Two-phase Problem Research Articles

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Delta )^s u = lambda _+ (u_+)^{q-1} - lambda _- (u_-)^{q-1}quad text {in }B_1, where lambda _+,lambda _->0, q in [1,2), and u_+ and u_- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q=1 as well as sublinear equations for 1<q<2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s=1, we prove that the admissible vanishing orders can not exceed the critical value k_q= 2s/(2- q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that k_q< 1, we prove a remarkable difference with the local case: solutions can only vanish with order k_q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

A multiscale Robin-coupled implicit method for two-phase flows in high-contrast formations

In the presence of strong heterogeneities, it is well known that the use of explicit schemes for the transport of species in a porous medium suffers from severe restrictions on the time step. This has led to the development of implicit schemes that are increasingly favored by practitioners for their computational efficiency. The transport equation requires knowledge of the velocity field, which results from an elliptic problem (Darcy problem) that is the most expensive part of the computation. When considering large reservoirs, a cost-effective way of approximating the Darcy problems is using multiscale domain decomposition (MDD) methods. They allow for the pressure and velocity fields to be computed on coarse meshes (large scale), while detailed basis functions are defined locally, usually in parallel, in a much finer grid (small scale). In this work we adopt the Multiscale Robin Coupled Method (MRCM, Guiraldello et al., 2018; Rocha et al., 2020), which is a generalization of previous MDD methods that allows for great flexibility in the choice of interface spaces. In this article we investigate the combination of the MRCM with implicit transport schemes. A sequentially implicit strategy is proposed, with different trust-region algorithms ensuring the convergence of the transport solver. The method is assessed on several very stringent 2D two-phase problems, demonstrating its stability even for large time steps. It is also shown that the best accuracy is achieved by considering recently introduced non-polynomial interface spaces, since polynomial spaces are not optimal for high-contrast channelized permeability fields.

Varifold solutions of the two-phase three-dimensional magnetohydrodynamic equations

In this paper we study the three-dimensional two-phase magnetohydrodynamic interface problem in a bounded domain. The two incompressible fluids are both Newtonian and the surface tension is considered. We shall use the Galerkin method to construct the approximate solutions in a bounded domain. Due to the magnetic field in the magnetohydrodynamic equations, we cannot use the method of monotone operators to solve the approximate equations. Instead, we will construct an iterating operator and solve the equations by finding the fixed-point of the operator. To deal with the free interface, we shall prove the compactness of the iterating operator and then use the Schauder fixed-point theorem. The existence of the varifold solution is established by the weak convergence method.

Gradient of the single layer potential and quantitative rectifiability for general Radon measures

We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on Rn+1 with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball B⊂Rn+1 such that μ(B)≈r(B)n. This result involves a flatness condition, formulated in terms of the so-called β1-number of B, and the L2(μ|B)-boundedness, as well as a control on the mean oscillation on the ball, of the operatorTμf(x)=∫∇xE(x,y)f(y)dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.

Similarity solution of two-phase cylindrical Stefan solidification problem

The mathematical model of determining temperature fields in cylindrical domain with solidification process is represented. The solidification process of cylinder due to cooling is constructed by two-phase cylindrical Stefan problem for liquid and solid zones with freezing interface. Respect to strength of the heat sink at the center of cylindrical material boundary condition &nbsp;is an important to determine temperature in solid domain. The analytical solution of the problem is introduced with method of similarity principle which enables us to reduce free boundary problem to ordinary differential equations. Temperature solutions of solid and liquid zones are represented by special function which called exponential integral equation. The free boundary at freezing interface and temperatures at two phases are determined. Lemmas about exponential integral functions are introduced and used to prove that obtained operator function is contraction operator. Upper boundness of the exponential integral function is checked graphically. It is shown that existence of uniqueness of solution exists.

Global solutions for the Muskat problem in the scaling invariant Besov space [formula omitted

The one-phase and two-phase Muskat problems with arbitrary viscosity contrast are studied in all dimensions. They are quasilinear parabolic equations for the graph free boundary. We prove that small data in the scaling invariant homogeneous Besov space B˙∞,11 lead to unique global solutions. The proof exploits a new structure of the Dirichlet-Neumann operator which allows us to implement a robust fixed-point argument. As a consequence of this method, the initial data is only assumed to be in B˙∞,11 and the solution map is Lipschitz continuous in the same topology. For the general Muskat problem, the only known scaling invariant result was obtained in the Wiener algebra (plus an L2 assumption) which is strictly contained in B˙∞,11.

