Prabir Daripa

Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.

We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward heat equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward heat equation. In particular, we numerically integrate the backward heat equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-Land U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.

This paper presents results of some numerical experiments on the backward heat equation. Two quasi-reversibility techniques, explicit filtering and structural perturbation, to regularize the ill-posed backward heat equation have been used. In each of these techniques, two numerical methods, namely Euler and Crank–Nicolson (CN), have been used to advance the solution in time. Crank–Nicolson method is very counter-intuitive for solving the backward heat equation because the dispersion relation of the scheme for the backward heat equation has a singularity (unbounded growth) for a particular wave whose finite wave number depends on the numerical parameters. In comparison, the Euler method shows only catastrophic growth of relatively much shorter waves. Strikingly we find that use of smart filtering techniques with the CN method can give as good a result, if not better, as with the Euler method which is discussed in the main text. Performance of these regularization methods using these numerical schemes have been exemplified.

We rigorously derive nonlinear instability of Hele-Shaw flows moving with a constant velocity in the presence of smooth viscosity profiles where the viscosity upstream is lower than the viscosity downstream. This is a single-layer problem without any material interface. The instability of the basic flow is driven by a viscosity gradient as opposed to conventional interfacial Saffman–Taylor instability where the instability is driven by a viscosity jump across the interface. Existing analytical techniques are used in this paper to establish nonlinear instability.

Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/ h 0 , and long-wavelength parameter, β = (h 0 /l) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ /ρgh 2 0 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | | β, and (2) | 1 3 − τ | = O(β). Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3. Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.

The problem of existence of trapped waves in fluids due to a cylinder is investigated for the hydrodynamic setup which involves a horizontal channel of infinite length and depth and of finite width containing three layers of incompressible fluids of different constant densities. The setup also contains a cylinder which is impermeable, fully immersed in the bottom (lower-most) fluid layer of infinite depth, and extends across the channel with its generators perpendicular to the side walls of the channel. When the ratios of the densities of the adjacent fluids differ from unity by sufficiently small quantities, the underlying mathematical problem reduces to a generalized nonlinear eigenvalue problem involving a cubic polynomial-cum-operator equation. The perturbation analysis of this eigenvalue problem suggests existence of three distinct modes with different frequencies: one of the order of one persisting at the free surface, and the other two of the order of the density ratio (except for modulo one) persisting at the two internal interfaces. The correlation between these results for the three-layer case and very recent numerical results of other authors in the two-layer case has also been addressed.

The Euler's equations describing the dynamics of capillary-gravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussi-nesq type of equation is derived involving the small-amplitude and long-wavelength parameters. A recently introduced sixth-order Boussinesq equation by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159– 207] is recovered from this equation in the 1/3 Bond number limit (from below) when the above parameters bear a certain relationship as they approach zero.

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a ¼ a=h 0 and wavelength parameter b ¼ ðh 0 =lÞ 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number s ¼ C=qgh 2 0 , where C is the surface tension coefficient, q is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) b (ð1=3 À sÞ 6 1=3 and (ii) ð1=3 À sÞ ¼ OðbÞ. The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for s ¼ 1=3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.: S 0 0 2 0-7 2 2 5 (0 2) 0 0 1 8 0-5

An analytical method is developed for solving an inverse problem for Helmholtz's equation associated with two semi-infinite incompressible fluids of different variable refractive indices, separated by a plane interface. The unknowns of the inverse problem are: (i) the refractive indices of the two fluids, (ii) the ratio of the densities of the two fluids, and (iii) the strength of an acoustic source assumed to be situated at the interface of the two fluids. These are determined from the pressure on the interface produced by the acoustic source. The effect of the surface tension force at the interface is taken into account in this paper. The application of the proposed analytical method to solve the inverse problem is also illustrated with several examples. In particular, exact solutions of two direct problems are first derived using standard classical methods which are then used in our proposed inverse method to recover the unknowns of the corresponding inverse problems. The results are found to be in excellent agreement.

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. Motivated by their work, we formally derive this equation from two-dimensional potential ¯ow equations governing the small amplitude long capillary-gravity waves on the surface of shallow water for Bond number very close to but less than 1/3. On the basis of far-®eld analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-®eld. We review various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (®fth-order) KdV equation. Using these methods, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-®eld.

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-field. Using various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-field.

