Viscous Approximation Research Articles

Characteristic Boundary Layers for Mixed Hyperbolic‐Parabolic Systems in One Space Dimension and Applications to the Navier‐Stokes and MHD Equations

AbstractWe provide a detailed analysis of the boundary layers for mixed hyperbolic‐parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so‐called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so‐called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic‐parabolic system and for the hyperbolic system obtained by neglecting the second‐order terms. Our analysis applies in particular to the compressible Navier‐Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems. © 2020 Wiley Periodicals, Inc.

An entropy satisfying two-speed relaxation system for the barotropic Euler equations: application to the numerical approximation of low Mach number flows

In the first part of this work, we introduce a new relaxation system in order to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Our relaxation system is a generalization of the well-known Suliciu relaxation system, and it is entropy satisfying. A Godunov-type finite volume scheme based on the exact resolution of the Riemann problem associated with the relaxation system is deduced, as well as its stability properties. In the second part of this work, we show how the new relaxation approach can be successfully applied to the numerical approximation of low Mach number flows. We prove that the underlying scheme satisfies the well-known asymptotic-preserving property in the sense that it is uniformly (first-order) accurate with respect to the Mach number, and at the same time it satisfies a fully discrete entropy inequality. This discrete entropy inequality allows us to prove strong stability properties in the low Mach regime. At last, numerical experiments are given to illustrate the behaviour of our scheme.

Inviscid limit for the compressible Euler system with non-local interactions

The collective behavior of animals can be modeled by a system of equations of continuum mechanics endowed with extra terms describing repulsive and attractive forces between the individuals. This system can be viewed as a generalization of the compressible Euler equations with all of its unpleasant consequences, e.g., the non-uniqueness of solutions. In this paper, we analyze the equations describing a viscous approximation of a generalized compressible Euler system and we show that its dissipative measure-valued solutions tend to a strong solution of the Euler system as viscosity tends to zero, provided the strong solution exists.

Lower Semi-continuity of the Index in the Viscosity Method for Minimal Surfaces

AbstractThe goal of the present work is two-fold. First we prove the existence of a Hilbert manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary compact submanifold $M^m$ of an Euclidian space ${{\mathbb{R}}}^Q$. Second, using this Hilbert manifold structure, we prove a lower semi-continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying a Struwe entropy estimate and a bubble tree strongly converging in $W^{1,2}$ to a limiting minimal surface as the viscous parameter is going to zero.

Study of mirror effect on scrape‐off layer‐divertor plasma based on a generalized fluid model incorporating ion temperature anisotropy

The mirror effect on scrape‐off layer (SOL)‐divertor plasma profiles was studied by relaxing the assumption of homogeneous magnetic fields used in a one‐dimensional (1D) plasma fluid model, which incorporates the anisotropic ion temperature. For collisionless plasmas, the perpendicular ion temperature proportional to the magnetic field strength is observed, which reflects the conservation of the magnetic moment. In diverging field cases, the plasma flow becomes supersonic, while it is kept subsonic in contracting field cases. Parallel momentum gains/losses and energy transfers between parallel and perpendicular components caused by the mirror effect are also observed. In addition, it is indicated that the viscous flux approximation is invalid in inhomogeneous magnetic fields regardless of the plasma collisionality.

A note on the $3$-D Navier-Stokes equations

We consider the Navier-Stokes model in a bounded smooth domain $\Omega\subset \mathbb R^3$. Assuming a smallness condition on the external force $f$, which does not necessitate smallness of $\| f\|_{[L^2(\Omega)]^3}$-norm, we show that for any smooth divergence free initial data $u_0$ there exists ${\mathcal T}={\mathcal T}(\|u_0\|_{[L^2(\Omega)]^3})$ satisfying $$ {\mathcal T} \to 0 \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to 0 \quad \text{and} \quad {\mathcal T} \to \infty \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to \infty, $$% and such that either a corresponding regular solution ceases to exist until $\mathcal T$ or, otherwise, it is globally defined and approaches a maximal compact invariant set $\mathbb A$. The latter set $\mathbb A$ is a global attractor for the semigroup restricted to initial velocities $u_0$ in a certain ball of fractional power space $X^{1/4}$ associated with the Stokes operator, which in turn does not necessitate smallness of the gradient norm $\|\nabla u_0\|_{[L^2(\Omega)]^3}$. Moreover, $\mathbb A$ attracts orbits of bounded sets in $X$ through Leray-Hopf type solutions obtained as limits of viscous parabolic approximations.

