DOI: https://doi.org/10.15588/1607-3274-2019-1-3

### NUMERICAL ANALYSIS OF SLOW STEADY AND UNSTEADY VISCOUS FLOW BY MEANS OF R-FUNCTIONS METHOD

A. V. Artiukh, S. N. Lamtyugova, M. V. Sidorov

#### Abstract

Objective. The purpose of this paper is to compare the previously developed methods of numerical analysis of the steady and unsteady flows of a viscous incompressible fluid.
Method. The flow of a viscous incompressible fluid can be described by the system of nonlinear Navier-Stokes equations. The variables of this system are velocity, pressure, density, volume forces, and fluid viscosity. Using the stream function, the Navier-Stokes equations can be transformed to the initial-boundary problem with the differential equation of the fourth order. To solve the problem the structural variational method of R-functions and Ritz method (steady problem) or Galerkin method (unsteady problem) are used. The R-functions method allows satisfying the boundary conditions accurately and transforming them to the homogeneous,
which are the prerequisite for application of Ritz or Galerkin method. The problem transforms to the solving the system of linear algebraic equations or the system of ordinary differential equations for steady and unsteady flows respectively. The matrices elements are the scalar products in the norms of the corresponding differential operators. Numerical integration was made by means of Gaussian quadratures with 16 points. Solutions of the system of linear algebraic equations and the system of ordinary differential equations were found with the help of the Gauss method and the Runge-Kutta method with an automatic step-size control
respectively. The existence of a unique solution of the problems is proved.
Results. The computational experiments for the problem of flows of a viscous incompressible fluid for the different rectangular domains carried out.
Conclusions. The conducted experiments have confirmed that the stream function, the flow velocity, and other flow characteristics are converging to the steady state when the time is increasing. This allows us to say that the obtained methods work as expected. The further research may be devoted to the comparison of the solution methods for the non-linear problems.

#### Keywords

Navier-Stokes equations; steady flow; unsteady flow; viscous fluid; stream function; R-functions method; successive approximations method; Ritz method; Galerkin method.

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