Session BK: Free Surface Flows II

Angling hydraulic jumps

We present an experimental and mathematical study of the normal impact of a jet onto an inclined solid surface, focusing on the characteristics of the hydraulic jump. The angle of the surface is varied between vertical and horizontal positions, using both flat and curved (patterned) surfaces. Comparisons of the outer envelope of the hydraulic jump are made with the ballistic theory and the model of Edwards, Howison, Ockendon, \& Ockendon.   

Nonlinear Dynamics in Viscoelastic Jets

Instabilities in free surface continuous jets of non-Newtonian fluids, although relevant for many industrial processes, remain poorly understood in terms of fundamental fluid dynamics. Inviscid, and viscous Newtonian jets have been studied in considerable detail, both theoretically and experimentally. Instability in viscous jets leads to regular periodic coiling of the jet, which exhibits a non-trivial frequency dependence with the height of the fall. Here we present a systematic study of the effect of viscoelasticity on the dynamics of continuous jets of worm-like micellar surfactant solutions of varying viscosities and elasticities. We observe complex nonlinear spatio-temporal dynamics of the jet, and uncover a transition from periodic to quasi-periodic to a multi-frequency, broad-spectrum dynamics. Beyond this regime, the jet dynamics smoothly crosses over to exhibit the ``leaping shampoo'' or the Kaye effect. We examine different dynamical regimes in terms of scaling variables, which depend on the geometry (dimensionless height), kinematics (dimensionless flow rate), and the fluid properties (elasto-gravity number) and present a regime map of the dynamics of the jet in terms of these dimensionless variables.   

Role of molecular phonons in water evaporation

The conditions existing at the interface of water and its vapor during evaporation have been controversial. Earlier thermocouple measurements of the temperatures in the liquid and vapor phases indicated the interfacial vapor temperature was as much as 7.8 $^\circ$C greater than that in the liquid. In more recent studies, when the vapor was heated electrically, interfacial temperature discontinuities of as much as 27.83 $^\circ$C were reported. Only thermocouple measurements have indicated temperature discontinuities. The validity of the reported temperature discontinuities has been investigated using the quantum-mechanically based statistical-rate theory (SRT) to predict the temperature discontinuities found in three different experimental investigations. It is found that from SRT the temperature discontinuities are correctly predicted up to 15.69 $^\circ$C, but larger values cannot be confirmed because of uncertainties in the pressure measurements. By taking the limit of $\hbar \omega / k_b T$ going to zero for all of the molecular phonons, SRT can be reduced to the classical result, but it no longer gives a valid prediction of the temperature discontinuities.   

A Model for Polygonal Hydraulic Jumps

We present a model for the shape of polygonal hydraulic jumps discovered by Ellegaard et al. 1998 and modeled there by a force balance (an inward push of gravity and an outward pull of viscous stresses) on the ``surface roller" in the jump-region, which is known to be a prerequisite for the occurrence of polygonal jumps. We develop this model, replacing their unexplained ``line tension" by a more detailed modeling of the flow in the roller, including the tangential flow. We solve a simplified model exactly in which each polygon exists in a finite region of parameter space with shapes very similar to those observed in experiments. The number of corners $N$ in the polygon scale as $N \sim (Q \nu)^2 h_o^{-4} h_i^{-3}$, in terms of the volumetric flux $Q$, the inner height $h_i$, the outer height $h_o$ and the viscosity $\nu$. In contrast to recent work by Bush et al. 2006, our model does not include surface tension explicitly, but since it affects the radius of the jump it will also affect $h_i$. A reduction of the surface tension will reduce $h_i$ and therefore increase $N$. If the reduction is strong enough, this could restore the circular symmetry.   

Stationary condensation and draining of a liquid film on a horizontal disk

On the top of a subcooled horizontal disk, a liquid film condenses from saturated vapor. The liquid is forcedly removed at the disk edge, to which it is conveyed in a radial flow. Stationary regimes of the flow are studied such that (i) gravity is negligible, being eclipsed by capillary forces; (ii) the maximum film thickness is much smaller than the disk radius; and (iii) the slow-flow lubrication approximation is valid. A highly nonlinear differential equation for the film thickness as a function of the radial coordinate is obtained. The (two-dimensional) fields of velocities, temperature, and pressure in the film are explicitly determined by the radial profile of its thickness. The equilibrium is controlled by two parameters: (i) the vapor-disk difference of temperatures and (ii) the mass transfer, or the liquid outflow, rate. For the flow regimes with a nearly uniform film thickness, the governing equation linearizes, and the film interface is analytically predicted to have a concave-up quartic parabola profile. Thus, perhaps counterintuitively, the liquid film is thicker at the edge and thinner at the center of the disk. This work is a part of research on low-gravity condensation-flow processes for conditioning cryogenic liquid acquisition devices to be used in space-based propulsion systems.   

