Paul H. Steen

Dynamics of inviscid capillary breakup : collapse and pinchoff of a film bridge

An axisymmetric film bridge collapses under its own surface tension, disconnecting at a pair of pinchoff points that straddle a satellite bubble. The free-boundary problem for the motion of the film surface and adjacent inviscid fluid has a finite-time blowup (pinchoff). This problem is solved numerically using the vortex method in a boundary-integral formulation for the dipole strength distribution on the surface. Simulation is in good agreement with available experiments. Simulation of the trajectory up to pinchoff is carried out. The self-similar behaviour observed near pinchoff shows a ‘conical-wedge’ geometry whereby both principal curvatures of the surface are simultaneously singular – lengths scale with time as t 2/3 . The similarity equations are written down and key solution characteristics are reported. Prior to pinchoff, the following regimes are found. Near onset of the instability, the surface evolution follows a direction dictated by the associated static minimal surface problem. Later, the motion of the mid-circumference follows a t 2/3 scaling. After this scaling ‘breaks’, a one-dimensional model is adequate and explains the second scaling regime. Closer to pinchoff, strong axial motions and a folding surface render the one-dimensional approximation invalid. The evolution ultimately recovers a t 2/3 scaling and reveals its self-similar structure.

Capillarity-based switchable adhesion.

Drawing inspiration from the adhesion abilities of a leaf beetle found in nature, we have engineered a switchable adhesion device. The device combines two concepts: The surface tension force from a large number of small liquid bridges can be significant (capillarity-based adhesion) and these contacts can be quickly made or broken with electronic control (switchable). The device grabs or releases a substrate in a fraction of a second via a low-voltage pulse that drives electroosmotic flow. Energy consumption is minimal because both the grabbed and released states are stable equilibria that persist with no energy added to the system. Notably, the device maintains the integrity of an array of hundreds to thousands of distinct interfaces during active reconfiguration from droplets to bridges and back, despite the natural tendency of the liquid toward coalescence. We demonstrate the scaling of adhesion strength with the inverse of liquid contact size. This suggests that strengths approaching those of permanent bonding adhesives are possible as feature size is scaled down. In addition, controllability is fast and efficient because the attachment time and required voltage also scale down favorably. The device features compact size, no solid moving parts, and is made of common materials.

Capillary Surfaces: Stability from Families of Equilibria with Application to the Liquid Bridge

A method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge. Axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The method delivers: (i) stability envelopes in the {L, V, B} parameter space for constant-pressure and constant-volume disturbances (generating new and recovering classical results), (ii) the number of unstable modes for any equilibrium (state of instability) once the stability of one equilibrium state is known (e. g. that of the sphere) based on (iii) a demonstration that all known families of equilibria are connected. The method uses ‘preferred’ bifurcation diagrams, a plot of volume V against pressure p. The state of instability of an equilibrium shape relative to its neighbours is immediate from this plot. In addition, an invariant wavenumber classification is introduced and used to label the numerous families of liquid bridge equilibria. The preferred diagram method, which is based on properties of the Jacobi equation, gives stronger results than classical bifurcation theory. Application to other capillary surfaces, including drops and non-axisymmetric shapes, is discussed.

Capillary oscillations of a constrained liquid drop

An inviscid spherical liquid drop held by surface tension exhibits linear oscillations of a characteristic frequency and mode shape (Rayleigh oscillations). If the drop is pinned on a circle of contact the mode shapes change and the frequencies are shifted. The linear problem of inviscid, axisymmetric, volume-preserving oscillations of a liquid drop constrained by pinning along a latitude is solved here. The formulation gives rise to an integrodifferential boundary value problem, similar to that for Rayleigh oscillations, and for oscillations of a drop in contact with a spherical bowl [M. Strani and F. Sabetta, J. Fluid Mech. 141, 233 (1984)], only more constrained. A spectral method delivers a truncated solution to the eigenvalue problem. A numerical routine has been used to generate the eigenfrequencies/eigenmodes as a function of the location of the pinned circle of constraint. The effect of pinning the drop is to introduce a new low-frequency eigenmode. The center-of-mass motion, important in applicat...

