In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work , we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.
In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gerard-Varet and M. Prestipino, <i>Z. Angew. Math. Phys.</i>, 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.
The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity . Under some conditions on the initial and boundary data, we show that the thickness is of the order | In |. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.
In this paper, we will survey some recent results on the study of the viscous and invisid compressible flow with vacuum. It is wellknown that the study on vacuum has significance in the investigation on some important physical phenomena. However, most of the important questions about vacuum are still open due to the singularities caused by vacuum which need new mathematical tools and techniques to handle.
In the long-time scale, we consider the fluid dynamical limits for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic velocity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equations are the incompressible Navier-Stokes equations with viscous heating. This is different from the case when the even and odd parts of the fluctuation have the same scaling where the standard incompressible Navier-Stokes equations without viscous heating are obtained. On the other hand, when the background is a local Maxwellian, it is shown that the above even-odd decomposition leads to a non-classical fluid dynamical system without viscous heating which has been used to describe the ghost effect in the kinetic theory. In addition, the above even-odd decomposition is justified rigorously for the Boltzmann equation for the former case when the background is an absolute Maxwellian.
In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations, EulerPoisson equations and NavierStokes equations. The main concern is the well-posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas. Furthermore, the relation of the solutions for the gas dynamics with vacuum to those of the Boltzmann equation will also be discussed. In fact, the results obtained so far for vacuum states are far from being complete and satisfactory. Therefore, this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems. Moreover, the problems considered here are limited to the author's interest and
We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set HamiltonJacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity <i>A</i> increases. Three distinct regimes are identified. When <i>A</i> is below the critical level, homogenization holds and the turbulent flame speed <i>s</i> <sub>T</sub> (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as <i>s</i> <sub>T</sub> <i>O</i>(<i>A</i>/log <i>A</i>). In the second regime where <i>A</i> is slightly above the critical value, homogenization breaks down, and <i>s</i> <sub>T</sub> is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When <i>A</i> is
We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity
The minimal speeds (c) of the KolmogorovPetrovskyPiskunov (KPP) fronts at small diffusion ( 1) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of c reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advectiondiffusion operator with spacetime periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of spectral methods and finite element, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling c= O ( 1/4) in steady cellular flows, a new relation c= O (1) as 1 is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub
The shape stability of the reaction interface for reactive flow in a layered porous medium is studied. This is done using a complete linearized stability analysis in the setting of a free boundary model of this phenomenon. The spectrum of the linearized problem is obtained in the general case, and it is compared with that obtained for homogeneous media in two typical cases.
We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0<<i></i><2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent <i>a</i>(<i>d</i>/<i></i>)+1 are statistically realizable, where <i>d</i> is space dimension and <i></i>=2<i></i>. An infinite sequence of invariants <i>J</i> <sub>p</sub>, <i>p</i>=0,1,2,..., is pointed out, where <i>J</i> <sub>0</sub> is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants <i>J</i> <sub>0</sub> or <i>J</i> <sub>1</sub> must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariantthe first being <i>J</i> <sub>p</sub>converges at long times to a scaling
We develop an efficient numerical method to compute single slit or double slit diffraction patterns from high frequency wave in inhomogeneous media. We approximate the high frequency asymptotic solution to the Helmholtz equation using the Eulerian Gaussian beam summation proposed in [20, 21]. The emitted rays from a slit are embedded in the phase space using an open segment. The evolution of this open curve is accurately computed using the recently developed Grid Based Particle Method  which results in a very efficient computational algorithm. Following the grid based particle method we proposed in [23, 24], we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The end-points of the open curve are tracked explicitly and consistently with interior particles. To construct the overall wavefield, each of these sampling particles also carry necessary quantities that are obtained by solving advection-reaction equations. Numerical experiments show that the resulting method can model diffraction patterns in inhomogeneous media accurately, even in the occurrence of caustics.
In this paper, our aim is to present (1) an embedded fracture model (EFM) for coupled flow and mechanics problem based on the dual continuum approach on the fine grid and (2) an upscaled model for the resulting fine grid equations. The mathematical model is described by the coupled system of equation for displacement, fracture and matrix pressures. For a fine grid approximation, we use the finite volume method for flow problem and finite element method for mechanics. Due to the complexity of fractures, solutions have a variety of scales, and fine grid approximation results in a large discrete system. Our second focus in the construction of the upscaled coarse grid poroelasticity model for fractured media. Our upscaled approach is based on the nonlocal multicontinuum (NLMC) upscaling for coupled flow and mechanics problem, which involves computations of local basis functions via an energy minimization
In this paper, we propose a rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods . Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) in  and derive appropriate local problems in oversampled regions once we identify important modes representing each continuum. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. Note that the use of non-local upscaled model for porous
Myxobacteria are social bacteria, that can glide in two dimensions and form counterpropagating, interacting waves. Here, we present a novel age-structured, continuous macroscopic model for the movement of myxobacteria. The derivation is based on microscopic interaction rules that can be formulated as a particle-based model and set within the Self-Organized Hydrodynamics (SOH) framework. The strength of this combined approach is that microscopic knowledge or data can be incorporated easily into the particle model, whilst the continuous model allows for easy numerical analysis of the diﬀerent eﬀects. However, we found that the derived macroscopic model lacks a diﬀusion term in the density equations, which is necessary to control the number of waves, indicating that a higher order approximation during the derivation is crucial. Upon ad hoc addition of the diﬀusion term, we found very good agreement between the age-structured model and the biology. In particular, we analyzed the inﬂuence of a refractory (insensitivity) period following a reversal of movement. Our analysis reveals that the refractory period is not necessary for wave formation, but essential to wave synchronization, indicating separate molecular mechanisms.
