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Hydraulic fracturing for oil and gas production from shale formations, as well as natural geological phenomena, involve the propagation of thin viscous films within elastic media. For viscous fluids, stress diverges as the thickness of the film tends to zero, arresting the propagation of the film, and thus implying the contact line paradox. For free-surface films, this paradox is resolved by considering a precursor film, leading to Tanner's law. This approach was extended recently for viscous films between a thin elastic plate and a rigid solid, allowing calculation of the film propagation rate. In this work, we examine the effect of a pre-wetting layer on the rate of propagation of a viscous flow within an infinitely deep and long domain. We analyse the linear and nonlinear dynamic problems, and perform a self-similarity analysis. We find that peeling front propagation scales as time to the power of $1/9$ and $1/3$ for thin and thick pre-wetting layer limits, respectively. Our results contribute to the understanding of the contact line paradox in elastic media and the crucial role of the pre-wetting layer in resolving it.
Redox flow batteries (RFBs) are an emerging electrochemical technology envisioned towards storage of renewable energy. A promising sub-class of RFBs utilizes single-flow membraneless architectures in an effort to minimize system cost and complexity. To support multiple functions, including reactant separation and fast reactant transport to electrode surfaces, electrolyte flow must be carefully designed and optimized. In this work, we propose adding a secondary channel adjacent to a permeable battery electrode, solving for the flow field and analysing the effects on the reactant concentration boundary layer at the electrode. We find that an adjacent channel with gradually changing thickness leads to a desired nearly uniform flow through the electrode to the adjacent channel. Consequently, the thickness of the concentration boundary layer is significantly reduced, increasing reactant transport to the electrode surface to 140% of the rate of a battery with a constant width adjacent channel, and 350% of the rate with no adjacent channel. Overall, this theory provides insight into the important role of flow physics for this promising sub-class of flow batteries, and can pave the way to improved energy efficiency of such flow batteries.
Viscous flows in hyperelastic chambers are relevant to many biological phenomena such as inhalation into the lung's acinar region, and medical applications such as the inflation of a small chamber in minimally invasive procedures. In this work, we analytically study the viscous flow and elastic deformation created due to inflation of such spherical chambers from one or two inlets. Our investigation considers the shell's constitutive hyperelastic law coupled with the flow dynamics inside the chamber. For the case of a narrow tube filling a larger chamber, the pressure within the chamber involves a large spatially uniform part, and a small-order correction. We derive a closed-form expression for the inflation dynamics, accounting for the effect of elastic bistability. Interestingly, the obtained pressure distribution shows that the maximal pressure on the chamber's surface is greater than the pressure at the entrance to the chamber. The calculated series solution of the velocity and pressure fields during inflation is verified by using a fully coupled finite element scheme, resulting in excellent agreement. Our results allow the estimation of the chamber's viscous resistance at different pressures, thus enabling us to model the process of inflation and deflation.
We present a theoretical model and experimental demonstration of thin liquid film deformations due to a dielectric force distribution established by surface electrodes. We model the spatial electric field produced by a pair of parallel electrodes and use it to evaluate the stress on the liquid–air interface through Maxwell stresses. By coupling this force with the Young–Laplace equation, we obtain the deformation of the interface. To validate our theory, we design an experimental set-up which uses microfabricated electrodes to achieve spatial dielectrophoretic actuation of a thin liquid film, while providing measurements of microscale deformations through digital holographic microscopy. We characterize the deformation as a function of the electrode-pair geometry and film thickness, showing very good agreement with the model. Based on the insights from the characterization of the system, we pattern conductive lines of electrode pairs on the surface of a microfluidic chamber and demonstrate the ability to produce complex two-dimensional deformations. The films can remain in liquid form and be dynamically modulated between different configurations or polymerized to create solid structures with high surface quality.
