Two-dimensional Free-boundary Problem Research Articles

Erosion of surfaces by trapped vortices

Two two-dimensional free boundary problems describing the erosion of solid surfaces by the flow of inviscid fluid in the presence of trapped vortices are considered. The first problem tackles an initially flat, infinite fluid-solid interface with uniform flow at infinity and a vortex in equilibrium above the surface. The second involves flow around a finite body with a trailing Föppl-type vortex pair. The conformal invariance of the complex potential permits both problems to be formulated as a Polubarinova–Galin (PG) type equation in which the time-dependent eroding surface in the physical z-plane is mapped to the fixed boundary of the ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-disk. The Hamiltonian governing the equilibrium position of the vortex (or vortex pair in the second problem) is also found from the same map. In each problem, the PG equation giving the conformal map is found numerically and the time-dependent evolution of the interface and vortex location is determined. Different models governing the erosion of the interface are investigated in which the normal velocity of the boundary depends on some given function of the fluid flow velocity at the boundary. Typically, in the infinite surface case, erosion leads to the formation of a symmetric valley beneath the vortex which, in turn, moves downward toward the interface. A finite body undergoes erosion which is asymmetric in the flow direction leading to a flattening of the lee surface of the body so displaying some similarity to the experiments and associated viscous theory of Ristroph et al, Moore et al (Proc Natl Acad Sci 109(48):19606–19609, 2012, Phys Fluids 25(11):116602, 2013).

Exact and numerical solutions of a free boundary problem with a reciprocal growth law

Abstract A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outwards is formulated both in terms of the Schwarz function of the boundary and a Polubarinova–Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova–Galin equation is used to compute the evolution of the boundary having other initial shapes.

(Quasi-)conformal methods in two-dimensional free boundary problems

In this paper, we relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and apply this result to the solutions of one-phase Bernoulli free boundary problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

Dissolution of plane surfaces by sources in potential flow

The two-dimensional free boundary problem for a surface dissolving in potential flow owing to concentrated sources of dissolving agent c is formulated and solved. The surface boundary evolves quasi-steadily, and the resulting steady advection–diffusion equation for c, and Laplace’s equation for the velocity potential form a coupled pair of conformally invariant PDEs. The dynamics of the coupled fluid flow and evolving surface depends on the Péclet number: Pe=UL/D, where U and L are typical velocity and lengthscales respectively, and D is the diffusivity of c. The conformally invariant property is exploited in finding an equation of the Polubarinova–Galin class for the conformal map from the unit ζ-disk to the evolving domain in physical space. The equation is solved numerically and the time-evolution of the dissolving surface determined. For a single concentrated source with flow initially parallel to a flat surface, a Pe-dependent scallop-like surface shape typically develops. The problem involving a periodic array of concentrated sources aligned parallel to an initially flat surface is also solved.

Biphasic co-flow through a sudden expansion or contraction of a Hele-Shaw channel

We study the biphasic co-flow through an expansion (or contraction) of the shallow (Hele-Shaw) microchannel. Assuming gentle streamwise variation of the flow, we derive the 3rd-order nonlinear differential equation governing the steady-state shape of the interface separating the two fluids. The interfacial profiles obtained by integrating this nonlinear equation are further compared to numerical solution of the two-dimensional free-boundary problem and experimental results. The amplitude of the capillary ridge emerging upstream from a sudden expansion and leading to narrowing of the thread of the inner phase is substantial, however, not large enough to trigger instability and breakup.

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

PurposeThe purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodologyFor the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

Solitary waves on constant vorticity flows with an interior stagnation point

Abstract

Effects of rotation on planetary inertial gravity waves initiated by the stratospheric polar vortex

We examine stability of the polar vortex that initiates planetary-scale westerly (west to east) atmospheric flow that encircles the pole in middle or high latitude of a planet. In our studies, the polar vortex is modeled as an annular flow (zonal flow) extending radially from a rigid inner boundary to a free outer boundary on a Ek− plane, which is parallel to the equatorial plane but generally displaced in latitude. The vortex is associated with a particular class of exact solution of a nonlinear two-dimensional free boundary problem on the nonstationary motion of a perfect fluid propagating around a solid circle with a sufficiently large radius with included Coriolis effect. The intensity of the polar vortex is parametrized by means of the “average” (in vertical) zonal flow that can be associated with atmospheric flow patterns that are parallel or nearly parallel to the lines of latitude. Perturbations are allowed in the zonal and nominally vertical (inward–outward) direction. A linear stability analysis purportedly suggests that increasing latitude decreases the maximum strength of the vortex that is permitted for stability. Three essential parameters that control stability of the vortex are identified.

On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of \begin{document}$ \pi/2 $\end{document} .