Investigation into Behavior of a Steam-Water Mixture Flow Through Holes in a Submerged Perforated Sheet at High Void Fractions

Processing of experimental data on the pressure difference across a submerged perforated sheet (SPS) revealed that, at sufficiently high void fractions under SPS, the pressure difference across it became less than the pressure difference for the pure steam stream with the same flowrate. To find the cause of this, the effect of a liquid film, which can be formed on the SPS upstream surface as a result of water droplets’ impact and can smooth over sharp edges of holes in SDS, was examined. This can decrease the pressure drop across the sharp edges of holes. This assumption was checked through numerical solution to several model problems in the axisymmetric formulation for a steam flow in a round pipe with an orifice. The flow of steam and water was modeled using the viscous incompressible liquid approximation, while the turbulence was described by the k–e model. The evolution of the interfacial area was modeled using the VOF model. The following model problems of steam flow through an orifice were studied: a single-phase flow, a flow through the orifice with a liquid film on its upstream surface, a flow through a chamfered hole, and a flow through the orifice with a liquid film on its upstream surface without liquid supply to the film. The predictions demonstrate that even the approximate account of the liquid film effect on the steam flow yields a considerable decrease in the pressure drop across the hole (from 8 to 24%) due to smoothing its sharp outlet edges over. This makes it possible to make a conclusion that the cause of a decrease in the pressure drop across SPS observed in the experiments at high void fractions is the formation of a liquid film, which smooths the sharp edges of the hole.

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor–Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier–Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number and the dimensionless wall regression ratio as primary and secondary perturbation parameters. In this manner, closed-form analytical solutions are obtained for both large and small Reynolds number with small-to-moderate . The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.

On the asymptotic behavior of a subcritical convection-diffusion equation with nonlocal diffusion

In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term and letting . Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with L1-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik’s inequality, we obtain the first term in the asymptotic behavior of the nonnegative solutions. Finally, the large time behavior of changing sign solutions is proved using the classical flux-entropy method and estimates for the nonlocal operator.

The Comparison of Viscous Force Approximations of Smoothed Particle Hydrodynamics in Poiseuille Flow Simulation

Poiseuille flows at two Reynolds numbers (Re) 2.5 × 10−2 and 5.0 are simulated by two different smoothed particle hydrodynamics (SPH) schemes on regular and irregular initial particles' distributions. In the first scheme, the viscous stress is calculated directly by the basic SPH particle approximation, while in the second scheme, the viscous stress is calculated by the combination of SPH particle approximation and finite difference method (FDM). The main aims of this paper are (a) investigating the influences of two different schemes on simulations and reducing the numerical instability in simulating Poiseuille flows discovered by other researchers and (b) investigating whether the similar instability exists in other cases and comparing results with the two viscous stress approximations. For Re = 2.5 × 10−2, the simulation with the first scheme becomes instable after the flow approaches to steady-state. However, this instability could be reduced by the second scheme. For Re = 5.0, no instability for two schemes is found.

Snow carrots after the Chelyabinsk event and model implications for highly porous solar system objects

