Andrejs Koliškins: Weakly nonlinear instability of non-isothemal incompressible flows

Date and time: 2025-05-20, 13:15-14:15

Location: U3-012 (Västerås)

Video link: Teams External link.

Speaker: Prof. Andrejs Koliškins, Riga Technical University, Latvia (host: Maksat Ashyralyyev)

Abstract:

Consider a convective flow between two parallel vertical planes induced by an internal heat sources. The flow is described by the system of the Navier-Stokes equations under the Boussinesq approximation. Two dimensionless parameters characterize the flow:

(a) the Grasshof number proportional to the intensity of the internal heat sources, and

(b) the Prandtl number describing the properties of the fluid.

Lınear stability analysis of a steady flow is used to determine the critical values of the parameters of the problem. Assuming that the Grasshof number is slightly larger than the critical value (the flow is linearly unstable, but the growth rate of a perturbation is small) we use the method of multiple scales to construct an amplitude evolution equation for the most unstable mode. The following steps are required to derive the equation:

(1) solution of the linear stability problem (the critical values of the parameters and the eigenfucntion representing the most unstable mode),

(2) solution of the corresponding adjoint problem (the eigenfunction of the adjoint problem),

(3) solution of three boundary-value problems one of which is resonantly forced,

(4) use of the Fredholm’s Alternative at order two in the small parameter to determine the group velocity,

(5) use of the Fredholm’s Alternative at order three to determine the amplitude evolution equation.

It is shown that the equation is the complex Ginzburg-Landau equation. The coefficients of the equation are calculated in terms of integrals containing the solution of problems in steps 1 – 4 described above. Results of numerical calculations are presented.