Two-stream interaction problem and its application to mass transport

Classical boundary layer theory is effectively extended to study two-stream interaction problems. In particular, by proper scaling we invent a universal curve which relates the second-derivative initial condition of the normalized stream function to the first-derivative free-stream condition. This relationship is independent of any fluid property, and the Blasius flow profiles in respective streams are connected by matching at the interface. Accurate approximations to the universal curve are constructed by piecewise third-order polynomials. This technique finds an excellent application to mass transport problem of solvents from one immiscible fluid to another, namely, extraction of fluid as an interfacial problem. Analytical formula is obtained to show explicit dependence of mass flux on the nonlinear Blasius profiles, partition coefficient, Schmidt numbers, ratio of mass transfer coefficients, ratio of kinematic viscosities as well as the contact length of the two immiscible fluids. In other words, given these physical parameters, we directly obtain the mass flux simply by numerical quadrature of integrals involving the Blasius profiles.