Viscous Equations Research Articles

The research on current rush to air mass in delivery pipeline system with YOUNGS-VOF method

Water hammer with gas often occurs in long delivery pipeline system lager than ordinary water hammer, surface tracking between air and water is important and difficult. In this paper, flow rush to interception air-mass in long water conveyance pipe system is researched on base on surface in between water and gas tracking method of staggered grid dispersed with YOUNGS-VOF, the limited compressibility method of viscous incompressible fluid control equations, and numerical method of ideal gas state equation. The numerical results show that YOUNGS-VOF method can be used to deal with the interface and water hammer is consistent with the theory analysis.

Lubrication state identification of the line contact tribo-pair under a pure rolling condition with ultrasonic method

A line contact tribo-pair is a key mechanism unit in rolling bearings, which is often characterized by ultra-high contact pressure and ultra-thin oil film. Elastohydrodynamic lubrication is often adopted to characterize the lubrication state of such a tribo-pair. As a primary parameter for elastohydrodynamic lubrication, the oil film thickness is often evaluated with simplified theoretical models or complicated measurements. So far, a comprehensive verification of the lubrication states in a real line-contact tribo-pair, however, is rarely reported. Focusing on the roller/ring tribo-pair of a wet-lubricated rolling bearing under pure rolling conditions, this study investigates the lubrication states by integrating multiple theories. Five regions including isoviscous hydrodynamic, piezoviscous hydrodynamic, elastohydrodynamic lubrication, mixture lubrication, and boundary lubrication regions can be identified using the framework. Then, validation experiments are carried out on a line contact tribo-pair test rig under the same operating conditions applied in the theoretical analysis. The oil film thickness is measured by the ultrasonic method. The analysis results demonstrate that only two regions, the elastohydrodynamic lubrication and mixture lubrication regions, can be identified using the experimental data. The identified elastohydrodynamic lubrication and mixture lubrication regions are consistent with theoretical analysis; and the Blok equation and elastohydrodynamic lubrication theory are suggested to calculate the oil film thickness in the elastohydrodynamic lubrication and mixture lubrication regions, respectively. Moreover, the oil film thickness calculated by the Dowson equation is larger than that based on the elastohydrodynamic lubrication theory due to a different viscous pressure equation.

Selection by vanishing common noise for potential finite state mean field games

The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality.

Viscosity approximation of the solution to Burgers' equations with shock layers

ABSTRACT Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, , as a viscosity parameter ε tends to zero. The inviscid model, i.e. when , possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of to as in ( in ) in the regions including shocks under an entropy condition.

A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients

A one-step new general mesh free scheme, which is based on radial basis functions, is presented for a viscous wave equation with variable coefficients. By constructing a simple extended radial basis function, it can be directly applied to wave propagation by using the strong form-based mesh free collocation method. There is no need to deal with the time-dependent variable particularly. Numerical results for a viscous wave equation with variable coefficients show that the proposed mesh free collocation method is simple with accurate solutions.

Numerical Simulation Coupling on Fluid Flow and Heat Transfer in Nozzle and Equipment Cabin of Solid Rocket Motor during Operation Process

Driven by the goal of studying the temperature changes in the nozzle and equipment cabin of solid rocket motor during the whole process of ignition starting, working, cooling and decompression, this paper uses the fluid dynamics software which uses the RNG k-ε turbulence model to simulate the mixing of the fuel and oxidant and the Sutherland law to describe the viscosity of gas and air, and the controlling equation is the two-dimensional axisymmetric viscous compressible URANS equation. In this paper, the temperature field of the nozzle and equipment cabin in the working process of solid rocket motor during 0 ∼ 60s is numerically simulated, and the ground experiment is carried out. The results show that (1) the numerical simulation results are in good agreement with the ground test results, and the absolute value of the maximum error appears at the time of 60s, which is 5.9k. indicating that the numerical calculation method and heat source equivalent method established in this paper are suitable for the temperature field simulation of composite structure nozzle and equipment cabin, (2) the unsmooth surface of the nozzle formed by high temperature ablation makes the flow situation in the divergent section of the nozzle complex and affects the heat transfer coefficient of the wall contacting with the equipment cabin, (3) due to the insulation design of the equipment cabin, only a small part of the heat is transferred to the equipment cabin. During the whole working process of solid rocket motor, the temperature field of equipment cabin increases sharply at both ends and gently in the middle.

