Theoretical study of a Benard-Marangoni problem

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:376

Article

Pardo, R.¹  Herrero, H.²  Hoyas, S.³ 

[1] Univ Complutense Madrid, Fac Ciencias Quim, Dept Matemat Aplicada, E-28040 Madrid, Spain

[2] Univ Castilla La Mancha, Dept Matemat, Fac Ciencias Quim, E-13071 Ciudad Real, Spain

[3] Univ Valencia, Dept Informat, ETS Ingn, E-46100 Valencia, Spain

关键词: Fluid dynamics;    Thermal convection;    Bifurcation;    Incompressible Boussinesq-Navier-Stokes equations;    Benard-Marangoni problem;   

DOI  :  10.1016/j.jmaa.2010.10.064

来源: Elsevier

【 摘 要 】

In this paper we prove the existence of strong solutions for the stationary Benard-Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Benard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier-Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Benard-Marangoni problem. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free