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# Construction of Inhomogeneous Velocity Fields Using Expansions in Terms of Eigenfunctions of the Laplace Operator

published 30 September 2022

2022, Vol. 18, no. 3, pp.  441-464

Author(s): Vetchanin E. V., Portnov E. A.

In this paper we present a method for constructing inhomogeneous velocity fields of an incompressible fluid using expansions in terms of eigenfunctions of the Laplace operator whose weight coefficients are determined from the problem of minimizing the integral of the squared divergence. A number of examples of constructing the velocity fields of plane-parallel and axisymmetric flows are considered. It is shown that the problem of minimizing the integral value of divergence is incorrect and requires regularization. In particular, we apply Tikhonov’s regularization method. The method proposed in this paper can be used to generate different initial conditions in investigating the nonuniqueness of the solution to the Navier – Stokes equations.
Keywords: inhomogeneous velocity field, expansion in terms of eigenfunctions, ill-conditioned system of linear algebraic equations
Citation: Vetchanin E. V., Portnov E. A., Construction of Inhomogeneous Velocity Fields Using Expansions in Terms of Eigenfunctions of the Laplace Operator, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp.  441-464
DOI:10.20537/nd220308