Asymptotic Bayesian Optimization: A Markov sampling-based framework for design optimization

This paper presents a Markov sampling-based framework, called Asymptotic Bayesian Optimization, for solving a class of constrained design optimization problems. The optimization problem is converted into a unified two-phase sample generation problem which is solved by an effective Markov chain Monte Carlo simulation scheme. First, an exploration phase generates designs distributed over the feasible design space. Based on this information, an exploitation phase obtains a set of designs lying in the vicinity of the optimal solution set. The proposed formulation can handle continuous, discrete, or mixed discrete-continuous design variables. Appropriate adaptive proposal distributions for the continuous and discrete design variables are suggested. The set of optimal solutions provides valuable sensitivity information of the different quantities involved in the problem with respect to the design variables. Representative examples including an analytical problem involving nonlinear benchmark functions, a classical engineering design problem, and a performance-based design optimization problem of a structural system under stochastic excitation are presented to show the effectiveness and potentiality of the proposed optimization scheme. Validation calculations show that the scheme is a flexible, efficient and competitive choice for solving a wide range of classical and complex engineering design problems.

An Unfitted Finite Element Method for Two-Phase Stokes Problems with Slip Between Phases

We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor--Hood finite element pair $\mathbf{P}_{k+1}-P_k$, $k\ge1$, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach with respect to the contrast in viscosity, slip coefficient value, and position of the interface relative to the fixed computational mesh.

On a two-phase free boundary problem ruled by the infinity Laplacian

In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in B1 are universally Lipschitz continuous in B1/2, which is the optimal regularity for the problem. We make a new use of the Ishii–Lions’ method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.

All-regime two-phase flow modeling using a novel four-field large interface simulation approach

Two-phase flows may occur in many regimes, ranging from dispersed flow, slug or plug flow to stratified flow. To capture all those regimes with a monolithic approach, a so-called all-regime two-phase model should be developed. This paper presents such an all-regime two-phase modeling approach called Four-Field Large Interface Simulation (FF-LIS), in which we consistently consider two phases (gas and liquid) which can both be either resolved or sub-grid, resulting in four fields. The efficacy of this approach hinges on the accuracy with which the inter-scale mass, momentum and energy transfer terms between dispersed and resolved regimes are captured. In this paper, we introduce three phenomenological mechanisms to control the inter-scale transfer: absorption, break-up and coalescence. The separation of multiphase scales allows for the explicit modeling of transfer between them, whereas other methods often rely on an algebraic regime map in which transfer between scales is accounted for implicitly. We apply FF-LIS to the simulation of both bubbly and fully resolved single bubble flow, as a way of validating FF-LIS in those two two-phase limits. Next, the transfer terms are illustrated by simulation of bubbles rising in a column. As the bubble concentration increases, it is shown that a transfer from sub-grid to resolved occurs. Finally, we apply FF-LIS to the simulation of two-phase horizontal pipe flow in which the several flow regimes, depending on the superficial velocities, are recovered, and to the simulation of a plunging jet in which FF-LIS is shown to produce an entrained bubble plume. FF-LIS presents a computational platform for faithful two-phase resolved and sub-grid simulation of relevant real-world two-phase problems, and allows two-phase modelers for the explicit control on what can be resolved, and what should be sub-grid.

Shape Optimization of an Imperfect Interface: Steady-State Heat Diffusion

In the context of a diffusion equation, this work is devoted to a two-phase optimal design problem where the interface, separating the phases, is imperfect, meaning that the solution is discontinuous, while the normal flux is continuous and proportional to the jump of the solution. The shape derivative of an objective function with respect to the interface position is computed by the adjoint method. Numerical experiments are performed with the level set method and an exact remeshing algorithm so that the interface is captured by the mesh at each optimization iteration. Comparisons with a perfect interface are discussed in the setting of optimal design or inverse problems.

Gas-liquid two-phase flows simulation based on weakly compressible scheme with interface-adapted AMR method

A weakly compressible scheme using the interface-adapted adaptive mesh refinement (AMR) method is proposed for simulating low Mach number gas-liquid two-phase flows in large-scale and high-resolution simulations. A pressure evolution equation is derived from the iso-thermal process and low-Mach-number assumptions. We solve the Navier-Stokes equation and the pressure equation explicitly including acoustic waves. In the conventional semi-implicit method to solve incompressible two-phase flows, the computational cost of the linear solver increases as the problem size becomes larger. The conservative phase-field equation was solved using the finite volume method (FVM) to capture the gas-liquid interface while conserving the total mass of each phase. We developed GPU code for a tree-based AMR that greatly reduces the computational cost of assigning a high-resolution mesh around the moving interface region. The data structure of the memory pool has extra space allocated to prevent frequent memory allocations and deallocations. Results for the two-phase flow benchmark problems, 2D rising-bubble and dam-breaking onto dry floor problem, are in good agreement with the results of incompressible solvers. The two-phase flow simulation of the dam-breaking onto a wet floor problem maintained the shape interface regardless of whether the AMR method was used or not. We have computed a two-dimensional soap bubble growing from 16mm to 120mm in diameter with about 0.9 million total cells. The final thickness of the liquid film was about 0.1mm with the finest cell size being 0.0195mm, equivalent to 8,192×8,192 uniform cells. This computation showed that high-resolution two-phase flow simulation with the explicit scheme and AMR method reproduced the behavior of the thin liquid film.