Zero Kinematic Viscosity-Magnetic Diffusion Limit of the Incompressible Viscous Magnetohydrodynamic Equations with Navier Boundary Conditions

We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω ⊂ ℝ3. It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient. The solution is uniformly bounded in a conormal Sobolev space and W1,∞ (Ω) which allows us to take the zero kinematic viscosity-magnetic diffusion limit. Moreover, we also get the rates of convergence in L∞(0, T; L2), L∞(0, T; W1, p) (2 ≤ p < ∞), and L∞((0, T) × Ω) for some T > 0.

Arbitrary initial energy blow up for fourth-order viscous damped wave equation with exponential-type growth nonlinearity

This letter is concerned with the following damped nonlinear wave equation with exponential nonlinearity utt+Δ2u+u+Δ2ut+ut=f(u)inΩ×(0,∞)where the nonlinearity f is a regular function satisfying f(0)=0 with an exponential growth to fix later. The above wave problem describes a class of essential nonlinear evolution equations appearing in the elastic–plastic-microstructure models. In this letter blow up result is established for arbitrary initial energy with the viscous damping terms.

An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

In this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Exponential stability of a pendulum in dynamic boundary feedback with a viscous damped wave equation

In this paper, we continue the earlier work [Lu, L., & Wang, D. L. (2017). Dynamic boundary feedback of a pendulum coupled with a viscous damped wave equation. In Proceedings of the 36th Chinese Control Conference(CCC) (pp. 1676–1680)] on study the stability of a pendulum coupled with a viscous damped wave equation model. This time we get the exponential stability result which is much better than the previous strong stability. By a detailed spectral analysis and operator separation, we establish the Riesz basis property as well as the spectrum determined growth condition for the system. Finally, the exponential stability of the system is achieved.

Space-time decay estimates of solutions to 3D incompressible viscous Camassa-Holm equations

In this paper, based on the parabolic interpolation inequality and inductive argument, we study the space-time decay estimates of higher-order time and spatial derivatives of strong solutions for the 3D incompressible viscous Camassa-Holm equations provided that the initial datum is well localized.

Impact of bulk viscosity on flow morphology of shock-accelerated cylindrical light bubble in diatomic and polyatomic gases

Shock-accelerated bubbles have long been an intriguing topic for understanding the fundamental physics of turbulence generation and mixing caused by the Richtmyer–Meshkov instability. In this study, the impact of bulk viscosity on the flow morphology of a shock-accelerated cylindrical light bubble in diatomic and polyatomic gases is investigated numerically. An explicit mixed-type modal discontinuous Galerkin scheme with uniform meshes is employed to solve a two-dimensional system of unsteady physical conservation laws derived rigorously from the Boltzmann–Curtiss kinetic equations. We also derive a new complete viscous compressible vorticity transport equation including the bulk viscosity. The numerical results show that, during the interaction between a planar shock wave and a cylindrical light bubble, the bulk viscosity associated with the viscous excess normal stress in diatomic and polyatomic gases plays an important role. The diatomic and polyatomic gases cause significant changes in flow morphology, resulting in complex wave patterns, vorticity generation, vortex formation, and bubble deformation. In contrast to monatomic gases, diatomic and polyatomic gases produce larger rolled-up vortex chains, various inward jet formations, and large mixing zones with strong, large-scale expansion. The effects of diatomic and polyatomic gases are explored in detail through phenomena such as the vorticity generation, degree of nonequilibrium, enstrophy, and dissipation rate. Furthermore, the evolution of the shock trajectories and interface features is investigated. Finally, the effects of bulk viscosity on the flow physics of shock-accelerated cylindrical light bubble are comprehensively analyzed.

An improved meshfree scheme based on radial basis functions for solving incompressible Navier–Stokes equations

AbstractIn this paper, an improved meshfree scheme based on radial basis functions (RBFs) is provided for solving the incompressible viscous Navier–Stokes equations and two enhancements are proposed to mitigate the typical numerical oscillations. The first one is the combination of the RBFs‐based finite difference (RBF‐FD) method with the semi‐Lagrangian RBFs (SLM‐RBF), with the former being used for the viscous diffusion term and pressure Poisson equation and the latter being used for the advection term. The second enhancement is a regularization term that constructs smooth constraints for RBFs interpolations instead of clipping operations. The capability of the proposed scheme in mitigating numerical fluctuations is demonstrated by validating it against the one‐dimensional (1‐D) advection problem and the advection–diffusion problem with step field functions. The overall performance of the proposed scheme is also validated by the lid‐driven cavity flow and laminar flow around a circular cylinder, showing good agreement with the existing results, indicating that the proposed scheme has good stability in both temporal and spatial domains.