Instability of rotund capillary bridges to general disturbances: Experiment and theory

Abstract Interfacial tension holds a fixed volume of incompressible fluid between coaxial circular endplates forming a capillary bridge. As the endplates are brought together very slowly the bridge fattens and eventually changes shape. In contrast to the much-studied configuration of slowly separating endplates (slender bridges), the rotund axisymmetric bridge is first unstable to nonaxisymmetric disturbances. We obtain the boundary of marginal stability (constant volume disturbances) as it depends on the aspect ratio and the volume of the bridge using an energy stability method. Furthermore, we map out the stability limit by experiment for a range of aspect ratios using neutrally buoyant fluids to simulate a zero-gravity environment. Theory and experiment are in excellent quantitative agreement and predict instability when the free interface meets each endplate tangent to its flat surface.

Computational efficiency and approximate inertial manifolds for a Bénard convection system

SummaryA computational comparison between classical Galerkin and approximate inertial manifold (AIM) methods is performed for the case of two-dimensional natural convection in a saturated porous material. For prediction of Hopf and torus bifurcations far from convection onset, the improvements of the AIM method over the classical one are small or negligible. Two reasons are given for the lack of distinct improvement. First, the small boundary layer length scale is the source of the instabilities, so it cannot be modeled as a “slave” to the larger scales, as the AIM attempts to do. Second, estimates based on the Gevrey class regularity of solutions to the governing equations show that the classical and AIM methods may be virtually equivalent. It is argued that these two reasons are physical and mathematical reflections of one another.

Heat transfer and solidification in planar-flow melt-spinning: high wheelspeeds

Abstract Operation of the planar-flow melt-spinning process in a regime where the wheelspeed is high relative to the (average) solidification rate is studied. A distinct solidification front occurs. An analysis based on the governing energy and momentum equations shows that heat transfer and solidification are only weakly coupled to fluid flow. For long puddles and high wheelspeeds the governing equations decouple, leaving a Stefan problem for the shape of the solidification front and temperature fields. This problem has an analytic solution. The linear front corresponding to thin ribbons is a limiting case. Nonequilibrium kinetics at the freeze interface and undercooling of the melt are included in the general solution. These influences on solidification are thereby examined. Results are compared to previous solidification models and experiments.

Time-periodic convection in porous media: transition mechanism

We resolve the disturbance structures that destabilize steady convection rolls in favour of a time-periodic pattern in two-dimensional containers of fluid-saturated porous material. Analysis of these structures shows that instability occurs as a travelling wave propagating in a closed loop outside the nearly motionless core. The travelling wave consists of five pairs of thermal cells and four pairs of vorticity disturbances in the case of a square container. The wave speed of the thermal disturbances is determined by an average base-state velocity and their structure by a balance between convection and thermal diffusion. Interpretation of the ‘exact’ solution is aided by a one-dimensional convection-loop model which correlates (i) point of transition, (ii) disturbance wavenumber, and (iii) oscillation frequency given the base-state temperature and velocity profiles. The resulting modified Mathieu-Hill equation clarifies the role of the vertical pressure gradient, induced by the impenetrable walls, and the role of the base-state thermal layer.

Low-dissipation capillary switches at small scales

A system of two coupled capillary surfaces is made to switch between its stable states via mechanical and electrochemical disturbances. The bistable switch is experimentally demonstrated using water droplets, where the mechanical activation or “toggle” is achieved by a momentary air-pressure change. Requirements for capillary switches to avoid viscous dissipation are described and strategies for utilizing capillary switches for transporting other liquids or solids are discussed. Addressability of individual switches is achieved using electrochemical activation via a redox surfactant, where surface tension of one element of the switch is changed relative to the other.

Transition to oscillatory convective heat transfer in a fluid-saturated porous medium

The transition from steady to time-periodic motion in the analog of Benard convection in a two-dimensional region of fluid-saturated porous media is studied by means of an eigenfunction expansion and a branch-tracing technique. This method leads to the location of the transition at Rayleigh number 9.9 times that at convection onset. The small-amplitude motion has a period 0.012 times shorter than the thermal diffusion time and comes into existence through a Hopf bifurcation. The structure and time progression of the destabilizing disturbance indicates that the dominant effect is an instability convected by the base flow whose strength is coupled to the steepening thermal boundary layers. The effects of truncation of the expansion on the prediction of the transition and its mechanism are discussed.