An analytical treatment of the formation of one-dimensional steady shock waves in uniform and diverging ducts
Boundary value problems of the form \((*)\quad \epsilon(d^ 2y/dx^ 2)=f(x,y)(dy/dx)+g(x,y), a<x<b\), y(a,\(\epsilon)\), y(b,\(\epsilon)\) prescribed, whose solutions model the behavior of one-dimensional steady shock waves in ducts of uniform and variable cross-section have been studied by singular perturbation techniques. The sufficient conditions for the existence of solutions which have a 'shock layer structure' at a point \(x_ 0\in(a,b)\) have been found, the position \(x_ 0\) of the shock layer has been located, and the 'states' separated by a solution of (*) having such an interior transition are determined.
Rankine-Hugoniot condition, Prandtl's relation
ducts of uniform and variable cross-section
scientific article; zbMATH DE number 3864826