Решение двухфазной задачи Стефана в энтальпийной постановке со сглаживанием коэффициентов

To simulate heat transfer processes with phase transitions, the classical enthalpy model of Stefan is used, accompanied by phase transformations of the medium with absorption and release of latent heat of a change in the state of aggregation. The paper introduces a solution to the two-phase Stefan problem for a one-dimensional quasilinear second-order parabolic equation with discontinuous coefficients. A method for smearing the Dirac delta function using the smoothing of discontinuous coefficients by smooth functions is proposed. The method is based on the use of the integral of errors and the Gaussian normal distribution with an automated selection of the value of the interval of their smoothing by the desired function (temperature). The discontinuous coefficients are replaced by bounded smooth temperature functions. For the numerical solution, the finite difference method and the finite element method with an automated selection of the smearing and smoothing parameters for the coefficients at each time layer are used. The results of numerical calculations are compared with the solution of Stefan’s two-phase self-similar problem --- with a mathematical model of the formation of the ice cover of the reservoir. Numerical simulation of the thawing effect of installing additional piles on the existing pile field is carried out. The temperature on the day surface of the base of the structure is set with account for the amplitude of air temperature fluctuations, taken from the data of the Yakutsk meteorological station. The study presents the results of numerical calculations for concrete piles installed in the summer in large-diameter drilled wells using cement-sand mortars with a temperature of 25 °С. The distributions of soil temperature are obtained for different points in time

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier–Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.

Preconditioned Jacobian-free Newton–Krylov fully implicit high order WENO schemes and flux limiter methods for two-phase flow models

Motivated by the high-resolution properties of high-order Weighted Essentially Non-Oscillatory (WENO) and flux limiter (FL) for steep-gradient problems and the robust convergence of Jacobian-free Newton-Krylov (JFNK) methods for nonlinear systems, the preconditioned JFNK fully implicit high-order WENO and FL schemes are proposed to solve the transient two-phase two-fluid models. Specially, the second-order fully-implicit BDF2 is used for the temporal operator and then the third-order WENO schemes and various flux limiters can be adopted to discrete the spatial operator. For the sake of the generalization of the finite-difference-based preconditioning acceleration methods and the excellent convergence to solve the complicated and various operational conditions, the random vector instead of the initial condition is skillfully chosen as the solving variables to obtain better sparsity pattern or more positions of non-zero elements in this paper. Finally, the WENO_JFNK and FL_JFNK codes are developed and then the two-phase steep-gradient problem, phase appearance/disappearance problem, U-tube problem and linear advection problem are tested to analyze the convergence, computational cost and efficiency in detailed. Numerical results show that WENO_JFNK and FL_JFNK can significantly reduce numerical diffusion and obtain better solutions than traditional methods. WENO_JFNK gives more stable and accurate solutions than FL_JFNK for the test problems and the proposed finite-difference-based preconditioning acceleration methods based on the random vector can significantly improve the convergence speed and efficiency.

On the Efficient Flow Restoration in Spectrally-Spatially Flexible Optical Networks

We focus on the efficient modeling and optimization of the flow restoration in the spectrally-spatially flexible optical networks (SS-FONs) realized using a single mode fiber bundle. To this end, we study a two-phase optimization problem, which consists of: (i) the network planning with respect to the spectrum usage and (ii) the flow restoration after a failure aiming at maximizing the restored bit-rate. Both subproblems we formulate using the integer linear programming with two modeling approaches—the node-link and the link-path. We perform simulations divided into: (i) a comparison of the proposed approaches, (ii) an efficient flow restoration in SS-FONs—case study. The case study focuses on the verification whether the spectral-spatial allocation may improve the restoration process (compared to the spectral allocation) and on the determination of the full restoration cost (the amount of additional resources required to restore whole traffic) in two network topologies. The results show that the spectral-spatial allocation allows us to restore up to 4% more traffic compared to the restoration with only the spectral channels. They also reveal that the cost of the full traffic restoration depends on plenty of factors, including the overall traffic volume and the network size, while the spectral-spatial allocation allows us to reduce its value about 30%.