We introduce a unified particle framework which integrates the phase-field method with multi-material simulation to allow modeling of both liquids and solids, as well as phase transitions between them. A simple elastoplastic model is used to capture the behavior of various kinds of solids, including deformable bodies, granular materials, and cohesive soils. States of
matter or phases, particularly liquids and solids, are modeled using the non-conservative Allen-Cahn equation. In contrast, materials—made of different substances—are advected by the conservative Cahn-Hilliard equation. The distributions of phases and materials are represented by a phase variable and a concentration variable, respectively, allowing us to represent commonly observed fluid-solid interactions. Our multi-phase, multi-material system is governed by a unified Helmholtz free energy density. This framework provides the first method in computer graphics capable of modeling a continuous interface between phases. It is versatile and can be readily used in many scenarios that are challenging to simulate. Examples are provided to demonstrate the capabilities and effectiveness of this approach.
In this paper, we present a novel pairwise-force smoothed particle hydrodynamics (PF-SPH) model to allow modeling of
various interactions at interfaces in real time. Realistic capture of interactions at interfaces is a challenging problem for SPH-based simulations, especially for scenarios involving multiple interactions at different interfaces. Our PF-SPH model can readily handle multiple kinds of interactions simultaneously in a single simulation; its basis is to use a larger support radius than that used in standard SPH. We adopt a novel anisotropic filtering term to further improve the performance of interaction forces. The proposed model is stable; furthermore, it avoids the particle clustering problem which commonly occurs at the free surface. We show how our model can be used to capture various interactions. We also consider the close connection between droplets and bubbles, and show how to animate bubbles rising in liquid as well as bubbles in air. Our method is versatile, physically plausible and easy-to-implement. Examples are provided to demonstrate the capabilities and effectiveness of our approach.
Surface flow phenomena, such as rain water flowing down a tree trunk and progressive water front in a shower room, are common in real life. However, compared with the 3D spatial fluid flow, these surface flow problems have been much less studied in the graphics community. To tackle this research gap, we present an efficient, robust and high-fidelity simulation approach based on the shallow-water equations. Specifically, the standard shallow-water flow model is extended to general triangle meshes with a feature-based bottom friction model, and a series of coherent mathematical formulations are derived to represent the full range of physical effects that are important for real-world surface flow phenomena. In addition, by achieving compatibility with existing 3D fluid simulators and by supporting physically realistic interactions with multiple fluids and solid surfaces, the new model is flexible and readily extensible for coupled phenomena. A wide range of simulation examples are presented to demonstrate the performance of the new approach.
Markus Ihmsen · Jens Cornelis · Barbara Solenthaler · Christopher J Horvath · Matthias Teschner. Implicit Incompressible SPH. 2014.
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Zhu B, Yang X, Fan Y, et al. Creating and Preserving Vortical Details in SPH Fluid[J]. Computer Graphics Forum, 2010, 29(7): 2207-2214.
Chenfanfu Jiang · Craig Schroeder · Andrew Selle · Joseph Teran · Alexey Stomakhin. The affine particle-in-cell method. 2015.
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Yusuke Tsuda · Yonghao Yue · Yoshinori Dobashi · Tomoyuki Nishita. Visual simulation of mixed-motion avalanches with interactions between snow layers. 2010.
Ren B, Yan X, Yang T, et al. Fast SPH simulation for gaseous fluids[J]. The Visual Computer, 2016, 32(4): 523-534.
Suntae Kim · Jeongmo Hong. Visual simulation of turbulent fluids using MLS interpolation profiles. 2013.
Shiguang Liu · Yixin Xu · Junyong Noh · Yiying Tong. Visual fluid animation via lifting wavelet transform: Fluid animation via lifting wavelet transform. 2014.
Gaseous fluids may move slowly, as smoke does, or at high speed, such as occurs with explosions. High-speed gas flow is always accompanied by low-speed gas flow, which produces rich visual details in the fluid motion. Realistic visualization involves a complex dynamic flow field with both low and high speed fluid behavior. In computer graphics, algorithms to simulate gaseous fluids address either the low speed case or the high speed case, but no algorithm handles both efficiently. With the aim of providing visually pleasing results, we present a hybrid algorithm that efficiently captures the essential physics of both low- and high-speed gaseous fluids. We model the low speed gaseous fluids by a grid approach and use a particle approach for the high speed gaseous fluids. In addition, we propose a physically sound method to connect the particle model to the grid model. By exploiting complementary strengths and avoiding weaknesses of the grid and particle approaches, we produce some animation examples and analyze their computational performance to demonstrate the effectiveness of the new hybrid method.