Exact analytical solutions are found for the steady state creeping flow in and around a vapor–liquid compound droplet, consisting of two orthogonally intersecting spheres of arbitrary radii a and b, submerged in axisymmetric extensional and paraboloidal flows of fluid with viscosity (1). The solutions are presented in singularity form with the images located at three points: the two centers of the spheres and their common inverse point. The important results of physical interest such as drag force and stresslet coefficient are derived and discussed. These flow properties are characterized by two parameters, namely the dimensionless viscosity parameter: (2) /((1) (2)), and the dimensionless parameter: b/a, where (2) is the viscosity of the liquid in the sphere part of the compound droplet with radius b. We find that for some extensional flows, there exists a critical value of c for each choice of in the interval 01 such that the drag force is negative, zero or positive depending on whether c , c , or c respectively. For other extensional flows, the drag force is always positive. The realization of these various extensional flows by simply changing the choice of the origin in our description of the undisturbed flow field is also discussed. In extensional flows where the drag force is always positive, we notice that this drag force D e for vapor–liquid compound droplet is maximum when 1 i.e., two spheres have almost the same radii. Moreover, we find the drag force D e is a monotonic function of , i.e., the drag force for vapor–liquid compound droplet lies between vapor–vapor and vapor-rigid assembly limits. We also find that the maximum value of the drag in paraboloidal flow depends on the viscosity ratio and significantly on the liquid volume in the dispersed phase.

The complicated nature of singularities associated with topological transition in the plane Taylor-bubble problem is briefly discussed in the context of estimating the speed of the fastest smooth Taylor-bubble in the absence of surface tension. Previous numerical studies were able to show the presence of a stagnation point at the tip of the bubbles for dimensionless speed F < 0.357 but were incomplete in characterizing the topology of these bubbles at the tip for values of F > 0.29 due to difficulties in obtaining numerical solutions with well-rounded profiles at the apex. These difficulties raise the question whether the bubbles rising at a speed F ∈ (0.29, 0.357) are smooth, pointed or spurious. This issue has led us to carefully scrutinize certain asymptotic behavior of the Fourier spectrums of the numerical solutions for a wide range of values of F and to extend these results in an appropriate limiting sense. Our findings indicate that these plane bubbles with F < 0.35784 (accurate up to four decimal places) have well-rounded profiles at the apex. The purpose of this paper is to describe our approach and its use in arriving at the above conclusion.

The mathematical foundation of an algorithm for fast and accurate evaluation of singular integral transforms was given by Daripa [9,10,12]. By construction, the algorithm offers good parallelization opportunities and a lower computational complexity when compared with methods based on quadrature rules. In this paper we develop a parallel version of the fast algorithm by redefining the inherently sequential recurrences present in the original sequential formulation. The parallel version only utilizes a linear neighbor-to-neighbor communication path, which makes the algorithm very suitable for any distributed memory architecture. Numerical results and theoretical estimates show good parallel scalability of the algorithm.

We consider an ill-posed Boussinesq equation which arises in shallow water waves and nonlinear lattices. This equation has growing and decaying modes in the linear as well as nonlinear regimes and its linearized growth rate r for short-waves of wave-number k is given by r $ k 2. Previous numerical studies have addressed numerical diculties and construction of approximate solutions for ill-posed problems with shortwave instability up to r $ k, e.g. Kelvin±Helmholtz …r $ k† and Rayleigh±Taylor …r $   k p † instabilities. These same issues are addressed and critically examined here for the present problem which has more severe shortwave instability. In order to develop numerical techniques for constructing good approximate solutions of this equation, we use a ®nite di€erence scheme to investigate the e€ect of this shortwave instability on the numerical accuracy of the exact solitary wave solution of this equation. Computational evidence is presented which indicates that numerical accuracy of the solutions is lost very quickly due to severe growth of numerical errors, roundo€ as well as truncation. We use both ®ltering and regularization techniques to control growth of these errors and to provide better approximate solutions of this equation. In the ®ltering technique, numerical experiments with three types of spectral ®lters of increasing order of regularity are performed. We examine the role of regularity of these ®lters on the accuracy of the numerical solutions. Numerical evidence is provided which indicates that the regularity of a ®lter plays an important role in improving the accuracy of the solutions. In the regularization technique, the ill-posed equation is regularized by adding a higher: S 0 0 9 6-3 0 0 3 (9 8) 1 0 0 7 0-X order term to the equation. Two types of higher order terms are discussed: (i) one that diminishes the growth rate of all modes below a cuto€ wavenumber and sets the growth rate of all modes above it to zero; and (ii) the other one diminishes the growth rate of all modes and the growth rate asymptotically approaches to zero as the wavenumber approaches in®nity. We have argued in favor of the ®rst type of regularization and numerical results using a ®nite di€erence scheme are presented. Numerical evidence is provided which suggests that regularization in combination with the most regular (g 2 here) spectral ®lter for small values of the regularization parameter can provide good approximate solutions of the ill-posed Boussinesq equation for longer time than possible otherwise. Some of the ideas presented here can possibly be utilized for solving other ill-posed problems with severe shortwave instabilities and may have an important role to play in numerical studies of their solutions.

