The effect of viscous relaxation on the spatiotemporal stability of capillary jets

Abstract

The linear spatiotemporal stability properties of axisymmetric laminar capillary jets with fully developed initial velocity profiles are studied for large values of both the Reynolds number, $Re=Q/(π\,a\,ν)$, and the Froude number, $Fr=Q^2/(π^2\,g\,a^5)$, where $a$ is the injector radius, $Q$ the volume flow rate, $ν$ its kinematic viscosity, and $g$ the gravitational acceleration. The downstream development of the basic flow and its stability are addressed with an approximate formulation that takes advantage of the jet slenderness. The base flow is seen to depend on two parameters, namely a Stokes number, $G=Re/Fr$, and a Weber number, $We=ρ\,Q^2/(π^2\,σ\,a^3)$, where $σ$ is the surface tension coefficient, while its linear stability depends also on the Reynolds number. When non-parallel terms are retained in the local stability problem, the analysis predicts a critical value of the Weber number, $We_c(G,Re)$, below which a pocket of local absolute instability exists within the near field of the jet. The function $We_c(Re)$ is computed for the buoyancy-free jet, showing marked differences with the results previously obtained with uniform velocity profiles. It is seen that, in accounting for gravity effects, it is more convenient to express the parametric dependence of the critical Weber number with use made of the Morton and Bond numbers, $Mo=ν^4 ρ^3 g/σ^3$ and $Bo=ρg a^2/σ$, as replacements for $G$ and $Re$. This alternative formulation is advantageous to describe jets of a given liquid for a known value of $g$, in that the resulting Morton number becomes constant, thereby leaving $Bo$ as the only relevant parameter. The computed function $We_c(Bo)$ for a water jet under Earth gravity is shown to be consistent with the experimental results of Clanet \& Lasheras (J. Fluid Mech. vol. 383, 1999, p. 307).