The nonlinear and non-stationary nature of Navier-Stokes equations produces fluid flows that can be noticeably different in appearance with subtle changes. In this paper we introduce a method that can analyze the intrinsic multiscale features of flow fields from a decomposition point of view, by using the Hilbert-Huang transform method on 3D fluid simulation. We show how this method can provide insights to flow styles and help modulate the fluid simulation with its internal physical information. We provide easy-toimplement algorithms that can be integrated with standard grid-based fluid simulation methods, and demonstrate how this approach can modulate the flow field and guide the simulation with different flow styles. The modulation is straightforward and relates directly to the flow¡¯s visual effect, with moderate computational overhead.
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Peer A, Ihmsen M, Cornelis J, et al. An implicit viscosity formulation for SPH fluids[J]. ACM Transactions on Graphics, 2015, 34(4).
Ando R, Thuerey N, Wojtan C, et al. A stream function solver for liquid simulations[J]. ACM Transactions on Graphics, 2015, 34(4).
Natsui S, Nashimoto R, Takai H, et al. SPH simulations of the behavior of the interface between two immiscible liquid stirred by the movement of a gas bubble[J]. Chemical Engineering Science, 2016: 342-355.
Takahashi T, Dobashi Y, Fujishiro I, et al. Implicit Formulation for SPH-based Viscous Fluids[J]. Computer Graphics Forum, 2015, 34(2): 493-502.
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Tao Yang · Ming C Lin · Ralph R Martin · Jian Chang · Shimin Hu. Versatile interactions at interfaces for SPH-based simulations. 2016.
Tao Yang · Jian Chang · Bo Ren · Ming C Lin · Jian J Zhang · Shimin Hu. Fast multiple-fluid simulation using Helmholtz free energy. 2015.
T Weaver · Z Xiao. Fluid Simulation by the Smoothed Particle Hydrodynamics Method: A Survey. 2016.
Seungho Baek · Kiwon Um · Junghyun Han. Muddy water animation with different details: Muddy water animation with different details. 2015.
This paper presents a versatile and robust SPH simulation approach for multiple-fluid flows. The spatial distribution of different phases or components is modeled using the volume fraction representation, the dynamics of multiple-fluid flows is captured by using an improved mixture model, and a stable and accurate SPH formulation is rigorously derived to resolve the complex transport and transformation processes encountered in multiple-fluid flows. The new approach can capture a wide range of realworld multiple-fluid phenomena, including mixing/unmixing of miscible and immiscible fluids, diffusion effect and chemical reaction etc. Moreover, the new multiple-fluid SPH scheme can be readily integrated into existing state-of-the-art SPH simulators, and the multiple-fluid simulation is easy to set up. Various examples are presented to demonstrate the effectiveness of our approach.
Tao YangTsinghua UniversityJian ChangBournemouth UniversityBo RenTsinghua UniversityMing C. LinUniversity of North Carolina at Chapel Hill Jian Jun ZhangBournemouth UniversityShi-Min HuTsinghua University
Fluid Dynamics and Shock Wavesmathscidoc:1608.22003
ACM Transactions on Graphics, 34, (6), 201, 2015.12
Multiple-fluid interaction is an interesting and common visual phenomenon we often observe. In this paper we present an energybased Lagrangian method that expands the capability of existing multiple-fluid methods to handle various phenomena, including extraction, partial dissolution, etc. Based on our user-adjusted Helmholtz free energy functions, the simulated fluid evolves from high-energy states to low-energy states, allowing flexible capture of various mixing and unmixing processes. We also extend the original Cahn-Hilliard equation to gain abilities of simulating complex fluid-fluid interaction and rich visual phenomena such as motionrelated mixing and position based pattern. Our approach is easy to be integrated with existing state-of-the-art smooth particle hydrodynamic (SPH) solvers and can be further implemented on top of the position based dynamics (PBD) method, improving the stability and incompressibility of the fluid during Lagrangian simulation under large time steps. Performance analysis shows that our method is at least 4 times faster than the state-of-the-art multiple-fluid method. Examples are provided to demonstrate the new capability and effectiveness of our approach.
This work extends existing multiphase-fluid SPH frameworks to cover solid phases, including deformable bodies and granular materials. In our extended multiphase SPH framework, the distribution and shapes of all phases, both fluids and solids, are uniformly represented by their volume fraction functions. The dynamics of the multiphase system is governed by conservation of mass and momentum within different phases. The behavior of individual phases and the interactions between them are represented by corresponding constitutive laws, which are functions of the volume fraction fields and the velocity fields. Our generalized multiphase SPH framework does not require separate equations for specific phases or tedious interface tracking. As the distribution, shape and motion of each phase is represented and resolved in the same way, the proposed approach is robust, efficient and easy to implement. Various simulation results are presented to demonstrate the capabilities of our new multiphase SPH framework, including deformable bodies, granular materials, interaction between multiple fluids and deformable solids, flow in porous media, and dissolution of deformable solids.