Some useful filtering techniques for computing approximate solutions of illposed problems 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.

In the past pointed bubbles have been obtained numerically in the presence of surface tension. In this paper it is proven that if such pointed bubbles do exist in the presence of surface tension, then the singularity at the corner must be an irregular singular point. The generality and significance of the result are discussed. In this work, we characterize the singularity at the tip of an unphysical pointed bubble in the presence of surface tension. These bubbles have been recently obtained by Vanden-Broeck' in numerical calculations. There are two primary motivations for characterizing this singularity. The first comes from the fact that construction of proper numerical methods in the presence of singularities may require the knowledge of the nature of these singularities. Even then, the numerical issues can be very delicate and may require very careful handling in the construction of numerical methods. The second comes from the fact that it has direct relevance to similar problems of pattern selection in the presence of surface tension (see concluding remarks). In the present context, these physical patterns are round bubbles. The determination of these round bubbles involves solving a nonlinear eigenvalue problem as in Vanden-Broeck' with the angle at the apex of the bubble as one of the free parameters. One solves this eigenvalue problem numerically to find the apex angle as a function of the speed of the bubbles for a fixed value of surface tension. The spectrum of round bubbles is found to be discrete and contained in a continuous family of bubbles. This continuous family of bubbles contains sets of a continuous family of pointed bubbles with the round bubbles separating these continuous families of pointed bubbles.' The physical situation here consists of an infinitely long symmetric bubble rising at a constant velocity U in a two-dimensional channel of width h (see Fig. 1) .' The interior angle at the tip of the bubble is denoted by 8,. The flow exterior to the bubble is considered inviscid and in-compressible. The flow is characterized by its Froude number F and the Weber number W F=U/@, W-p U2h/T. (1) Here g is the gravitational acceleration, T is the surface tension, and p is the density of the fluid. The Froude number F refers to the dimensionless speed of the bubble. The principal issue has been the speed of the bubble. This problem is usually solved in a moving reference frame attached to the bubble. Theoretically this problem has been intractable due to severe nonlinearities in the interface condition. There are no existence or uniqueness theorems for this problem. Birkhoff and Carter3 were the first to formulate and solve this problem numerically with zero surface tension. They considered only the existence of a unique round bubble and numerically obtained an approximate solution with FzO.23. This is consistent with experimental results of Collins4 with small surface tension. Birkhoff and Carter encountered difficulties with their numerical method on this problem and their numerical results were not very consistent. They attributed these difficulties to the presence of singularities at the tails of the bubble. Garabedian' subsequently applied asymptotic methods to this problem and provided analytical evidence that the solution is not unique. He suggested the existence of a continuous family of round bubbles for F <F, where F,zO.23. Subsequently Vanden-Broeck6 solved this problem using a numerical method similar to that of Birkhoff and Carter.3 He obtained the following results: smooth bubbles for F <F,, cusped for F > F,, and pointed with 13,= 120 " for F=F,, where F,=O.357 75. His results contain the results of Garabedian.' In obtaining these results, Vanden-Broeck put special effort in representing the solutions near the tip of the bubble where singularities may appear. For nonzero surface tension, this problem was also solved by Vanden-Broeck' numerically. In fact, his computation shows that surface tension makes the problem more singular. He finds that surface tension makes the round zero surface tension bubbles pointed except for a discrete set of velocities. Obviously, the source of these unphysical bubbles in the presence of surface tension is either in the equations or in their discrete analogs used for computations. The purpose here is to explain the origin of this nonphys-ical behavior from a theoretical standpoint. At this point it is worthwhile to classify these singu-larities at the tip. When the problem is formulated in the circle plane, 1 u I< 1, an apex angle of 6, corresponds to an analytic function g(a) having a singularity of order y= 0Jrr at a=i [see Eq. (4) below]. This analytic function f(a) then admits the following representation near this singularity: gb)=(l+ovxa). (2) The singularity is termed regular if f(a) is analytic there, otherwise the singularity is termed irregular. Identification of the nature of these singularities may be useful in devising appropriate numerical methods which can handle such singularities effectively. In the case of zero surface tension, it can be shown that a corner with 120 " interior angle is admissible and that such a comer is probably not a regular singular point. Support for the complicated nature of the singularity at such a corner is based on the analysis of Grant' at the crest

Some iterative adaptive grid generators, developed by the author, are numerically explored in detail to assess their relative merits against conventional grid generators, based on a direct method of integration and interpolation. We find that some of these iterative adaptive grid generators are preferable to a direct method of integration and interpolation In contrast with a direct method, appropriate use of these iterative adaptive grid generators produces adaptive grids with a smooth variation of the grid spacing ratio and resolution. All adaptive grid generators are a subclass of a more general iterative map. General features of this iterated map which are related to the function to be resolved are briefly discussed. The results obtained here supplements recent investigations on these adaptive grid generators by the author.