Blow-ups of caloric measure in time varying domains and applications to two-phase problems

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in Rn+1, n≥2, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems:1.Let Ω1 and Ω2 be disjoint domains in Rn+1, n≥2, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set E⊂∂Ω1∩∂Ω2 of mutual absolute continuity of the associated caloric measures ωi with poles at p¯i=(pi,ti)∈Ωi, i=1,2. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of ω1|E is n+1 and the tangent measures of ω1 at ω1-a.e. point of E are equal to a constant multiple of the parabolic (n+1)-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis.2.If, additionally, ω1|E and ω2|E are doubling, log⁡dω2|Edω1|E∈VMO(ω1|E), and E is relatively open in the support of ω1, then their tangent measures at every point of E are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that if Ω1 is a δ-Reifenberg flat domain for δ small enough and Ω2=Rn+1∖Ω‾1, and log⁡dω2dω1∈VMO(ω1), then Ω1∩{t<t2} is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian.3.Finally, we establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure. Assuming that Ωi, 1≤i≤3, are quasi-regular domains for both the heat and the adjoint heat equations, the set of points on ∩i=13∂Ωi, where the three caloric measures are mutually absolutely continuous has null caloric measure. In the course of proving our main theorems, we obtain new results on heat potential theory, parabolic geometric measure theory, nodal sets of caloric functions, and Optimal Transport, that may be of independent interest.

Simplified method for simulation of incompressible viscous flows inspired by the lattice Boltzmann method.

The lattice Boltzmann method (LBM) has gained increasing popularity in incompressible viscous flow simulations, but it uses many distribution functions (far more than the flow variables) and is often memory demanding. This disadvantage was overcome by a recent approach that solves the more actual macroscopic equations obtained through Taylor series expansion analysis of the lattice Boltzmann equations [Lu etal., J. Comput. Phys. 415, 109546 (2020)JCTPAH0021-999110.1016/j.jcp.2020.109546]. The key is to keep some small additional terms (SATs) to stabilize the numerical solution of the weakly compressible Navier-Stokes equations. However, there are many SATs that complicate the implementation of their method. Based on some analyses and numerous tests, we ultimately pinpoint two essential ingredients for stable simulations: (1) suitable density (pressure) diffusion added to the continuity equation and (2) proper numerical dissipation related to the velocity divergence added to the momentum equations. Then we propose a simplified method that is not only easier to implement but noticeably faster than the original method and the LBM. It contains much simpler SATs that only involve the density (pressure) derivatives, and it requires no intermediate steps or variables. As well, it is extended for thermal flows with small temperature variations and for two-phase flows with uniform density and viscosity. Several test cases, including some two-phase problems under two-dimensional, axisymmetric, and three-dimensional geometries, are presented to demonstrate its capability. This work may help pave the way for the simplest simulation of incompressible viscous flows on collocated grids based on the artificial compressibility methodology.

Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials

Stefan problems provide one of the most fundamental frameworks to capture phase change processes. The problem in cylindrical coordinates can model outward solidification, which ensures the thermal design and operation associated with phase change materials (PCMs). However, this moving boundary problem is highly nonlinear in most circumstances. Exact solutions are restricted to certain domains and boundary conditions. It is therefore vital to develop approximate analytical solutions based on physically tangible assumptions, e.g., a small Stefan number. A great amount of work has been done in one-phase Stefan problems, where the initial state is at its fusion temperature, yet very few literature has considered two-phase problems particularly in cylindrical coordinates. This paper conducts an asymptotic analysis for a two-phase Stefan problem for outward solidification in a hollow cylinder, consisting of three temporal and four spatial scales. The results are compared with the enthalpy method that simulates a mushy region between two phases by numerical iterations, rather than a sharp interface in Stefan problems. After studying both mathematical models, the role of mushy-zone thickness in the enthalpy method is also unveiled. Moreover, a wide range of geometric ratios, thermophysical properties and Stefan numbers are selected from the literature to explore their effects on the developed model with regards to interface motion and temperature profile. It can be concluded that the asymptotic solution is capable of tracking the moving interface and evaluating the transient temperature for various geometric ratios and thermophysical properties in PCMs. The accuracy of this solution is found to be affected by Stefan number only, and the computational cost is much less compared with the enthalpy method and other numerical schemes.

Reexamining the one-fluid formulation for two-phase flows

The one-fluid formulation for two-phase problems has been firmly established as a solution methodology, which applies everywhere in the domain, including at the gas-liquid interface. The physical consistency of this formulation is examined by considering whether it satisfies the governing equations in the bulk region of the domain and the jump conditions at the interface. An analysis of various versions of the momentum and energy one-field formulations published in the literature reveals a frequent inconsistency, particularly for problems involving phase change. The issue is in part attributed to a lack of equivalence between conservative and non-conservative forms of the equation when m˙Γ≠0, although other problems are also uncovered. A proposed generalized one-field formulation is advanced that is able to exhibit the desired level of physical consistency by satisfying interfacial and bulk phase relations. Additionally, as an illustration of the use of the proposed one-field formulation, a derivation of the VoF transport equation is presented under phase change conditions.