Jets into liquid under gravity

We study the flow of a heavy, viscous, possibly non-Newtonian axisymmetric jet of liquid of density ρ falling under gravity g into a lighter liquid of density . If the change in the momentum of the entrained lighter liquid is neglected the jet will ultimately reach a modified Torricelli limit with a speed given by \[ U(x) = \left[2\frac{\delta\rho}{\rho}gx \right]^{\frac{1}{2}} \] and an asymptotic radius \begin{equation} a(x) = \left[\frac{2Q^2}{(\delta\rho /\rho)gx} \right]^{\frac{1}{4}}, \end{equation} where x is the downstream distance, and 2Q is the volume flow. An exact asymptotic solution perturbing the Torricelli limit with effects of surface tension, viscosity and elasticity is given in powers of x−¼. An extended unsteady problem including effects of entrainment is formulated in terms of nonlinear ordinary differential equations which also account for weak radial variations of the velocity across the cross-section of the jet. These equations are solved in a boundary-layer approximation which gives \begin{equation} a(x)\approx 1.171\left(\frac{\tilde{\rho}}{x}\right)^{\frac{1}{20}}\frac{Q^{\frac{3}{10}}\tilde{\mu}^{\frac{1}{5}}}{(\delta\rho g)^{\frac{1}{4}}}, \end{equation} where is the viscosity of the ambient fluid. Equation (1) is in agreement with experimental observations of jets of liquid into air. Equation (2) is in agreement with experimental observations of jets of liquids into liquids.