Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis

The linear stability of miscible displacement for radial source flow at infinite P\'eclet number in a Hele-Shaw cell is calculated theoretically. The axisymmetric self-similar flow is shown to be unstable to viscous fingering if the viscosity ratio $m$ between ambient and injected fluids exceeds $3\over2$ and to be stable if $m<{3\over2}$. If $1<m<{3\over2}$ small disturbances decay at rates between $t^{-3/4}$ and $t^{-1}$ relative to the $t^{1/2}$ radius of the axisymmetric ...

The first stages of finger formation in a Hele-Shaw cell with lifting plates are investigated by means of linear stability analysis. The equation of motion for the pressure field (growth law) results to be that of the directional solidification problem in some unsteady state. At the beginning of lifting the square of the wavenumber of the dominant mode results to be proportional to the lifting rate (in qualitative agreement with the experimental data), to the square of the le...

A pair of concentric spheres separated by a small gap form a spherical Hele-Shaw cell. In this cell an interfacial instability arises when two immiscible fluids flow. We derive the equation of motion for the interface perturbation amplitudes, including both pressure and gravity drivings, using a mode coupling approach. Linear stability analysis shows that mode growth rates depend upon interface perimeter and gravitational force. Mode coupling analysis reveals the formation of...

The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this letter we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise constant pumping protocol. It results in a reduction in the size of the viscous fingers by one o...

We make a theoretical study of the behavior of a simple fluid displacing a shear thinning fluid confined in a Hele-Shaw cell. To study the Saffman-Taylor instability when the displaced fluid is non Newtonian we face the problem of having a field which is non laplacian. By means of an hodographic transformation we are able to solve the problem in the case of weak shear thinning while taking into account the non laplacian character of the equation. Our results predict that the ...

Traditional mathematical models of Hele--Shaw flow consider the injection (or withdrawal) of an air bubble into (or from) an infinite body of viscous fluid. The most commonly studied feature of such a model is how the Saffman-Taylor instability drives viscous fingering patterns at the fluid/air interface. Here we consider a more realistic model, which assumes the viscous fluid is finite, covering a doubly connected two-dimensional region bounded by two fluid/air interfaces. F...

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 p...

Using air to displace a viscous fluid contained in Hele-Shaw cell can create a fingering pattern at the interface between the fluids, if the capillary number exceeds a critical value. This Saffman-Taylor instability is revisited for the inverse case of a viscous fluid displacing air, when partially wettable hydrophilic particles are lying on the walls. Though the inverse case is otherwise stable, the presence of the particles results in a fingering instability at low capillar...

The onset of viscous fingering in the presence of a non monotonic viscosity profile is investigated theoretically for two immiscible fluids. Classical fluid dynamics predicts that no unstable behavior may be observed when a viscous fluid pushes a less viscous one in a Hele-Shaw cell. Here, we show that the presence of a viscosity gradient at the interface between both fluids destabilize the interface facilitating the spread of the perturbation. The influence of the viscosity ...

A hierarchy of mathematical models describing viscosity-stratified flow in a Hele-Shaw cell is constructed. Numerical modelling of jet flow and development of viscous fingers with the influence of inertia and friction is carried out. One-dimensional multi-layer flows are studied. In the framework of three-layer flow the interpretation of the Saffman--Taylor instability is given. A kinematic-wave model of viscous fingering taking into account friction between the fluid layers ...