An algorithm is provided for the fast and accurate computation of the solution of nonhomogeneous Cauchy-Riemann equations in the complex plane in the interior of a unit disk. The algorithm is based on the representation of the solution in terms of a double integral, some recursive relations in Fourier space, and fast Fourier transforms. The numerical evaluation of the solution at N 2 points on a polar coordinate grid by straightforward summation for the double integral would require O(N2) floating point operations per point. Evaluation of these integrals has been optimized in this paper giving an asymptotic operation count of O(ln N) per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of O(1) per point. The algorithm has the added advantage of working in place, meaning that no additional memory storage is required beyond that of the initial data. The performance of the algorithm has been demonstrated on several prototype problems. The algorithm has applications in many areas, particularly fluid mechanics, solid mechanics, and quasi-conformal mappings.

The theory of a new approach to adaptive grid generation in one dimension is developed. The approach is based on approximating either the resolution or the grid spacing ratio on discrete lattice points by continuous variables. The order of accuracy of these approximations in a suitable reference frame characterizes the various methods. Approximations that are first-or second-order accurate in a suitable reference coordinate are derived in this paper. The free parameters associated with these methods provide flexibility in generating a large family of adaptive grids with smooth grid spacing ratio and high resolution. A selected group of this family of adaptive grids may prove very useful in adaptive computations of partial differential equations. The adaptive grids are numerically generated using these approximations. Numerical examples are given that exemplify the usefulness of these adaptive grids.

A new method for one dimensional adaptive grid generation is introduced based on defining the grid spacing ratio as a continuous variable. In this paper we validate our theoretical results in order to justify their use in numerical construction of adaptive grids in one dimension.

In this note we generate some quasi-conformal grids by using fast Fourier transforms and path integrals. Thus this method is rapid as opposed to other elliptic methods. In particular we generate such grids around airfoils. The method uses potential flow equations of fluid flow in the construction of these grids. We briefly discuss about embedding this method within a general framework. The main purpose of this note is to discuss this specific but fast method and generate the grids which may have practical applications. This note is not meant to be an exhaustive treatment on quasi-conformal grid generation and thus we do not discuss their applications here.

The inverse problem in the tangent gas approximation is considered. An exact method for designing airfoils is presented. Constraints on the speed distribution are easily implemented. A simple numerical algorithm which is fast and accurate is presented. Comparison of designed airfoils using the tangent gas method with exact Euler results is found to be excellent for sub-critical flows.

Steady, inviscid, irrotational flow of a perfect gas in two dimensions is considered in the tangent gas approximation. A fast and accurate method of solution is proposed and solved numerically. Comparison of tangent gas and exact flows are presented. Tangent gas solutions when used as the first step in the iterative solution of the exact flowfield are shown to give substantial reduction in computational time.

Standard finite difference approximations to Burgers' equation are considered from the point of view of dynamical systems theory. Phase plane analyses for discretizations with a few grid points are presented. These show the existence of initial conditions leading to spurious solutions with unlimited amplitude growth due to nonconservation of kinetic energy by the nondissipative terms in the discretizations. It is shown that such solutions may be found even for arbitrarily fine pointwise resolution, i.e., for arbitrarily many grid points. On the other hand, an energy conserving discretization of the nondissipative terms removes all spurious solutions of this kind. The results obtained seem to complement recent investigations of the steady state problem.

The pulsatile blood flow in an eccentric catheterized artery is studied numerically by making use of an extended version of the fast algorithm of Borges and Daripa [J. Comp. Phys., 2001]. The mathematical model involves the usual assumptions that the arterial segment is straight, the arterial wall is rigid and impermeable, blood is an incompressible Newtonian fluid, and the flow is fully developed. The flow rate (flux) is considered as a periodic function of time (prescribed). The axial pressure gradient and velocity distribution in the eccentric catheterized artery are obtained as solutions of the problem. Through the computed results on axial pressure gradient, the increases in mean pressure gradient and frictional resistance in the artery due to catheterization are estimated. These estimates can be used to correct the error involved in the measured pressure gradients using catheters.

