Analysis of the bifurcation structure near the excitation threshold of convection during uniform fluid infiltration through a slightly asymmetric porous chip

Abstract

Fluid-saturated porous scaffolds are crucial components of flow perfusion bioreactors, which act as microfluidic chips. These devices are used to grow cell cultures in a constant, steady-state environment through maintaining a continuous flow of nutrient-rich solution.  Cell proliferation often occurs in non-isothermal conditions, which under gravity can lead to the onset of thermal convection. In this paper, we consider the problem of convection excitation in a porous chip heated from below, across which uniform pumping of an incompressible fluid is carried out. It has been known that incorporating a fluid-saturated porous structure into highly conductive impermeable medium leads to the fact that the boundary value problem takes on the cosymmetry property for any simply connected 2-D shape of the domain. This causes branching of one-parameter families of heterogeneous equilibrium states. Note however that even weak fluid pumping across the porous medium leads to cosymmetry breaking and the type of dynamic behavior of the system in this case depends on the domain symmetry. If under reflection the pumping domain is symmetric about the vertical axis, the resulting convection will be stationary. In the asymmetric domain, periodic oscillations are excited. In this paper, we investigate the bifurcation structure near the convection excitation threshold occurring in a weakly asymmetric domain.  The main focus is on the transition from stationary to oscillatory convection. The study of the system includes the following issues: finding the basic state of the system analytically; linear analysis of its stability using the Galerkin method; analysis of weakly nonlinear solutions near the first bifurcation point using the method of multiple time scales. It has been shown that branching of solutions near the excitation threshold of convection in a weakly non-cosymmetric system of equations defined in a weakly asymmetric domain is of surprisingly complex nature. The transition process includes the Andronov--Hopf bifurcation, the birth and death of limiting cycles at the backward and forward tangential bifurcations of a pair of equilibria, respectively; and the formation of homoclinic trajectories. Stability maps are constructed on the planes of control parameters of the problem: the Rayleigh number for the porous medium, the thermal Péclet number, and the chip tilt angle, which acts as a symmetry imperfection parameter. Bifurcation diagrams of a dynamic system on its slow manifold are presented.