A Basic System of Equations for the Motion of a Fluid with Variable Flow Rate in the Bilateral Flow Offtake

Abstract

All theoretical studies conducted in modern hydrodynamics are based on a system of Eulerian equations, a continuity equation, and a characteristic equation [1]. In the area of hydraulics that is concerned with a one-dimensional motion, the basis for all theoretical investigations is a system of three equations: (i) a Bernoulli equation, which is a particular solution of the Eulerian system, (ii) a flow equation, which is an analog of the continuity equation, and (iii) a characteristic equation. These equations are used to study the motion of a fluid at a constant flow rate. For the case of motion of a fluid at a variable flow rate, these equations must be modified and augmented with new equations to make the system complete. Most studies were mainly concerned with formulating a “basic equation” for the motion of a fluid at a variable flow rate. For the first time, the problem of constructing a general complete system of equations for the motion of a fluid at a variable flow rate for unilateral division of the flow was brought to a logical conclusion by Kiselev [2]. To follow suit, it was our goal in this study to set up a general complete system of equations for the motion of a fluid at a variable flow rate for bilateral division of the flow. General equations for the fluid motion at a variable flow rate for bilateral flow offtake. We consider a model for constructing general equations. From the main stream flowing through a constraint space (a river, a channel, etc.), part of the fluid is uniformly and simultaneously diverted. Within an infinitesimal time interval At, the masses m lo and m ro moving with the velocities u lo and u ro are instantaneously diverted from the main stream of mass m moving with the velocity u (here the subscripts “lo” and “ro” denote the left-side offtake and right-side offtake, respectively). At the instant of diversion, the remaining mass m – m lo – m ro has the velocity u + Au = u. The lapse of time from At to dt is is a continuous process involving the change in mass. One can assume that at any instant of the unit time interval within the volume of fluid AW, the system in question can be partitioned into two parts: one is the mass of the main stream m, and the other is the diverted masses m lo and m ro . The space of volume AW is filled with fluid masses of the main and diverted flows. The fluid motion is assumed to be uninter