Amir Gat

Dynamics of freely moving plates connected by a shallow liquid bridge

We study the dynamics of freely moving plates connected by a shallow liquid bridge via analytic and experimental methods. The gap between the plates is used as a small parameter within a lubrication approximation, reducing the problem to an Abel equation of the second kind. Analysis of the governing differential equation yields two novel physical phenomena: (1) An impulse-like peak in the force applied by the liquid bridge on the plates, obtained from a uniform asymptotic solution for small capillary numbers. (2) Both linear and non-linear oscillations of the system for the case of surfaces with low wettability, obtained from small perturbations of the system around the equilibrium point. An experimental setup examining the motion of freely moving plates was constructed, yielding experimental data which compared favorably with the analytic results and specifically displayed the predicted oscillations and impulse-like peak of the applied force. The application of the current analysis to the manipulation of solid bodies and possible future research directions are discussed.

A higher-order Hele-Shaw approximation with application to gas flows through shallow micro-channels

The classic hydrodynamic Hele-Shaw problem is revisited in the context of evaluating the viscous resistance to low-Mach compressible viscous gas flows through shallow non-uniform micro-fluidic configurations. Our recent study of gas flows through constricted shallow micro-channels indicates that the failure of the standard HeleShaw approximation to satisfy the no-slip boundary condition at the sidewalls severely restricts its applicability. To overcome this we have extended the asymptotic scheme to incorporate an inner solution in the vicinity of the sidewalls (which, in turn, allows for the characterization of the effects of channel cross-section geometry) and its matching to an outer correction. We have compared the results of the present asymptotic analysis to existing exact analytic and numerical results for straight and uniform channels and to finite-element simulations for a 90 ◦ turn and a symmetric Tjunction, which demonstrate a remarkably improved accuracy relative to the standard Hele-Shaw approximation. This suggests the present scheme as a viable alternative for the rapid performance estimate of micro-fluidic devices.

Gas flows through constricted shallow micro-channels

We study the viscous compressible flow through micro-channels of non-uniform cross-section. A lubrication approximation is applied to analyse the flow through shallow configurations whose gap width is small in comparison with the other characteristic dimensions. Focusing on channels with a symmetric constriction (or cavity) we obtain the solution to the problem by means of a Schwarz–Christoffel transformation. This analytic solution is verified by examining the convergence of numerical simulations with diminishing Reynolds number and gap width. Explicit closed-form expressions for the pressure-head and mass-flow-rate losses in terms of the geometrical parameters characterizing the constriction are presented and discussed in the context of experimental data existing in the literature.

Elastic deformations driven by non-uniform lubrication flows

The ability to create dynamic deformations of micron-sized structures is relevant to a wide variety of applications such as adaptable optics, soft robotics, and reconfigurable microfluidic devices. In this work we examine non-uniform lubrication flow as a mechanism to create complex deformation fields in an elastic plate. We consider a Kirchoff-Love elasticity model for the plate and Hele-Shaw flow in a narrow gap between the plate and a parallel rigid surface. Based on linearization of the Reynolds equation, we obtain a governing equation which relates elastic deformations to gradients in non-homogenous physical properties of the fluid (e.g. body forces, viscosity, and slip velocity). We then focus on a specific case of non-uniform Helmholtz-Smoluchowski electroosmotic slip velocity, and provide a method for determining the zeta-potential distribution necessary to generate arbitrary static and quasi-static deformations of the elastic plate. Extending the problem to time-dependent solutions, we analyze transient effects on asymptotically static solutions, and finally provide a closed form solution for a Greens function for time periodic actuations.

Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube

We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. The study focuses on viscous-elastic time-scales for which the rate of solid deformation is of the same order-of-magnitude as the velocity of the fluid. We employ an elastic shell model and the lubrication approximation to obtain a forced nonlinear diffusion equation governing the viscous-elastic interaction. In the case of an advancing liquid front into a configuration with a negligible film layer (compared with the radial deformation of the elastic tube), the governing equation degenerates into a forced porous medium equation, for which several closed-form solutions are presented. In the case where the initial film layer is non-negligible, self-similarity is used to devise propagation laws for a pressure driven liquid front. When advancing external forces are applied on the tube, the formation of dipole structures is shown to dominate the initial stages of the induced flow and deformation regimes. These are variants of the dipole solution of the porous medium equation. Finally, since the rate of pressure propagation decreases with the height of the liquid film, we show that isolated moving deformation patterns can be created and superimposed to generate a moving wave-like deformation field. The presented interaction between viscosity and elasticity may be applied to fields such as soft-robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary.

