Direct numerical method for the velocity profile and the form of a laminar jet in a liquid-liquid system

An exact numerical solution for the velocity profiles and the form of a laminar jet in immiscible'Newtonian liquid-liquid systems is obtained. The main difficulty connected with the simultaneous integration of the equations of motion of the jet and of the continuous phase in the presence of a unknown interface is overcome by using an indivisible finite-difference scheme. Thus is avoided any additional iteration for defining the unknown velocities of the points of the interface. The solution of the equations of motion as written in a boundary layer approximation is a function of the following nondimensional numbers: the Reynolds numbers of both phases, the Weber and Froude numbers and the ratio of the densities. The influence of some of these parameters on the jet behaviour is illustrated as well as the influence of the initial velocity profiles at the nozzle exit. A comparison is made with some known results. Important differences are found to exist between the exact and the approximate velocity profiles and their gradients at the interface. It seems that these differences result from the comparatively inexact description of the boundary layer of the continuous phase when using moment methods.Such a conclusion limits the applicability of the approximate moment solutions to heat and mass transfer problems as well as to the jet stability analysis.