Investigation of the equation of motion of turbulent fluid under long-wave approximation (Q2703288)

The author considers two-dimensional flow of inhomogeneous fluid in a narrow channel. The corresponding boundary value problem has the form: \(\Delta\psi= F(\psi_{y},\psi,y)\) with \(\psi=0\) on the upper boundary \(y=h_1(x)\), and \(\psi= \psi_{b}\) on the lower boundary \(y=h_2(x), \psi\to\psi_{\infty}, \psi_{x}\to0\), as \(x\to-\infty\), where \(\psi\) is the stream-function. Using long-wave approximation, the above equation is reduced to the equation \(\psi_{yy}=F(\psi_{y},\psi,y)\). The proposed method allows to establish the existence of solution, and to examine the non-uniqueness of solutions for different flow conditions.