Investigation of the boundary value problem with partially unknown boundary for exponential vorticity (Q2703266)

The authors study the flow of incompressible fluid in curvilinear channel with an unknown part of boundary. The equation for stream-function \(\Psi\) has the form \(\Psi_{xx}+\Psi_{yy}-{i\over y}\Psi_{y}= -y^{2i}\gamma e^{\alpha\Psi},\;i=1,2.\) The pressure \(P(x,y)\) is determined by the equation \((\Psi_{x}^2+\Psi_{y}^2)/2y^{2i}+P/\rho=H(\Psi)\), where \(\rho\) is the density. The boundary condition on the known part of the boundary \(y=f_{2}(x)\) is \(\Psi(x,f_2)=\Psi_{w}(x)\), the position of other boundary \(y=f_1(x)\) is unknown, and the corresponding boundary conditions are \(\Psi(x,f_1)= 0\) and \(P_1(x,f_1)=P_2(x,f_2)\), where \(P_1(x,y)=P(x,y)\) for \(y>f_1(x)\), \(p_2(x,y)=P(x,y)\) for \(x>0\) and \(f_2(x)\leq y\leq f_1(x)\). A solution of the problem is obtained in analytical form.