Instability of Vertical Constant Through Flows in Binary Mixtures in Porous Media with Large Pores

Florinda Capone,Roberta De Luca,Isabella Torcicollo

doi:10.1155/2019/7379597

Abstract

A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical fluid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability are found. When the layer is salted from above, a condition guaranteeing the occurrence of “cold” instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.

Highlights

Convection in fluid mixtures saturating porous media has attracted—in the past as nowadays—the attention of many scientists due to its practical applications like, for example, in geothermal energy exploitation, extraction of oil from underground reservoirs, ground-water pollution, underground flows movement, thermal engineering, crystal growth, polymer engineering, and ceramic processing
The models describing the fluid motion in porous media are reaction-diffusion dynamical systems of P.D.Es, which, as it is well known, play an important role in the modeling and studying of many phenomena {see, for instance, [11, 12] and references therein}
Several geophysical and technological applications involve nonisothermal flow of fluids through porous media called throughflow which affects the stability of the system significantly

Summary

Introduction

Convection in fluid mixtures saturating porous media has attracted—in the past as nowadays—the attention of many scientists due to its practical applications like, for example, in geothermal energy exploitation, extraction of oil from underground reservoirs, ground-water pollution, underground flows movement, thermal engineering, crystal growth, polymer engineering, and ceramic processing. In some situations, a vertical motion (throughflow) is observable in a horizontal porous layer heated from below and the problem to determine until this motion is stable is of fundamental importance especially in applications involving cloud physics, hydrological/geophysical studies, seabed hydrodynamics, subterranean pollution, and many industrial and technological processes {see [1, 13,14,15,16] }. This is because, after the loss of stability, a secondary motion arises and this motion can be steady or oscillatory.

Statement of the Problem

Instability Analysis

Conclusions