On One Method for Solving Heat Exchange Problems When a Fluid Flows in a Cylindrical Channel

Abstract

The paper presents a high-precision solution of the problem is obtained for laminar fluid motion in a pipe with a variable boundary condition of the first kind by an additional unknown function and additional boundary conditions in the integral method. The function characterizing the temperature change in the center of the channel along a longitudinal spatial variable is considered as an additional one. Its introduction is based on the infinite velocity of heat distribution laid by the parabolic heat exchange equation. According to this equation the temperature at any point of the cylindrical channel will change immediately after the application of the boundary condition on its surface. The use of the additional function allows one to reduce the solution of the equation in partial derivatives to the solution of an ordinary equation. Additional conditions are built so that their satisfaction is adequate to the implementation of the equation at the boundary points. It is shown that the satisfaction of the equation at the points of the boundary leads to its fulfillment in the entire area with accuracy depending on the number of approximations (the number of additional conditions). The performed studies allow one to conclude that already in the third approximation, the found analytical solution in the range of the longitudinal variable practically coincides with the solution by the numerical method (finite difference method). Due to the simplicity of the defined solutions that include only algebraic polynomials, it is possible to perform research in the fields of isotherms and the velocities of their movement.