Showing papers on "Hele-Shaw flow published in 2021"

Viscoplastic fingers and fractures in a Hele-Shaw cell

Abstract: A study is presented of the instabilities that may arise in radial displacement flows of yield-stress fluid in a Hele-Shaw cell. Theoretically, the viscoplastic version of the Saffman–Taylor interfacial instability is predicted to arise when the yield-stress fluid is displaced by a Newtonian one. The interface is expected to remain stable, however, if the yield-stress fluid displaces the Newtonian one. A variety of experiments are then performed using an aqueous suspension of Carbopol. As predicted theoretically, the Saffman–Taylor instability is observed when the Carbopol is displaced by either air or an immiscible oil, and no instabilities are observed when the displacement is the other way around. However, when water is used in the displacement experiments, other instabilities appear that take the form of localized fractures of the Carbopol over the sections of the interface that are under tension. The fractures arise in both the stable and unstable Saffman–Taylor configurations, leading to a rich range of patterns within the Hele-Shaw cell. We argue that this pattern formation cannot be explained by a recently proposed instability of shear-thinning extensional flow, whatever the degree of effective slip over the plates of the cell. Instead, we attribute the fractures to a reduction in the fracture energy of the suspension when placed in contact with water.

Saffman-Taylor instability in a radial Hele-Shaw cell for a shear-dependent rheological fluid

Abstract: We present the study of viscous fingering in the radial hele shaw cell, using the shear-dependent rheological fluid which transitions from the Newtonian to shear thinning behaviour, at low shear rates. The experimental observations show that the characteristic features of the fingering phenomena in the case of shear-thinning flow (such as side branching, growth of second generation fingers, inhibition of tip splitting), is observed at late times, from an initial Newtonian regime. On account of the asymmetry of the fingering structure, the local shear rate is non-uniform. This leads to different rheological behaviour in different sections of the network locally, revealing contrasting local fingering patterns, specific to the respective fluid rheological state based on the local shear rate. This is also explained using the numerical simulations for such rheological fluid, in the two-phase porous media flow. The fingering features, such as the length, width are quantified as a function of the flowrate. We have also observed detachment of the fingers beyond a certain critical size of the fingering network. Finally, the stability of the fingering is also investigated at ultra low flow rates, which suggests that the duration of stability exists much beyond what is predicted using the linear stability analysis for classical Newtonian flows.

A review of one-phase hele-shaw flows and a level-set method for nonstandard configurations

Abstract: The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity. doi:10.1017/S144618112100033X

Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

Abstract: The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman–Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap .

Laminar flow and pressure loss in planar Tee joints: Numerical simulations and flow analysis

Abstract: This paper presents a numerical investigation of the laminar flow and pressure drop characteristics of planar Tee joints, a canonical flow of interest for the thermal-hydraulic design of oil-immersed power transformer windings. After formulating the problem in nondimensional form, the steady, constant property flow in planar 90 ∘ Tee joints is computed numerically by integrating the Navier–Stokes equations with fully developed upstream and downstream boundary conditions. The analysis assumes a straight-through configuration in which the straight duct holds flow in the same direction before and after the junction, whereas the flow in the side branch can either divide from the incoming flow or combine with it. The analysis starts with the description of the flow patterns that emerge in the dividing and combining cases for all mass split ratios, 0 ≤ m 1 ≤ 1 , and a wide range of straight duct to side branch width ratios, 1 ≤ α ≤ 3 , and Reynolds numbers of the common branch, 0 < Re3 ≤ 200, values that are representative of the cooling oil flow in oil-immersed transformer winding. Flow maps for planar Tee joints are then presented, showing the existence of different regions in the (Re3, m 1 )-plane that exhibit different number and location of recirculation zones. Pressure distributions and secondary loss coefficients are then computed and analyzed, providing a numerical database that is used to develop new local pressure loss correlations for planar Tee joints in an accompanying paper.

