# Splitting a Liquid Jet

"Splitting of a Liquid Jet"

Srinivas Paruchuri and Michael P. Brenner

Physical Review Letters **98**(13) 134502 (2007)

## Contents

## Soft Matter Keywords

dominant balance, liquid jet splitting, Navier Stokes

## Summary

Paruchuri and Brenner develop a theoretical explanation for the splitting of a circular cross section liquid jet into two separate filaments. Beginning from Navier Stokes in the limit that jet cross section changes slowly as it moves downstream from the nozzle, they demonstrate that only when sufficiently large tangential stresses are present does the jet split. A full numerical simulation is compared to analytical approximations with full agreement. Looking at the simulation results, the authors find that there are three distinct regions as one passes from the center of the jet to the outside: 1) a lamellar region characterized by a flat thickness profile and linear velocity field, 2) an outer region characterized by a nearly time independent thickness and velocity, and 3) a jump region with a large jump in the velocity field connecting the lamellar and outer regions. Dominant force balances for these regions are proposed and tested against simulation.

## Practical Application of Research

The authors note that already, filament splitting events have been observed in electrospinning systems. Though these events have been documented, little had been done to understand the mechanism behind their occurrence and they typically appear in uncontrolled situations. This work proposes a mechanism for the splitting and paves the way for future experimental and theoretical work aimed at better controlling jets and splitting events. If jet splitting were to become controllable, it could provide a new route to producing very small fibers.

## Mechanism for Splitting Events

The authors begin from Navier Stokes for a fluid moving out of a circular nozzle. Assuming an in-plane force, <math>f</math>, large enough to make velocities in the cross section of the jet (denoted by <math>u_\parallel</math>) much larger than the axial velocity (denoted by <math>u_z</math>), and that the radius of the jet is much smaller than the length scale of the deformation in the axial direction, the equation of motion reduces to:

<math>\rho(u_z\delta_zu_\parallel + u_\parallel \cdot \nabla_\parallel u_\parallel) = -\nabla_\parallel p + \mu\nabla^2_\parallel u_\parallel + f</math>

These assumptions also allow the axial velocity to be considered constant and replaced by <math>u_z \approx Q/(\pi a^2)</math>. The authors also assume that near a splitting event, the jet cross section becomes long and narrow with a thickness <math>h(x,t)</math>. <math>x</math> is taken to represent the long axis along the cross section, and the assumption that variations in the <math>y</math> direction are large compared to variations in the <math>x</math> direction.

written by Donald Aubrecht