A water jet of radius emerges horizontally into still air

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Instability, transition and turbulence

Fluid Dynamics, Ch Hydrodynamic instability: the normal-mode method

Problem

A water jet of radius

a = 0.80\,\mathrm{mm}

emerges horizontally into still air. Very close to the nozzle exit, the jet's outer edge is a thin annular shear layer of thickness

\delta = 0.050\,\mathrm{mm}

separating the jet core from the still air; treat this local layer as the plane

\tanh

mixing layer of the free shear flows studied for planar jets and wakes, itself fed by the boundary layer that grew along the inside of the nozzle. Surface tension against air is

\gamma = 0.0728\,\mathrm{N\,m^{-1}}

. ((a)) The

\tanh

profile has an inflection point, which by Rayleigh's theorem permits inviscid instability without guaranteeing it; Michalke's numerical solution of Rayleigh's equation shows that the layer is in fact unstable for

0 < k\delta < 1

and neutral beyond. Treating the layer, as a rough model, as a shear layer within a single fluid, find the shortest unstable wavelength

\lambda_\delta^\ast

, at the edge of this band where

k\delta = 1

. ((b)) The jet as a whole breaks up by the axisymmetric capillary instability, whose growth rate is fastest at

ka = 0.697

, the same maximisation (from the tabulated ratio

I_1(ka)/I_0(ka)

) that gives a wavelength of

9.02a

. Find this fastest-growing wavelength

\lambda_{\rm cap}

and, using conservation of the liquid volume in one wavelength when it collects into a single spherical drop, the drop radius

R

. ((c)) Compare

\lambda_{\rm cap}

with

\lambda_\delta^\ast

. Does the capillary wavelength lie inside the shear layer's unstable band? Say whether these two wavelengths alone can tell you which instability sets the size of the drops, and what further information you would need to decide.

Browse the problem library