On pressure-driven Hele–Shaw flow of power-law fluids

Abstract: We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, ie a thin three-dimensional domain which is confined between two curved surfaces and contains a cyl

Nonstationary problem of morphological stability of radially displaced fluid in a Hele–Shaw cell

Abstract: For the first time, the unsteady Navier–Stokes equation with an inertial term is taken into consideration to study the interface stability of a radially displaced fluid in a finite Hele–Shaw cell. The linear order perturbation theory is used. An equation for the perturbation amplitude is obtained. The dependences of the critical size of morphological stability on the cell size, fluid properties, and displacement rate are obtained and analyzed. In the case of high displacement rates, previously unknown an unusual (reentrant) behavior is discovered in which displacement is initially unstable, and then is stable and then again unstable. The possibility of increasing the critical size of stability in the cell with an increase in the displacement rate is also theoretically demonstrated for the first time.

Gas-Liquid Flow in Small Channels: Artificial Neural Network Classifiers for Flow Regime Prediction

Abstract: To design and operate multiphase apparatus with mini- and microchannels, it is important to know how the fluids stream inside. Most literature reports the occurrence of flow regimes in dependence on gas and liquid superficial velocities in flow maps that are valid for fluids with similar properties and channels with similar geometry. Attempts to develop universally applicable flow maps show limitations in the number and variation of considered model parameters or in the number of considered flow regimes. This paper presents artificial neural network classifiers able to predict all relevant flow regimes: (a) Taylor flow, (b) bubbly flow, (c) Taylor-annular flow, (d) churn flow, (e) dispersed flow, (f) annular flow, (g) rivulet flow, and h) parallel flow in dependence on geometric and operational parameters as well as fluid properties with a high precision ( R = 0.92 . . . 0.95 and classification rates were generally above 80 % ). The classifiers were developed and validated by using more than 13,000 experimental data on gas-liquid flows extracted from 97 flow maps and are based on 7 significant dimensionless groups, namely, R e G , R e L , W e G , W e L , C a L , Θ * , and the channel form factor F C .

Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell

Abstract: Hele-Shaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on an interfacial flow with kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain the local existence of analytic solution of the moving boundary problem when the initial data is analytic. The methodology is to use complex analysis and reduce the free boundary problem to a Riemann-Hilbert problem and an abstract Cauchy-Kovalevskaya evolution problem.

A macrotransport equation for the Hele-Shaw flow of a concentrated suspension

Abstract: A depth-averaged, convection-dispersion equation is derived for the particle volume fraction distribution in the pressure-driven flow of a concentrated suspension of neutrally buoyant, non-colloidal particles between two parallel plates, by implementing a two-time-scale perturbation expansion of the suspension balance model (Nott & Brady, J. Fluid Mech., vol. 275, 1994, pp. 157–199) coupled with the constitutive equations of Zarraga et al. (J. Rheol., vol. 52, issue 2, 2000, pp. 185–220). The Taylor-dispersion coefficient in the macrotransport equation scales as , respectively.

Instability of co-flow in a Hele-Shaw cell with cross-flow varying thickness

Abstract: We analyse the stability of the interface between two immiscible fluids both flowing in the horizontal direction in a thin cell with vertically varying gap width. The dispersion relation for the growth rate of each mode is derived. The stability is approximately determined by the sign of a simple expression, which incorporates the density difference between the fluids and the effect of surface tension in the along- and cross-cell directions. The latter arises from the varying channel width: if the non-wetting fluid is in the thinner part of the channel, the interface is unstable as it will preferentially migrate into the thicker part. The density difference may suppress or complement this effect. The system is always stable to sufficiently large wavenumbers owing to the along-flow component of surface tension.

Experimental Study of the Use of Tracing Particles for Interface Tracking in Primary Cementing in an Eccentric Hele–Shaw Cell

Abstract: We present the results of the displacement flows of different Newtonian and Herschel–Bulkley non-Newtonian fluids in a new-developed eccentric Hele–Shaw cell with dynamic similarly to real field wellbore annulus during primary cementing. The possibility of tracking the interface between the fluids using particles with intermediate or neutral buoyancy is studied. The behaviors and movements of particles with different sizes and densities against the primary vertical flow and strong secondary azimuthal flow in the eccentric Hele–Shaw cell are investigated. The effects of fluid rheology and pumping flow rate on the efficiency of displacement and tracing particles are examined. Moreover, the behavior of pressure gradients in the cell is described and analyzed. Successful results of tracing the interface using particles give us this opportunity to carry out a primary cementing with high quality for the cases that the risk of leakage is high, e.g., primary cementing in wells penetrating a CO2 storage reservoir.

