Hele-Shaw Flow Research Articles

Pattern Formation in Miscible Rotating Hele-Shaw Flows Induced by a Neutralization Reaction

Pattern Formation in Miscible Rotating Hele-Shaw Flows Induced by a Neutralization Reaction

A review of one-phase Hele-Shaw flows and a level-set method for nonstandard configurations

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

Viscous normal stresses and fingertip tripling in radial Hele-Shaw flows.

Viscous fingering in radial Hele-Shaw cells is markedly characterized by the occurrence of fingertip splitting, where growing fingered structures bifurcate at their tips, via a tip-doubling process. A much less studied pattern-forming phenomenon, which is also detected in experiments, is the development of fingertip tripling, where a finger divides into three. We investigate the problem theoretically, and employ a third-order perturbative mode-coupling scheme seeking to detect the onset of tip-tripling instabilities. Contrary to most existing theoretical studies of the viscous fingering instability, our theoretical description accounts for the effects of viscous normal stresses at the fluid-fluid interface. We show that accounting for such stresses allows one to capture the emergence of tip-tripling events at weakly nonlinear stages of the flow. Sensitivity of fingertip-tripling events to changes in the capillary number and in the viscosity contrast is also examined.

Growing Interface with Phase Separation and Spontaneous Convection during Hydrodynamically Stable Displacement.

The displacement of one fluid by another is an important process, not only in industrial and environmental fields, such as chromatography, enhanced oil recovery, and CO2 sequestration, but also material processing, such as Lost Foam Casting. Even during hydrodynamically stable fluid displacement where a more viscous fluid displaces a less viscous fluid in porous media or in Hele-Shaw cells, the growing interface fluctuates slightly. This fluctuation is attributed to thermodynamic conditions, which can be categorized as the following systems: fully miscible, partially miscible, and immiscible. The dynamics of these three systems differ significantly. Here, we analyze interfacial fluctuations under the three systems using Family–Vicsek scaling and calculate the scaling indexes. We discovered that the roughness exponent, , and growth exponent, , of the partially miscible case are larger than those of the immiscible and fully miscible cases due to the effects of the Korteweg convection as induced during phase separation. Moreover, it is confirmed that fluctuations in all systems with steady values of and are represented as a single curve, which implies that accurate predictions for the growing interface with fluctuations in Hele-Shaw flows can be accomplished at any scale and time, regardless of the miscibility conditions.

Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

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

Abstract

A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS

AbstractThe 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.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects.

In this paper, the interfacial motion between two immiscible viscous fluids in the confined geometry of a Hele-Shaw cell is studied. We consider the influence of a thin wetting film trailing behind the displaced fluid, which dynamically affects the pressure drop at the fluid-fluid interface by introducing a nonlinear dependence on the interfacial velocity. In this framework, two cases of interest are analyzed: The injection-driven flow (expanding evolution), and the lifting plate flow (shrinking evolution). In particular, we investigate the possibility of controlling the development of fingering instabilities in these two different Hele-Shaw setups when wetting effects are taken into account. By employing linear stability theory, we find the proper time-dependent injection rate Q(t) and the time-dependent lifting speed b[over ̇](t) required to control the number of emerging fingers during the expanding and shrinking evolution, respectively. Our results indicate that the consideration of wetting leads to an increase in the magnitude of Q(t) [and b[over ̇](t)] in comparison to the nonwetting strategy. Moreover, a spectrally accurate boundary integral approach is utilized to examine the validity and effectiveness of the controlling protocols at the fully nonlinear regime of the dynamics and confirms that the proposed injection and lifting schemes are feasible strategies to prescribe the morphologies of the resulting patterns in the presence of the wetting film.

Stability results on radial porous media and Hele-Shaw flows with variable viscosity between two moving interfaces

Abstract We perform a linear stability analysis of three-layer radial porous media and Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change of variables results in an eigenvalue problem that has time-dependent coefficients and eigenvalue-dependent boundary conditions. We study this eigenvalue problem and find upper bounds on the spectrum. We also give a characterization of the eigenvalues and prescribe a measure for which the eigenfunctions form an orthonormal basis of the corresponding $L^2$ space. This allows for any initial perturbation of the interfaces and viscosity profile to be easily expanded in terms of the eigenfunctions by using the inner product of the $L^2$ space, thus providing an efficient method for simulating the growth of the perturbations via the linear theory. The limit as the viscosity gradient goes to zero is compared with previous results on multi-layer radial flows. We then numerically compute the eigenvalues and obtain, among other results, optimal profiles within certain classes of functions.

Time-dependent injection strategies for multilayer Hele-Shaw and porous media flows

Universality in the behavior of multilayer radial Hele-Shaw flows is discovered by semi-analytical methods. In particular, it is found numerically that the maximum injection rate for a stable flow decreases proportional to ${t}^{\ensuremath{-}1/3}$ regardless of the number of interfaces and increases at a rate proportional to the number of interfaces to the two-thirds power at large time $t\ensuremath{\gg}1$. However, at earlier times the number of interfaces can increase the maximum stable injection rate by a much greater amount.

