shear stress to be generated in the thermal boundary layer which joins the lid to the isoviscous region underneath (though a large velocity is not generated); consequently, the layer does not “see” the shear stress exerted by the interior flow (at leading order), and thereforethe thermal boundary layer structure is totally self‐determined: it even has a similarity structure (as a consequence). This fact makes it easy to analyse the problem, since the boundary layer uncouples from the rest of the flow. In addition, we find an alternative scaling (in which the lid base is “almost” flat), but it seems that the resulting boundary layer equations have no solution, though this is certainly open to debate: the results quoted above are not for this case. When a free slip boundary condition is applied at the top surface, one finds that there exists a thin “skin” at the top of the lid which is astressboundary layer. The shear stress changes rapidly to zero, and there exists a huge longitudinal stress (compressive/tensile) in this skin. For earthlike parameters, this stress far exceeds

ISSN:0022-2526    

DOI:10.1002/sapm1985723189

年代:1985

数据来源: WILEY

摘要:

A method is presented that combines phase‐plane techniques with the ideas of multiple scale and matched asymptotic expansions to explain the behavior of solutions to second‐order, nonlinear, nonautonomous, singular boundary‐value problems. It is shown that if one is willing to give up the detailed information provided by a procedure such as matched asymptotic expansions, then complete qualitative information can be obtained by the much simpler method given here. (“Complete” here means that the method provides a way of categorizing all possible solutions of such problems.) In addition, the similarities and differences between the present method and that of Melnikov, which has been useful in the study of dynamical systems,

ISSN:0022-2526    

DOI:10.1002/sapm1985723221

年代:1985

数据来源: WILEY

摘要:

This paper describes some applications of a transformation which takes a system of two, first order, quasilinear, partial differential equations in two dependent and two independent variables into a similar system. Typically, the equations govern wave propagation in dispersive and dissipative systems. It is shown that certain nonlinear equations which are of current practical interest can be transformed into linear equations.

ISSN:0022-2526    

DOI:10.1002/sapm1985723241

年代:1985

数据来源: WILEY

摘要:

It is well‐known how the representation theory of the Lie algebra sl(2, ℂ) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the latticeL(m,n) of Young diagrams contained in anm × nrectangle) which have the Sperner

ISSN:0022-2526    

DOI:10.1002/sapm1985723263

年代:1985

数据来源: WILEY