Nonlinear interfacial dynamics and mechanisms of liquid transport in a gas-liquid core-annular flow

An experimental and theoretical study of two-phase core-annular flow in a cylindrical pipe is carried out with purpose of illustrating fundamental mechanisms of mucus propulsion by air flow in lung airways. A highly viscous fluid lining the inner wall of the pipe is driven by high pressure air flow at constant volumetric flux. We derive a nonlinear evolution equation based on the lubrication approximation for the interface under a certain closure turbulence model for the air flow. We study numerically the interface evolution of an initially axisymmetric disturbance of the annular film of viscous liquid and compare with the preliminary data collected from the experimental setup. The mean thickness of the liquid layer in the experiment can be predicted using this model, and the existence of the ring-like waves observed in the experiments is confirmed by the interfacial dynamics of the model.   

A theoretical model for the prediction of fingering in the flow upstream of a circular hydraulic jump

Experiments and numerical simulations of a circular hydraulic jump in a liquid metal have indicated, under certain conditions, the presence of ``fingering'' in the high-speed region upstream of the jump location. Rather than proceeding in the usual thin sheet to the jump, the liquid forms radial fingers (rivulets) that transport fluid to the jump location. The phenomenon appears to be due to the reduced wettability of the substrate by the high-surface-tension liquid metal. A theoretical model for the formation of fingers is presented, based on the use of the Bernoulli and continuity equations to obtain the pressure and flow rate, respectively, in the fingers, and the assumption that the interface curvatures at the symmetry line between fingers are equal and opposite to yield constant pressure there, allowing the number of fingers to be determined. For a sufficiently wettable (contact angle of roughly $\pi $/4) substrate and low jet velocities, no fingering solutions exist. A similar treatment determines the transition point between fingers and the circular hydraulic jump. Comparisons with experiments and numerical simulations will be presented.   

The dynamics of corona formation

Experiments show that the corona, the thin liquid sheet ejected by liquid impact, can have a variety of shapes, ranging from a concave bowl to an open tube with nearly straight wall and even a closed bell. Here we examine the formation process via a simple model that tracks how the centerline of the ejected sheet evolves over time. We restrict to axisymmetric shapes. Viscous dissipation and airflow effects are assumed to be negligible. During impact, the sheet-ejection location expands radially and the ejection speed decreases over time. To account for these effects in a simple way, we assume that the kinetic energy of the impact dominates over the surface energy and that the projectile is a sphere. As a result, the ejection location $R_J(t)$ expands as $\sqrt{a_0 U_0 t}$, where $a_0$ is the projectile radius, $U_0$ the impact speed and $t$ the time elapsed since impact. The ejection speed is simply $dR_J/dt$. Our calculation shows that, as a result of this kinematic condition, a curved profile is generated even without surface tension. More intricate shapes are possible when surface tension effects are included.   

Dynamics of complete wetting liquid under evaporation

The dynamics of a contact line under evaporation and total wetting conditions is studied taking into account the divergent nature of evaporation near the border of the liquid, as evidenced by Deegan et al.~[Nature \textbf{389}, 827 (1997)]. Complete wetting is assumed to be due to Van der Waals interactions. The existence of a precursor film at the edge of the liquid is shown analytically and numerically. The length of the precursor film is controlled by Hamacker constant and evaporative flux. Past the precursor film, Tanner's law is generalized accounting for evaporative effects.   

Capillary rise of a liquid between two vertical plates making a small angle

The penetration of a wetting liquid in the narrow gap between two vertical plates making a small angle is analyzed in the framework of the lubrication approximation. At the beginning of the process, the liquid rises independently at different distances from the line of intersection of the plates, except in a small region around this line where the effect of the gravity is negligible. The maximum height of the liquid initially increases as the cubic root of time and is attained at a point that reaches the line of intersection only after a certain time. At later times, the motion of the liquid is confined to a thin layer around the line of intersection whose height increases as the cubic root of time and whose thickness decreases as the inverse of the cubic root of time. The evolution of the liquid surface is computed numerically and compared with the results of a simple experiment.