Rayleigh — Benard problem for Polymer Solution

Abstract

There are three mathematical models describing the motion of aqueous solutions of polymers: the second grade fluid model (Rivlin — Eriksen), the hereditary model (Voitkunsky — Amfilokhiev — Pavlovsky), and its asymptotic simplification (Pavlovsky). This work considers the problem of fluid equilibrium stability in a horizontal fluid layer heated from below or from above. Also, equations of thermal gravitational convection for all three models are derived. Three types of boundary conditions are considered: two solid boundaries; the lower solid boundary and the upper free boundary; two free boundaries (the Rayleigh problem). For the case of heating from below, the principle of perturbation monotonicity is established that ensures the spectral problem eigenvalues to be of real type. This greatly simplifies the determination of the critical Rayleigh numbers. It turned out that these numbers coincide with the critical Rayleigh numbers in the classical Rayleigh — Benard problem. In the case of heating from above at large temperature gradients, the perturbation decrements become complex, but their real parts are negative. The conclusion that the relaxation properties of a second grade fluid and an aqueous solution of polymers do not lead to a change in the critical Rayleigh number may seem strange at first glance. According to our assumption, it is explained by the base state of the liquid being a state of rest.