A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele-Shaw cell

Abstract: In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele-Shaw cell Fingering instabilities initiated at the interface between a low viscosity fluid and a high viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO$_2$ sequestration, but does not pretend to analyse this complex problem Instead, we present a detailed study of the analogous problem occurring in a Hele-Shaw cell, giving indications of possible plume patterns that can develop during the CO$_2$ injection The numerical scheme utilises a boundary element method in which the normal velocity at the interface of the two fluids is directly computed through the evaluation of a hypersingular integral The boundary integral equation is solved using a Neumann convergent series with cubic B-Spline boundary discretisation, exhibiting 6th order spatial convergence The convergent series allows the long term non-linear dynamics of growing viscous fingers to be explored accurately and efficiently Simulations in low mobility ratio regimes reveal large differences in fingering patterns compared to those predicted by previous high mobility ratio models Most significantly, classical finger shielding between competing fingers is inhibited Secondary fingers can possess significant velocity, allowing greater interaction with primary fingers compared to high mobility ratio flows Eventually, this interaction can lead to base thinning and the breaking of fingers into separate bubbles

A simple numerical method for Hele-Shaw type problems by the method of fundamental solutions.

Abstract: Hele-Shaw flows with time-dependent gaps create fingering patterns, and magnetic fluids in Hele-Shaw cells create intriguing patterns.We propose a simple numerical method for Hele-Shaw type problems by the method of fundamental solutions.The method of fundamental solutions is one of the mesh-free numerical solvers for potential problems, which provides a highly accurate approximate solution despite its simplicity.Moreover, combining with the asymptotic uniform distribution method, the numerical method satisfies the volume-preserving property.We use Amano's method to arrange the singular points in the method of fundamental solutions.We show several numerical results to exemplify the effectiveness of our numerical scheme.

L 1 -theory for reaction-diffusion hele-shaw flow with linear drift

Abstract: The main goal of this paper is to prove L 1-comparison and contraction principles for weak solutions (in the sense of distributions) of Hele-Shaw flow with a linear Drift. The flow is considered with a general reaction term including the Lipschitz continuous case, and subject to mixed homogeneous boundary conditions : Dirichlet and Neumann. Our approach combines a renormalization proceedings a la DiPerna-Lions with Kruzhkov device of doubling and de-doubling variables. The L 1contraction principle allows to handle the problem in a general framework of nonlinear semigroup theory in L 1 , taking thus advantage of this strong theory to study existence, uniqueness, comparison of weak solutions, L 1-stability as well as many further questions.

Shape of spreading and leveling gravity currents in a Hele-Shaw cell with flow-wise width variation

Abstract: We study the spreading and leveling of a gravity current in a Hele-Shaw cell with flow-wise width variations as an analog for flow in a porous medium with horizontally heterogeneous permeability. Using phase-plane analysis, we obtain second-kind self-similar solutions to describe the evolution of the gravity current's shape during both the spreading (pre-closure) and leveling (post-closure) regimes. The self-similar theory is compared to numerical simulations of the partial differential equation governing the evolution of current's shape (under the lubrication approximation) and to table-top experiments. Specifically, simulations of the governing partial differential equation from lubrication theory allow us to compute a pre-factor, which is \textit{a priori} arbitrary in the second-kind self-similar transformation, by estimating the time required for the memory of the initial condition to be forgotten (and for the current to enter the self-similar regime). With this pre-factor calculated, we show that theory, simulations and experiments agree well near the propagating front. In the leveling regime, the current's memory resets, and another self-similar behavior emerges after an adjustment time, which we estimate from simulations. Once again, with the pre-factor calculated, both simulations and experiments are shown to obey the predicted self-similar scalings. For both the pre- and post-closure regimes, we provide detailed asymptotic (analytical) characterization of the universal current profiles that arise as self-similarity of the second kind.

The Hele-Shaw flow as the sharp interface limit of the Cahn-Hilliard equation with disparate mobilities

Abstract: In this paper, we study the sharp interface limit for solutions of the Cahn--Hilliard equation with disparate mobilities.This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn--Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele--Shaw flow with optimal energy-dissipation rate.