Primal-dual active set method for pricing American better-of option on two assets

An American better-of option is a rainbow option with two underlying assets. It can be described by a parabolic variational inequality (VI) on a two-dimensional unbounded domain, which can also be characterized by a two-dimensional free boundary problem. Based on the numeraire transformation and the known information on the free boundary, we derive a one-dimensional linear complementarity problem (LCP) related to options on a bounded domain. Moreover, the full discretization scheme of LCP is constructed by finite difference and finite element methods in temporal and spatial directions, respectively. The primal-dual active-set (PDAS) method is adopted for solving the resulting large-scale discretized system. In each PDAS iteration, a unique index set of primal-dual variables will be computed by solving a linear system. A systematical convergent analysis is presented on our proposed method for pricing American better-of options. One of the desirable features of our method is that we can get the option value and two free boundaries simultaneously. Numerical simulations are performed to verify the efficiency of our proposed method.

Capillary Waves Joint to Super-Hydrophobic Bars

Super-hydrophobic coating reduces hydrodynamic drag of rigid surfaces due to separation of a part of the surface from the flow by air. The drag reduction ratio is proportional to the ratio of the surface area covered by air to the whole surface area. The maximum ratio may be achieved for coating with a regular spanwise super-hydrophobic bar. The air–water boundary over such bar would be a capillary wave with wavelength equal the distance between bar apexes. The numerical analysis of such waves was carried out by solving a two-dimensional nonlinear free-boundary problem of ideal fluid theory. Besides several wave shapes, the main computational results include dependencies of wavelengths and dimensionless pressure coefficient necessary for wave maintenance on Weber number. These dependencies make it possible to select the bar size and inflow speed allowing for existence of such waves and the highest drag reduction ratios.

The generalized Riemann problem and instability of delta shock to the chromatography equations

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

Bounds for Solutions to the Problem of Steady Water Waves with Vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the ...

Primal-Dual Active Set Method for American Lookback Put Option Pricing

AbstractThe pricing model for American lookback options can be characterised as a two-dimensional free boundary problem. The main challenge in this problem is the free boundary, which is also the main concern for financial investors. We use a standard technique to reduce the pricing model to a one-dimensional linear complementarity problem on a bounded domain and obtain a corresponding variational inequality. The inequality is discretised by finite differences and finite elements in the temporal and spatial directions, respectively. By enforcing inequality constraints related to the options using Lagrange multipliers, the discretised variational inequality is reformulated as a set of semi-smooth equations, which are solved by a primal-dual active set method. One of the major advantages of our algorithm is that we can obtain the option values and the free boundary simultaneously, and numerical simulations show that our approach is as efficient as some other methods.

Poisson growth

The two-dimensional free boundary problem in which the field is governed by Poisson’s equation and for which the velocity of the free boundary is given by the gradient of the field—Poisson growth—is considered. The problem is a generalisation of classic Hele-Shaw free boundary flow or Laplacian growth problem and has many applications. In the case when the right hand side of Poisson’s equation is constant, a formulation is obtained in terms of the Schwarz function of the free boundary. From this it is deduced that solutions of the Laplacian growth problem also satisfy the Poisson growth problem, the only difference being in their time evolution. The corresponding moment evolution equations, a Polubarinova–Galin type equation and a Baiocchi-type transformation for Poisson growth are also presented. Some explicit examples are given, one in which cusp formation is inhibited by the addition of the Poisson term, and another for a growing finger in which the Poisson term selects the width of the finger to be half that of the channel. For the more complicated case when the right hand side is linear in one space direction, the Schwarz function method is used to derive an exact solution describing a translating circular blob with changing radius.

Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.

Sharp interfaces in two-dimensional free boundary problems: interface calculation via matched conformal maps.

We use conformal maps to study a free boundary problem for a two-fluid electromechanical system, where the interface between the fluids is determined by the combined effects of electrostatic forces, gravity, and surface tension. The free boundary in our system develops sharp corners or singularities in certain parameter regimes, and this is an impediment to using existing "single-scale" numerical conformal mapping methods. The difficulty is due to the phenomenon of crowding, i.e., the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary, leaving the smooth regions of the boundary poorly resolved. A natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately and then stitch the solutions together. However, this is not straightforward as conformal maps are rigid "global" objects, and it is not obvious how one would patch two conformal maps together to obtain a new conformal map. We develop a "multiscale" (i.e., adaptive) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together. We successfully apply our method to the electromechanical model problem.

No steady water waves of small amplitude are supported by a shear flow with a still free surface

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

Strongly Φ-like functions of order α in two-dimensional free boundary problems

In this paper we apply certain results in the theory of univalent functions to investigate the time evolution of the free boundary of a viscous fluid for a planar flow problem in the Hele–Shaw cell model under injection. To this end, we prove that the property of strongly Φ-likeness of order α∈(0,1] is preserved in time for both inner and outer problems, under the assumption of nonzero small surface tension.