scholarly journals Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary

2017 ◽  
Vol 834 ◽  
pp. 1-4 ◽  
Author(s):  
Scott W. McCue
Keyword(s):  
Viscous Fluid ◽  
Porous Media Flow ◽  
Elastic Membrane ◽  
Adhesive Tape ◽  
Interfacial Instabilities ◽  
Elastic Boundary ◽  
Viscous Fingers ◽  
Molar Teeth ◽  
Set Up ◽  
Surface Tension Forces

Injecting a less viscous fluid into a more viscous fluid in a Hele-Shaw cell triggers two-dimensional viscous fingering patterns which are characterised by increasingly long fingers undergoing tip splitting and branching events. These complex structures are considered to be a paradigm for interfacial pattern formation in porous media flow and other related phenomena. Over the past five years, there has been a flurry of interest in manipulating these interfacial fingering patterns by altering the physical components of the Hele-Shaw apparatus. In this Focus on Fluids article, we summarise some of this work, concentrating on a very recent study in which the alterations include replacing one of the two bounding plates with an elastic membrane (Ducloué et al., J. Fluid Mech., vol. 826, 2017, R2). The resulting experimental set-up gives rise to a wide variety of novel interfacial patterns including periodic sideways fingers, dendritic-like patterns and short, flat-tipped viscous fingers that appear to resemble molar teeth. These latter fingers are similar to those observed in the printer’s instability and when peeling off a layer of adhesive tape. This delightful work brings together a number of well-studied themes in interfacial fluid mechanics, including how viscous and surface tension forces compete to drive fingering patterns, how interfaces are affected by fluid–solid interactions and, finally, how novel strategies can be implemented to control interfacial instabilities.

scholarly journals Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

2019 ◽  
Vol 877 ◽  
pp. 1063-1097 ◽  
Author(s):  
Liam C. Morrow ◽  
Timothy J. Moroney ◽  
Scott W. McCue
Keyword(s):  
Viscous Fluid ◽  
Linear Prediction ◽  
Applied Mathematics ◽  
Viscous Fingering ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Inviscid Fluid ◽  
Interfacial Instabilities ◽  
Hele Shaw Cell ◽  
Injection Rates

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

Axisymmetric Motion of a Viscous Fluid Inside a Slender Surface of Revolution

1987 ◽  
Vol 54 (1) ◽  
pp. 190-196 ◽  
Author(s):  
D. A. Caulk ◽  
P. M. Naghdi
Keyword(s):  
Viscous Fluid ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Momentum Equation ◽  
Elastic Membrane ◽  
Surface Of Revolution ◽  
Rigid Tube ◽  
Axisymmetric Motion ◽  
Case Comparison ◽  
Starting Flow

Starting with the exact three-dimensional equations for an incompressible linear viscous fluid, an approximate system of one-dimensional nonlinear equations is derived for axisymmetric motion inside a slender surface of revolution. These equations are obtained by introducing an approximate velocity field into weighted integrals of the momentum equation over the circular cross-section of the fluid. The general equations may be specialized to reflect specific conditions on the lateral surface of the fluid, such as the presence of surface tension, a confining elastic membrane, or a rigid tube. Two specific examples are considered which involve flow in a rigid tube: (1) unsteady starting flow in a nonuniform tube, and (2) axisymmetric swirl superimposed on Poiseuille flow. In each case comparison is made with earlier, more restricted results derived by perturbation methods.

Turbulent drag reduction by compliant lubricating layer

2019 ◽  
Vol 863 ◽  
Author(s):  
Alessio Roccon ◽  
Francesco Zonta ◽  
Alfredo Soldati
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Drag Reduction ◽  
Fluid Viscosity ◽  
Two Phase ◽  
Main Layer ◽  
Two Fluids ◽  
Lubricating Layer ◽  
Light Oil ◽  
Set Up

We propose a physically sound explanation for the drag reduction mechanism in a lubricated channel, a flow configuration in which an interface separates a thin layer of less-viscous fluid (viscosity $\unicode[STIX]{x1D702}_{1}$) from a main layer of a more-viscous fluid (viscosity $\unicode[STIX]{x1D702}_{2}$). To single out the effect of surface tension, we focus initially on two fluids having the same density and the same viscosity ($\unicode[STIX]{x1D706}=\unicode[STIX]{x1D702}_{1}/\unicode[STIX]{x1D702}_{2}=1$), and we lower the viscosity of the lubricating layer down to $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D702}_{1}/\unicode[STIX]{x1D702}_{2}=0.25$, which corresponds to a physically realizable experimental set-up consisting of light oil and water. A database comprising original direct numerical simulations of two-phase flow channel turbulence is used to study the physical mechanisms driving drag reduction, which we report between 20 and 30 percent. The maximum drag reduction occurs when the two fluids have the same viscosity ($\unicode[STIX]{x1D706}=1$), and corresponds to the relaminarization of the lubricating layer. Decreasing the viscosity of the lubricating layer ($\unicode[STIX]{x1D706}<1$) induces a marginally decreased drag reduction, but also helps sustaining strong turbulence in the lubricating layer. This led us to infer two different mechanisms for the two drag-reduced systems, each of which is ultimately controlled by the outcome of the competition between viscous, inertial and surface tension forces.

scholarly journals Electric-field-induced interfacial instabilities of a soft elastic membrane confined between viscous layers

2012 ◽  
Vol 86 (4) ◽  
Author(s):  
Mohar Dey ◽  
Dipankar Bandyopadhyay ◽  
Ashutosh Sharma ◽  
Shizhi Qian ◽  
Sang Woo Joo
Keyword(s):  
Electric Field ◽  
Elastic Membrane ◽  
Interfacial Instabilities

Instabilities of the upstream meniscus in directional viscous fingering

1996 ◽  
Vol 312 ◽  
pp. 125-148 ◽  
Author(s):  
Sylvain Michalland ◽  
Marc Rabaud ◽  
Yves Couder
Keyword(s):  
Viscous Fluid ◽  
Viscous Fingering ◽  
Finite Amplitude ◽  
Dynamical Behaviour ◽  
Solid Surfaces ◽  
Time Dependent ◽  
Narrow Gap ◽  
Propagating Waves ◽  
Spatio Temporal ◽  
Set Up

New instabilities affecting the meniscus of a viscous fluid are presented. They occur in an experimental set-up introduced previously by Rabaud et al. (1990) in which a small quantity of a viscous fluid is placed in the narrow gap between two rotating cylinders. In this geometry the downstream meniscus located in the region where the two solid surfaces move away from each other is known to be unstable and to exhibit directional viscous fingering. In the present article it is shown that the upstream meniscus can also be unstable. Two types of instabilities are observed. In the first supercritical transition the front becomes time-dependent with either standing or propagating waves. In a second transition, which is subcritical, parallel fingers of finite amplitude are formed. The various types of spatio-temporal dynamical behaviour are discussed.

scholarly journals Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

2015 ◽  
Vol 784 ◽  
pp. 487-511 ◽  
Author(s):  
Draga Pihler-Puzović ◽  
Anne Juel ◽  
Gunnar G. Peng ◽  
John R. Lister ◽  
Matthias Heil
Keyword(s):  
Numerical Simulations ◽  
Stokes Equations ◽  
Theoretical Models ◽  
Navier Stokes ◽  
Elastic Membrane ◽  
Two Phase ◽  
Constant Flow Rate ◽  
Viscous Forces ◽  
Elastic Membranes ◽  
Set Up

The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Föppl–von Kármán equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier–Stokes equations or (ii) lubrication theory. We identify the dominant physical effects that control the behaviour of the system and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work, where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.

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