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Linear stability of convective motion in a tall vertical annulus was analysed in the paper. The base flow was generated by a non-uniform distribution of heat sources in the radial direction. The base flow velocity and temperature were obtained analytically solving the system of Navier-Stokes equations under the Boussinesq approximation. The linear stability problem was solved for axi-symmetric and asymmetric perturbations by a collocation method based on the Chebyshev polynomials. Numerical results showed that there were three destabilising factors: (1) increase of the gap between the cylinders, (2) increase of the density of internal heat sources towards to the outer boundary of the annulus and (3) increase of the Prandtl number.

Linear stability of a steady flow of a chemically reacting fluid located in a vertical fluid layer bounded by two infinite parallel planes is investigated. Steady convective flow in the vertical direction is initiated due to the combined effect of internal heat generation and the temperature difference between the planes. Imposing small perturbations on the base flow, linearizing equations of thermal convection under the Boussinesq approximation in the neighbourhood of the base flow and using the method of normal modes we obtain an eigenvalue problem for a system of ordinary differential equations. Collocation method is used to discretize the problem. Numerical calculations are performed in Matlab. Fluid velocity, pressure, and temperature are the solutions of a nonlinear boundary value problem. Properties of the nonlinear boundary value problem for the base flow are investigated numerically using bifurcation analysis. It is shown that both the temperature difference between the planes and intensity of internal heat generation have a destabilizing influence on the base flow. The intensity of heat transfer in the direction perpendicular to the main flow can promote instability and leads to more intensive mixing. This fact can be used in design of bioreactors for biomass thermal conversion.

Linear stability of a convective flow generated by nonhomogeneous internal heat sources between two parallel vertical planes is investigated in the paper. A uniform magnetic field is applied in the direction perpendicular to the planes. Analytical solution for the base flow velocity and temperature distribution is obtained. Linear stability of the base flow is analyzed numerically. Numerical results show stabilizing effect of the magnetic field and destabilizing effect of inhomogeneity of internal heat sources.

Consider a layer of a viscous incompressible fluid bounded by two vertical planes. There exists a steady flow in the vertical layer caused by internal heat generation. The heat sources are distributed within the fluid in accordance with the Arrhenius law. A magnetic field of constant strength is applied in the direction perpendicular to the planes. The flow is characterized by four dimensionless parameters: the Grashof number, the Prandtl number, the Hartmann number and the Frank-Kamenetsky parameter. This problem is important in applications such as biomass thermal conversion. The objective of the study is to determine the factors that enhance mixing and lead to more efficient energy conversion. The problem is described by a system of magnetohydrodynamic equations under the Boussinesq approximation. The nonlinear system of ordinary differential equations describing the steady flow is solved numerically. Linear stability of the steady flow is investigated using the method of normal modes. The corresponding linear stability problem is solved numerically by means of a collocation method. The solution is found for different values of the parameters characterizing the problem. It is found that the increase of the Frank-Kamenetsky parameter destabilizes the flow. On the other hand, the increase of the Hartmann number stabilizes the flow.

Semi-analytical method is used in the paper to solve direct eddy current testing problem for the case where a conducting plate has a flaw in the form of a circular cylinder located on top of the plate. The system of the Maxwell’s equations for the vector potential is solved by the method of separation of variables. The coefficients of the modal expansions are evaluated numerically. In addition, complex eigenvalues are also computed numerically using the methods of theory of complex variables. Such a model can be used to access the effect of corrosion in reinforced concrete where the corroded part often has larger volume than the uncorroded part.

Linear and weakly nonlinear analysis of shallow mixing layers with free surface is performed in the present paper. It is assumed that the resistance force is not constant in the transverse direction of the flow. Linear stability problem is solved numerically for different values of the parameters of the problem. Assuming that the bed friction number is slightly smaller than the critical value we derive an amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Schrodinger equation.

Linear stability of a steady non-isothermal Couette flow between two cylinders for the case where the inner cylinder is rotating with constant angular velocity ω while the outer cylinder is at rest is investigated. The base velocity vector has two components: the azimuthal component due to rotation and the vertical component due to internal heat sources distributed within the fluid in accordance with the Arrhenius law. The base flow velocity and temperature satisfy a nonlinear boundary value problem which is solved numerically. Linearized equations for the perturbed quantities are solved by means of the collocation method based on the Chebyshev polynomials. Numerical results show that the increase in ω destabilizes the flow. Thus, rotation can be considered as one of the factors that enhances instability.

Linear stability analysis of a chemically reacting fluid motion in a pipe is performed in the present paper. The reaction rate has an Arrhenius form. The base flow and temperature distribution is obtained from the nonlinear heat equation coupled with the equations of motion. The stability of the flow with respect to asymmetric (spiral) perturbations is investigated numerically. The critical Grasshof number of the flow depends on two dimensionless parameters: the Prandtl number and the Frank-Kamenetsky parameter. The increase of both parameters has a destabilizing influence on the flow. It is shown that the second branch of a marginal stability curve corresponding to smaller critical Grasshof numbers appears as the Prandtl number increases.