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ordinary differential
A study is made of a method of numerical solution of the system of ordinary differential equations describing the flow of a two-phase medium in a Laval nozzle in the one-dimensional approximation.
      
Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2].
      
After acceleration the rotation rate is determined from the ordinary differential equation describing the process of deceleration of the system as a whole.
      
For periodic perturbations this equation reduces to an infinite system of ordinary differential equations for the amplitudes of the Fourier harmonics.
      
The behavior of the disturbances is finally described by a system of ordinary differential equations with right sides in the form of power series in the amplitudes.
      
These generally involve self-similar solutions for which the corresponding ordinary differential equation can be integrated exactly.
      
In order to introduce arbitrary functions into the solutions of the equations of the axisymmetric boundary layer the problem is reduced to an overdetermined system of ordinary differential equations.
      
The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations.
      
It is shown that the system of equations for the two-phase multicomponent flow process, together with the equations of phase equilibrium, reduces to a system of two ordinary differential equations for the pressures in the gas and liquid phases.
      
A numerical solution of the system of second-order ordinary differential equations gives the ranges of the governing parameters on which self-similar solutions for the gas flow in a rotary channel can exist.
      
The problem is reduced to the numerical solution of ordinary differential equations.
      
A method for determining the eigenvalues of the linear stability problem is developed on the basis of Floquet theory, spectral representation of the variables, and multistep methods of integration of ordinary differential equations.
      
By introducing self-similar radial velocity and temperature profiles, the problem is reduced to a system of ordinary differential equations which are solved numerically.
      
By a self-similar change of variables the problem can be reduced to a second-order ordinary differential equation.
      
A solution form which makes it possible to reduce the complete system of partial differential equations to a system of ordinary differential equations is found.
      
As a result, the problem of calculating the longitudinal distributions of the amplitudes of the normal modes is reduced to the solution of an infinite-dimensional inhomogeneous system of ordinary differential equations.
      
For the next two examples, an analytical solution is found, while the solution of the last problem is reduced to a system of ordinary differential equations.
      
In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front.
      
For example, in the case of the problem of expansion flow in the vicinity of the exterior obtuse angle treated in this paper, the Bernoulli integral may be used to reduce the problem to a set of ordinary differential equations.
      
The model is based on the method of trajectories, in the case of which a set of partial equations can be reduced to a set of ordinary differential equations written for derivatives along ion trajectories.
      
 

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