Splitting a Liquid Jet
Original entry: Donald Aubrecht, APPHY 226, Spring 2009
"Splitting of a Liquid Jet"
Srinivas Paruchuri and Michael P. Brenner
Physical Review Letters 98(13) 134502 (2007)
Soft Matter Keywords
dominant balance, liquid jet splitting, Navier Stokes
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 excellent 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 filament splitting events have already 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 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 is made that variations in the <math>y</math> direction are large compared to variations in the <math>x</math> direction.
A splitting event is defined as the vanishing of the cross sectional thickness, <math>h(x,t)</math>, in finite time. With an applied tangential stress, results depend on the magnitude of the stress. For a small applied stress, surface tension balances the stress and the jet evolves to a steady state noncircular cross section. For a tangential stress greater than <math>0.39 \gamma/a^2</math> (where <math>\gamma</math> is the surface tension and <math>a</math> is the radius of the nozzle), no noncircular steady state exists and the jet splits into two filaments. See figure 2 for plots of the time evolution of the cross sectional thickness and velocity.
To construct approximate analytical solutions to the numerical results shown in figure 2, the authors consider different dominant balances in the three regimes of the solution. For the lamellar region, fluid inertia and the applied tangential stress are the dominant forces. With this balance of terms and assuming the ansatz <math>h(x) = f(x) \times (t^*-t)^p</math> and <math>v(x) = (t^*-t)^g(x)</math>, the analytical solution converges to the theoretical value of <math>2/3(\tau_0 / \rho)</math> predicted by the numerical simulation (<math>\tau_0</math> is the applied tangential stress and <math>\rho</math> is the fluid density). See figure 3(a) for comparison of the analytical approximation with the numerical predictions.
The jump region and outer region are approximated by a similarity solution of the form
where <math>\xi</math> is the similarity variable <math>(x-x_0)/l(t)</math>. <math>x_0</math> is the location of the jump region and <math>l(t) = \mu/(\rho a) \times (t^*-t)</math>. Figure 3(b) shows that the numerical results converge nicely to the similarity solution. These results demonstrate that a finite tangential stress, large enough to overcome surface tension, can cause a liquid jet to split into two filaments.
The authors also explore whether an applied normal stress can cause the splitting of jet. Numerical results indicate the three characteristic regions exist for this case, but there is no finite time solution that leads to splitting. This arises from dominant balances in the the lamellar and jump/outer regions that are different from the tangential stress case. The balance between surface tension and viscous stress in the jump region result in a solution that has the minimum thickness of the lamellar region decreasing exponentially in time. Hence, splitting of the jet does not occur in finite time for the case of an applied normal stress.
written by Donald Aubrecht