SMALL PERTURBATION METHOD TO SOLVE THE PROBLEM OF LAMINAR BOUNDARY LAYER OF INCOMPRESSIBLE VISCOUS FLUID FLOWS

The recurrent system of second order partial differential equations serving for asymptotic approximation to the Stokes equation has been derived by the asymptotic method. The zeroth order and basic approximation is the Prandtl equation used to study the non-steady-state boundary layer. The exact analytical solutions of the Prandtl equation with a pressure gradient have been obtained.

QUSI-LAGRANGIAN INTEGRATION OF TWO-DIMENSIONAL PRANDTL AND BOUSSINESQ EQUATIONS IN THE INVISCID LIMIT

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

To the numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials

In the paper, computational schemes for solving the Cauchy problem for the singular integro-differential Prandtl equation with a singular integral over a segment of the real axis, understood in the sense of the Cauchy principal value, are constructed and justified. This equation is reduced to equivalent Fredholm equations of the second kind by inversion of the singular integral in three classes of Muskhelishvili functions and applying spectral relations for the singular integral. At the same time, we investigate the conditions for the solvability of integral Fredholm equations of the second kind with a logarithmic kernel of a special form and are approximately solved. The new computational schemes are based on applying the spectral relations for the singular integral to the integral entering into the equivalent equation. Uniform estimates of the errors of approximate solutions are obtained.