AbstractAfter the catastrophic disruption of the Chelyabinsk meteoroid, small fragments formed funnels in the snow layer covering the ground. We constrain the pre‐impact characteristics of the fragments by simulating their atmospheric descent with the atmospheric entry model. Fragments resulting from catastrophic breakup may lose about 90% of their initial mass due to ablation and reach the snow vertically with a free‐fall velocity in the range of 30–90 m s−1. The fall time of the fragments is much longer than their cooling time, and, as a consequence, fragments have the same temperature as the lower atmosphere, i.e., of about −20 °C. Then, we use the shock physics code iSALE to model the penetration of fragments into fluffy snow, the formation of a funnel and a zone of denser snow lining its walls. We examine the influence of several material parameters of snow and present our best‐fit model by comparing funnel depth and funnel wall characteristics with observations. In addition, we suggest a viscous flow approximation to estimate funnel depth dependence on the meteorite mass. We discuss temperature gradient metamorphism as a possible mechanism which allows to fill the funnels with denser snow and to form the observed “snow carrots.” This natural experiment also helps us to calibrate the iSALE code for simulating impacts into highly porous matter in the solar system including tracks in the aerogel catchers of the Stardust mission and possible impact craters on the 67P/Churyumov‐Gerasimenko comet observed recently by the Rosetta mission.

Quasistatic crack growth based on viscous approximation: a model with branching and kinking

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

Optical Feedback Interferometry for Velocity Measurement of Parallel Liquid-Liquid Flows in a Microchannel.

Optical feedback interferometry (OFI) is a compact sensing technique with recent implementation for flow measurements in microchannels. We propose implementing OFI for the analysis at the microscale of multiphase flows starting with the case of parallel flows of two immiscible fluids. The velocity profiles in each phase were measured and the interface location estimated for several operating conditions. To the authors knowledge, this sensing technique is applied here for the first time to multiphase flows. Theoretical profiles issued from a model based on the Couette viscous flow approximation reproduce fairly well the experimental results. The sensing system and the analysis presented here provide a new tool for studying more complex interactions between immiscible fluids (such as liquid droplets flowing in a microchannel).

The existence of global weak solutions to the shallow water wave model with moderate amplitude

The existence of global weak solutions to the shallow water model with moderate amplitude, which is firstly introduced in Constantin and Lannes’s work (2009), is investigated in the space C([0,∞)×R)⋂L∞((0,∞);H1(R)) without the sign condition on the initial value by employing the limit technique of viscous approximation. A new one-sided lower bound and the higher integrability estimate act a key role in our analysis.

An overview on the approximation of boundary Riemann problems through physical viscosity

This note aims at providing an overview of some recent results concerning the viscous approximation of so-called boundary Riemann problems for nonlinear systems of conservation laws in small total variation regimes.

Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions.

A Hyperbolic–Elliptic Model Problem for Coupled Surface–Subsurface Flow

Abstract We consider a model problem for coupled surface–subsurface flow. The model consists of a nonlinear kinematic wave equation for the surface fluid’s height and a Brinkman model that governs fluid velocity and pressure for subsurface dynamics. For this coupled hyperbolic–elliptic model we establish the existence of weak solutions. The proof is based on a viscous approximation and the method of compensated compactness by virtue of appropriate energy estimates. To solve the coupled problem numerically, a finite volume method is applied. The numerical scheme is used to illustrate the influence of the Brinkman parameter on the coupled flow pattern for infiltration scenarios.

On weak solutions to a shallow water wave model of moderate amplitude

The existence of global weak solutions for a dissipative model equation for shallow water wave of moderate amplitude is studied in the space without the sign condition on the initial value by employing the limit technique of viscous approximation. A new one-sided lower bound and the higher integrability estimate act a key role in our analysis. Our results partly extend the work of Coclite et al. on the existence of global weak solutions to the generalized hyperlastic-rod equation.

Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation

We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three dimensional de Sitter space with a line. The resulting solution respects SO(3)_q x SO(1,1) x Z_2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations. The macroscopic solutions are obtained in de Sitter space and are subject to the same symmetries used to obtain the exact kinetic solution.

Global weak solutions for a generalized Dullin-Gottwald-Holm equation in the space H 1 ( R )

The existence of global weak solutions of the Cauchy problem for a generalized Dullin-Gottwald-Holm equation is established under the assumption that the initial value merely lies in the space . The limit of the viscous approximation for the equation is used to prove the global existence in the space . The elements in our study include a one-sided super bound estimate and a space-time higher-norm estimate on the first order derivative of the solution with respect to the space variable. MSC: 35Q35, 35Q51.