On a novel fully-decoupled, linear and second-order accurate numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele–Shaw flow

We construct a novel fully-decoupled and second-order accurate time marching numerical scheme with unconditional energy stability for the Cahn–Hilliard–Darcy phase-field model of the two-phase Hele–Shaw flow, in which, the key idea to realize the full decoupling structure is to use the so-called “zero-energy-contribution” function and design a special ordinary differential equation to deal with the nonlinear coupling terms between the flow field and the phase-field variable. Compared with the existing decoupling type schemes, the scheme developed here is more effective, efficient and easy to implement. At each time step, one only needs to solve a few fully-decoupled linear equations only with constant coefficients. We also strictly prove the solvability and unconditional energy stability of the scheme, and implement various numerical simulations in 2D and 3D to show the efficiency and stability of the proposed scheme numerically.

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

We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent . By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the -Laplace equation, , with a coefficient called ‘flow factor’, which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle.

Experimental and Numerical Simulation Study of Devolatilization in a Self-Wiping Corotating Parallel Twin-Screw Extruder.

Devolatilization is an important process for separating and removing unnecessary residual volatile substances or solvents during the production of polymers using twin-screw extruders. Latinen proposed a surface renewal model to determine the concentration of volatile components in the extrudate of a single-screw extruder. When a twin-screw extruder is used to calculate the concentration, it is necessary to use the exposed surface area of the resin in the starved region of Latinen’s model, which, however, is difficult to estimate. In our previous work, we numerically determined resin profiles of the screws using the 2.5D Hele–Shaw flow model and the finite element method, which helps in estimating the surface area of devolatilization. In this study, we numerically analyzed the volatile concentration of the extrudate in a self-wiping corotating twin-screw extruder using Latinen’s surface renewal model along with our resin profile calculation method. The experimental results of the concentrations of the volatile component (toluene) in the extrudate of polypropylene agreed well with its numerical calculation with a relative error of 6.5% (except for the data of the lowest rotational speed). Our results also showed that decreasing the flow rate and increasing the pump capacity were effective for removing the volatile component. The screw pitch of a full-flight screw was not affected by the devolatilization efficiency with a fixed flow rate and screw speed.

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Abstract

Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow

The displacement of a less viscous fluid by a more viscous fluid in a radial Hele-Shaw cell makes a circular pattern because the interface is hydrodynamically stable in this condition. Very recently, it has been experimentally reported that the hydrodynamically stable displacement in a partially miscible system induces fingering patterns while stable circular patterns are made at fully miscible and immiscible systems. The fingering instability in the partially miscible system results from complex and entangled elements involving viscous dissipation, molecular diffusion, and phase separation. The analyzing mechanism requires a quantitative relationship between the hydrodynamic interfacial fingering patterns and underlying physicochemical properties. Here, we experimentally investigated the change in fluid patterns formed by the progression of phase separation in the partially miscible systems and categorized them into three patterns: finger-like pattern, annular-like pattern, and circular pattern. Moreover, we propose the mechanism of the pattern formation by an interfacial tension measurement and evaluate the patterns by modified capillary number and newly defined body force ratio, Bf. Our analysis revealed that the deformation index of the pattern can be expressed as a function of Bf on a single curve regardless of the miscibility.

A fully discrete curve-shortening polygonal evolution law for moving boundary problems

We consider the numerical integration of moving boundary problems with the curve-shortening property, such as the mean curvature flow and Hele-Shaw flow. We propose a fully discrete curve-shortening polygonal evolution law. The proposed evolution law is fully implicit, and the key to the derivation is to devise the definitions of tangent and normal vectors and tangential and normal velocities at each vertex in an implicit manner. Numerical experiments show that the proposed method allows the use of relatively large time step sizes and also captures the area-preserving or dissipative property in good accuracy.

Resin distribution along axial and circumferential directions of self‐wiping co‐rotating parallel twin‐screw extruder

AbstractA self‐wiping co‐rotating twin‐screw extruder (TSE) is operated in a starved state in which the screws are partially filled with resin. Understanding the resin distribution on the screw surface of a TSE in this state is essential for the design, operation, and maintenance of the twin‐screw extrusion process. Accordingly, in this study, the circumferential and axial distribution of resin in a TSE were simulated using a novel method combining the mathematical formulation of Hele–Shaw flow, the finite element method, and a newly developed down‐wind pressure updating scheme. The results of the simulation were found to be in good agreement with experimental measurements. The proposed simulation method enables the detailed visualization of resin distribution in the entire axial and circumferential directions over the length of a TSE, improving the ability to determine both the devolatilization and fiber attrition during the extrusion process.

Weakly nonlinear analysis of the Saffman-Taylor problem in a radially spreading fluid annulus

Viscous fingering formation in a three-layer radial Hele-Shaw flow is studied. The inclusion of a third layer of fluid adds a new outer interface that couples with the inner one. It is found that the initial distance between these two interfaces has a great impact on the final morphologies of the patterns.

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

The paper analyzes the Morley element method for the Cahn–Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on 1/ε polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on 1/ε polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825–847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L∞(H2) error order, or we can merely obtain the error bounds which are exponentially dependent on 1/ε. Numerical results are presented to validate the optimal L∞(H2) error order and the asymptotic behavior of the solutions of the Cahn–Hilliard equation.

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Cahn–Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn–Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele–Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element method. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn–Hilliard equation, the Holder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele–Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.