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 shortwave 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. Keywords Three-layer Hele-Shaw flows · Improved oil recovery · Stability of flows · Sturm-Liouville problem · Upper bound

Hydrodynamic instability in immiscible porous media flows in the presence of capillarity is investigated here. The analysis and arguments presented here show that the slowdown of instabilities due to capillarity is usually very rapid which makes the flow almost, but not entirely, stable. The profiles of the stable and unstable waves in the far-field are characterized using a novel but very simple approach.

This paper evaluates the relevance of Hele-Shaw (HS) model based linear stability results to fully developed flows in enhanced oil recovery (EOR). In a recent exhaustive study [Transport in Porous Media, 93, 675-703 (2012)] of the linear stability characteristics of unstable immiscible three-layer " Hele-Shaw " flows involving regions of varying viscosity, an optimal injection policy corresponding to the smallest value of the highest rate of growth of instabilities was identified among several injection policies. Relevance of this HS model based result to EOR is established by performing direct numerical simulations of fully developed tertiary displacement in porous media. Results of direct numerical simulation are succinctly summarized including characterization of the optimal flooding scheme that leads to maximum oil recovery. These results have been compared with the HS model based linear stability results. The scope for potential application of the HS model based results to the development of fast methods for optimization of various chemical flooding schemes is discussed. Numerical experiments with more complex flooding schemes in both homogeneous and heterogeneous reservoir are also performed and results analyzed to test the universality of the generic optimal viscous families in a broader setting.

Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential
equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for
further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and
parallel implementation of these algorithms and their applications to some pure and applied problems are also
briefly reviewed.

ABSTRACT
The upper bound results on the growth rates in unstable multi-layer Hele-Shaw flows will be derived. The cases treated are constant viscosity layers and variable viscosity layers. The upper bound provides a way to assess cumulative effects of many layers and many interfaces on the growth rates of unstable waves. As an application of the bound, we obtain some sufficient conditions for suppressing instability of two-layer flows by introducing arbitrary number of constant viscosity fluid layers in between. This sufficient condition has very practical relevance because it narrows the choice of internal layer fluids based on surface tensions of all interfaces and viscosities of fluids. Importance of this condition which has been hitherto unknown is also discussed. Other consequences of these upper bounds and sufficient conditions are discussed. The case of internal fluid layers having unstable viscous profiles is also treated for three-layer and four-layer flows only. Implications of these stability results for these various multi-layer flows are discussed and compared from practical standpoint.

... The main result we obtain is a formula of the film thickness in terms of $M$ and $\Gamma$ where $M$ is the Marangoni number. A comparison with Bretherton&amp;#x27;s ``clean&amp;#x27;&amp;#x27; case shows the thickening effect of surfactant. This talk is partially based on an going work with Gelu Pasa. ...

Stability theory plays a major role from fundamental science to applied sciences. It is useful in the design of many processes and engineering instruments as well as in explaining many phenomena. In this paper we review some of the author&#39;s and his collaborator&#39;s recent works on the extension of Saffman-Taylor instability which occurs at an interface between two immiscible fluids in porous media and Hele-Shaw cells when displacing fluid is less viscous than the displaced one. The growth rate of interfacial disturbances is given by a formula called Saffman-Taylor formula which plays a very important role in many areas including flows in porous media and oil recovery among many others. In this talk, we will present our results on the generalization of this formula to multi-layer flows involving many interfaces. As an application of the generalized Saffman-Taylor formula, we will derive necessary conditions for suppressing instability of two-layer flows by introducing arbitrary...

Saffman-Taylor instability is a well known viscosity driven instability of an interface separating two immiscible fluids. We study linear stability of displacement processes in a Hele-Shaw cell involving an arbitrary number of immiscible fluid phases. This is a problem involving many interfaces. Universal stability results have been obtained for this multi-phase immiscible flow in the sense that the results hold for arbitrary number of interfaces. These stability results have been applied to design displacement processes that are considerably less unstable than the pure Saffman-Taylor case. In particular, we derive universal formula which gives specific values of the viscosities of the fluid layers corresponding to smallest unstable band. Other similar universal results will also be presented. The talk is based on the following paper.[4pt] [1] Prabir Daripa and Xueru Ding, ``Universal Stability Properties for Multi-Layer Hele-Shaw Flows and Application to Instability Control,&#39;&#...

We will present numerical solutions from initial value calculations of a model of shallow water wave equation. For small data, numerical solution develops singularity in a finite time. Driven by the structure of solutions, we carry out analysis based on numerical results to prove singularity formation. Numerical and theoretical results will be shown.