This work analyses the viscous flow and elastic deformation created by the forced axial motion of a rigid cylinder within an elastic liquid-filled tube. The examined configuration is relevant to various minimally invasive medical procedures in which slender devices are inserted into fluid-filled biological vessels, such as vascular interventions, interventional radiology, endoscopies and laparoscopies. By applying the lubrication approximation, thin shell elastic model, as well as scaling analysis and regular and singular asymptotic schemes, the problem is examined for small and large deformation limits (relative to the gap between the cylinder and the tube). At the limit of large deformations, forced insertion of the cylinder is shown to involve three distinct regimes and time scales: (i) initial shear dominant regime, (ii) intermediate regime of dominant fluidic pressure and a propagating viscous-peeling front, (iii) late-time quasi-steady flow regime of the fully peeled tube. A uniform solution for all regimes is presented for a suddenly applied constant force, showing initial deceleration and then acceleration of the inserted cylinder. For the case of forced extraction of the cylinder from the tube, the negative gauge pressure reduces the gap between the cylinder and the tube, increasing viscous resistance or creating friction due to contact of the tube and cylinder. Matched asymptotic schemes are used to calculate the dynamics of the near-contact and contact limits. We find that the cylinder exits the tube in a finite time for sufficiently small or large forces. However, for an intermediate range of forces, the radial contact creates a steady locking of the cylinder inside the tube.
The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.
We study pressure-driven propagation of gas into a two-dimensional microchannel bounded by linearly elastic substrates. Relevant fields of application include lab-on-a-chip devices, soft robotics and respiratory flows. Applying the lubrication approximation, the flow field is governed by the interaction between elasticity and viscosity, as well as weak rarefaction and low-Mach-number compressibility effects, characteristic of gaseous microflows. A governing equation describing the evolution of channel height is derived for the problem. Several physical limits allow simplification of the governing equation and solution by self-similarity. These limits, representing different physical regimes and their corresponding time scales, include compressibility–elasticity–viscosity, compressibility–viscosity and elasticity–viscosity dominant balances. Transition of the flow field between these regimes and corresponding exact solutions is illustrated for the case of an impulsive mass insertion in which the order of magnitude of the deflection evolves in time. For an initial channel thickness which is similar to the elastic deformation generated by the background pressure, a symmetry between compressibility and elasticity allows us to obtain a self-similar solution which includes weak rarefaction effects. The presented results are validated by numerical solutions of the evolution equation.
The characteristic time of low-Reynolds-number fluid–structure interaction scales linearly with the ratio of fluid viscosity to solid Young’s modulus. For sufficiently large values of Young’s modulus, both time and length scales of the viscous-elastic dynamics may be similar to acoustic time and length scales. However, the requirement of dominant viscous effects limits the validity of such regimes to micro-configurations. We here study the dynamics of an acoustic plane wave impinging on the surface of a layered sphere, immersed within an inviscid fluid, and composed of an inner elastic sphere, a creeping fluid layer and an external elastic shell. We focus on configurations with similar viscous-elastic and acoustic time and length scales, where the viscous-elastic speed of interaction between the creeping layer and the elastic regions is similar to the speed of sound. By expanding the linearized spherical Reynolds equation into the relevant spectral series solution for the hyperbolic elastic regions, a global stiffness matrix of the layered elastic sphere was obtained. The maximal pressure difference induced by the acoustic wave on the creeping region was found to occur for identical viscous-elastic and acoustic length scales. Comparing an elastic sphere with an embedded creeping layer to a fully elastic sphere, a significant reduction in magnitude and fluctuations (with regard to wavelength) are observed for both the displacements of the solid and target strength of the sphere. This effect was most significant for identical viscous-elastic and acoustic time scales. This work relates viscous-elastic dynamics to acoustic scattering and may pave the way to the design of novel metamaterials with unique acoustic properties.
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 Kirchhoff–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-homogeneous 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 electro-osmotic 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 analyse transient effects on asymptotically static solutions, and finally provide a closed form solution for a Green’s function for time periodic actuations.
We analyse flow of non-Newtonian fluids in a Hele-Shaw cell, subjected to spatially non-uniform electro-osmotic slip. Motivated by their potential use for increasing the characteristic pressure fields, we specifically focus on power-law fluids with wall depletion properties. We derive a $p$-Poisson equation governing the pressure field, as well as a set of linearized equations representing its asymptotic approximation for weakly non-Newtonian behaviour. To investigate the effect of non-Newtonian properties on the resulting fluidic pressure and velocity, we consider several configurations in one and two dimensions, and calculate both exact and approximate solutions. We show that the asymptotic approximation is in good agreement with exact solutions even for fluids with significant non-Newtonian behaviour, allowing its use in the analysis and design of microfluidic systems involving electrokinetic transport of such fluids.
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