Abstract
The object of research is obtaining general integrals and some particular solutions for two common flow conditions of incompressible liquid – laminar and averaged turbulent flow. Mathematical description is based on the system of equations of motion in stresses (Navier) and its special case for the Newtonian liquid. A condition of integrating the equations is the constancy of pressure drop and viscosity along the flow. The block schemes of obtaining the general integrals for flow in a pipe and turbulent flow on a plate are represented. Are as a result, three new general integrals and four particular solutions, which are compared with the known equations, were found. It was shown that the integrals of the Navier equation describe the distribution of tangential stress for turbulent flow. An analysis of solutions for the distribution of velocity showed that the Poiseuille equation for laminar flow in a pipe and the Blasius curve for laminar flow on a plate are particular solutions of one general integral. An analysis of the particular solutions made it possible to estimate the thickness of the laminar sublayer under turbulent flow condition. The results of the work create prerequisites for a more detailed further analysis of laminar and turbulent flows.
Highlights
The problems of hydrodynamics are categorized into internal and external, depending on the position of hard surface relative to the flow
The first problem includes the flows of working media in the equipment for thermal power engineering, hydraulics and hydrotechnology
The flow in a round pipe refers to the internal problem, and the flow around a plate – to the external
Summary
The problems of hydrodynamics are categorized into internal and external, depending on the position of hard surface relative to the flow. Both problems have great significance because of their wide spread in technology, which led to a detailed study of these flows for a long period of time In both problems, they search for the distribution of speed and tangential stresses along the radius of a pipe or near the flat surface of a plate under the laminar and turbulent condition [1,2,3,4,5]. The turbulent flowing condition, for which there are no exact solutions, is of great practical value In this case, the calculation is conducted on the basis of experimental data and different physical models, implemented in the form of semi-empirical equations and computer programs [2,3,4, 7]. Relevant appears the search for differential equations and their exact solutions, carried out according to the classical scheme and implemented, for example, in problems of thermal conductivity, theory of elasticity and others
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