On estimates for short wave stability and long wave instability in three-layer Hele-Shaw flows

We consider the linear stability of three-layer Hele-Shaw flows with each layer having constant viscosity and viscosity increasing in the direction of a basic uniform flow. While the upper bound results on the growth rate of long waves are well known from our earlier works, lower bound results on the growth rate of short stable waves are not known to date. In this paper, we obtain such a lower bound. In particular, we show the following results: (i) the lower bound for stable short waves is also a lower bound for all stable waves, and the exact dispersion curve for the most stable eigenvalue intersects the dispersion curve based on the lower bound at a wavenumber where the most stable eigenvalue is zero; (ii) the upper bound for unstable long waves is also an upper bound for all unstable waves, and the exact dispersion curve for the most unstable eigenvalue intersects the dispersion curve based on the upper bound at a wavenumber where the most unstable eigenvalue is zero. Numerical results are provided which support these findings.     

A Domain Embedding Method Using the Optimal Distributed Control and a Fast Algorithm

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. The method is based on formulating the problem as an optimal distributed control problem inside a disc in which the arbitrary domain is embedded. The optimal distributed control problem inside the disc is solved rapidly using a fast algorithm developed by Daripa et al. [3,7,10–12]. The arbitrary domains can be simply or multiply connected and the proposed method can be applied, in principle, to a large number of elliptic problems. Numerical results obtained for Dirichlet problems associated with the Poisson equation in simply and multiply connected domains are presented. The computed solutions are found to be in good agreement with the exact solutions with moderate number of grid points in the domain.     

Some useful filtering techniques for illposed problems

Some useful filtering techniques for computing approximate solutions of illposed are presented. Special attention is given to the role of smoothness of the filters and the choice of time-dependent parameters used in these filtering techniques. Smooth filters and proper choice of time-dependent parameters in these filtering techniques allow numerical construction of more accurate approximate solutions of illposed problems. In order to illustrate this and the filtering techniques, a severely illposed fourth-order nonlinear wave equation is numercally solved using a three time-level finite difference scheme. Numerical examples are given showing the merits of the filtering techniques.     

On Stabilization of Multi-Layer Hele-Shaw and Porous Media Flows in the Presence of Gravity

Stabilization of multi-layer Hele-Shaw flows is studied here by including the influence of Rayleigh?CTaylor instability in our earlier work (Daripa, J. Stat. Mech. 12:28, 2008a) on stabilization of multi-layer Saffman?CTaylor instability. Furthermore, this article goes beyond our previous work with few extensions, improvements, new interpretations, and clarifications on the use of some terminologies. Results of two complete studies have been presented: the first investigates the effect of individually unstable interfaces on the overall stability of the flow, and the second studies the cumulative effect of unstable interfaces as well as unstable internal viscous layers. In each case, modal and absolute upper bounds on the growth rate are reported. Next, these bounds are used to investigate (i) stabilization of long waves on various interfaces; (ii) stabilization of all waves on all interfaces in comparison to pure Taylor instability; (iii) stabilization of disturbances on interior interfaces instead of exterior interfaces. In the first study, notions of partial and total stabilization with respect to the pure Taylor growth rate are introduced. Then necessary and sufficient conditions for partial and total stabilizations are found. Proof of stabilization of long waves on one of the two external interfaces in multi-layer flows is also proved. In the second study, an absolute upper bound is obtained in the presence of stabilizing density stratification across each internal interface even though all interfaces and layers have unstable viscous profiles. Exact results on the upper bounds, and necessary and sufficient conditions for control of instabilities driven by stable/unstable density stratification, unstable viscous layers and unstable interfaces are new and may be relevant to explain observed phenomena in many complex flows generating these kinds of viscous profiles and density stratification as they evolve. The present work builds upon and goes much further in details and new results than our previous work. The gravity effect included here brings with it restrictions which have not been addressed before in this multi-layer context.     

An efficient and novel numerical method for quasiconformal mappings of doubly connected domains

A numerical method for quasiconformal mapping of doubly connected domains onto annuli is presented. The ratio R of the radii of the annulus is not known a priori and is determined as part of the solution procedure. The numerical method presented in this paper requires solving iteratively a sequence of inhomogeneous Beltrami equations, each for a different R. R is updated using a procedure based on the bisection method. The new method is an extension of Daripas method for the quasiconformal mapping of the exterior of simply connected domains onto the interior of unit disks [15]. It uses fast and accurate algorithms for evaluating certain singular integrals and is, thus, very efficient and accurate. Its performance is demonstrated for several doubly connected domains.     

Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient

In the presence of diffusion, stability of three-layer Hele-Shaw flows which models enhanced oil recovery processes by polymer flooding is studied for the case of variable viscosity in the middle layer. This leads to the coupling of the momentum equation and the species advection-diffusion equation the hydrodynamic stability study of which is presented in this paper. Linear stability analysis of a potentially unstable three-layer rectilinear Hele-Shaw flow is used to examine the effects of species diffusion on the stability of the flow. Using a weak formulation of the disturbance equations, upper bounds on the growth rate of individual disturbances and on the maximal growth rate over all possible disturbances are found. Analytically, it is shown that a short-wave disturbance if unstable can be stabilized by mild diffusion of species, where as an unstable long-wave disturbance can always be stabilized by strong diffusion of species. Thus, an otherwise unstable three-layer Hele-Shaw flow can be completely stabilized by a large enough diffusion, i.e., by increasing enough the magnitude of the species diffusion coefficient. The magnitude of this diffusion coefficient required to completely stabilize the flow will depend on the magnitude of interfacial viscosity jumps and the viscosity gradient of the basic viscous profile of the middle layer.     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green’s functions of these problems and compare the numerical results with the “decomposition” algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824–850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the “decomposition algorithm” as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824–850 [17]) for solving certain biharmonic problems uses the “direct algorithm” without which the problem can not be solved. Using classical Green’s function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z?plane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N ² is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.