Flow patterning in Hele-Shaw configurations using non-uniform electro-osmotic slip

We present an analytical study of electro-osmotic flow in a Hele-Shaw configuration with non-uniform zeta potential distribution. Applying the lubrication approximation and assuming thin electric double layer, we obtain a pair of uncoupled Poisson equations for the pressure and depth-averaged stream function, and show that the inhomogeneous parts in these equations are governed by gradients in zeta potential parallel and perpendicular to the applied electric field, respectively. We obtain a solution for the case of a disk-shaped region with uniform zeta potential and show that the flow field created is an exact dipole, even in the immediate vicinity of the disk. In addition, we study the inverse problem where the desired flow field is known and solve for the zeta potential distribution required in order to establish it. Finally, we demonstrate that such inverse problem solutions can be used to create directional flows confined within narrow regions, without physical walls. Such solutions are equivalent to flow within channels and we show that these can be assembled to create complex microfluidic networks, composed of intersecting channels and turns, which are basic building blocks in microfluidic devices.

Asymmetric wicking and reduced evaporation time of droplets penetrating a thin double-layered porous material

We study numerically and experimentally the penetration and evaporation dynamics of droplets wicking into a thin double-layered porous material with order-of-magnitude difference in the physical properties between the layers. We show that such double-layered porous materials can be used to create highly asymmetrical wicking properties, preventing liquid droplets wicking from one surface to the other, while allowing wicking in the reverse direction. In addition, these double-layered porous materials are shown to reduce the evaporation time of droplets penetrating into the porous material, compared with a single-layered porous material of equal thickness and physical properties similar to either of the layers.

Gas flows through shallow T-junctions and parallel microchannel networks

We apply a recent extension of the Hele-Shaw scheme to analyze steady compressible viscous flows through micro T-junctions. The linearity of the problem in terms of an appropriately defined quadratic form of the pressure facilitates the definition of the viscous resistance of the configuration, relating the gas mass-flow rate to entrance and exit conditions. Furthermore, under rather mild restrictions, the performance of complex microchannel networks may be estimated through superposition of the contributions of multiple basic junction elements. This procedure is applied to an optimization model problem of a parallel microchannel network. The analysis and results are readily adaptable to incompressible flows.

Interaction forces between microfluidic droplets in a Hele-Shaw cell

Various microfluidic systems, such as chemical and biological lab-on-a-chip devices, involve motion of multiple droplets within an immersing fluid in shallow micro-channels. Modeling the dynamics of such systems requires calculation of the forces of interaction between the moving droplets. These forces are commonly approximated by superposition of dipoles solutions, which requires an assumption of sufficiently large distance between the droplets. In this work we obtain exact solutions (in the Hele-Shaw limit) for two moving droplets, and a droplet within a droplet, located within a moving immersing fluid, without limitation on the distance between the droplets. This is achieved by solution of the pressure field in a bi-polar coordinate system and calculation of the force in a Cartesian coordinate system. Our results are compared with numerical computations, experimental data and the existing dipole-based models. We utilize the results to calculate the dynamics of a droplet within a droplet, and of two close droplets, located within an immersing fluid with oscillating speed. The obtained results may be used to study the dynamics of dense droplet lattices, common to many microfluidic systems.

Transient Dynamics of Elastic Hele-Shaw Cell Due to External Forces with Application to Impulse Mitigation

We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure-field. We model the flow-field via the lubrication approximation, and the plate deformation by the Kirchhoff-Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed form expressions. We examine a uniform external pressure acting on the elastic plate over a finite region and during a finite time period, similar to the viscous-elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure, thus suggesting such configurations may be used for impact mitigation.