A lattice Boltzmann model for the viscous shallow water equations with source terms

In this paper, a lattice Boltzmann (LB) model is proposed for a class of viscous shallow water equations, in which a second-order moment of the source term is applied to recover the viscosity in the governing equation and eliminate the additional errors generated during the Chapman-Enskog analysis. There are three different schemes based on different treatments of the source term. Through numerical simulations of several classical benchmark problems, we find that the second-order moment of the source term can not only improve the accuracy of the LB model, but also ensure the conservation of the system. In addition, the influence of rainfall intensity on shallow water flow is also taken into account, and the results show that present LB model can also accurately study such problems as overland flows.

High-order numerical solution of viscous Burgers' equation using an extended Cole–Hopf barycentric Gegenbauer integral pseudospectral method

This paper presents an extension to the Cole–Hopf barycentric Gegenbauer integral pseudospectral (PS) method (CHBGPM) presented in Elgindy and Dahy [High-order numerical solution of viscous Burgers' equation using a Cole–Hopf barycentric Gegenbauer integral pseudospectral method, Math. Methods Appl. Sci. 41 (2018), pp. 6226–6251] to solve an initial-boundary value problem of Burgers' type when the boundary function k defined at the right boundary of the spatial domain vanishes at a finite set of real numbers or on a single/multiple subdomain(s) of the solution domain. We present a new strategy that is computationally more efficient than that presented in [12] in the former case, and can be implemented successfully in the latter case when the method of [12] fails to work. Moreover, fully exponential convergence rates are still preserved in both spatial and temporal directions if the boundary function k is sufficiently smooth. Numerical comparisons with other traditional methods in the literature are presented to confirm the efficiency of the proposed method. A numerical study of the condition number of the linear systems produced by the method is included.

Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

We study low regularity behavior of the nonlinear wave equation in $\mathbb {R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb {R}^2)\times H^{s-1}(\mathbb {R}^2)$ is ill-posed whenever $0 < s < s_{cr}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb {R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.

Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic derivation and study of the viscous layer equations and the $L^2$ to $L^\infty$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations.

Color-electric conductivity in a viscous quark-gluon plasma * *Supported by the National Natural Science Foundation of China (11465007, 11735007, 11890710, 11890711, 11703005)

Several different transport processes, such as heat, momentum, and charge transports, may occur simultaneously in a thermal plasma system. The corresponding transport coefficients are heat conductivity, shear viscosity, and electric conductivity. In the present study, we investigate the color-electric conductivity of the quark-gluon plasma (QGP) in the presence of shear viscosity, focusing on the connection between the charge transport and momentum transport. To achieve this goal, we solve the viscous chromohydrodynamic equations obtained from the QGP kinetic theory associated with the distribution function modified by shear viscosity. According to the solved color fluctuations of hydrodynamic quantities, we obtain the induced color current through which the color-electric conductivity is derived. Numerical analysis shows that the conductivity properties of the QGP are mainly demonstrated by the longitudinal part of the color-electric conductivity. Shear viscosity has an appreciable impact on real and imaginary parts of the color-electric conductivity in some frequency regions.

Discrete approximation of the viscous HJ equation

We consider a stochastic discretization of the stationary viscous Hamilton-Jacobi equation on the flat d–dimensional torus \(\mathbb {T}^d\) associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on \(\mathbb {T}^d\), up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton–Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.

Relativistic non-resistive viscous magnetohydrodynamics from the kinetic theory: a relaxation time approach

We derive the relativistic non-resistive, viscous second-order magnetohydrodynamic equations for the dissipative quantities using the relaxation time approximation. The Boltzmann equation is solved for a system of particles and antiparticles using Chapman-Enskog like gradient expansion of the single-particle distribution function truncated at second order. In the first order, the transport coefficients are independent of the magnetic field. In the second-order, new transport coefficients that couple magnetic field and the dissipative quantities appear which are different from those obtained in the 14-moment approximation [1] in the presence of a magnetic field. However, in the limit of the weak magnetic field, the form of these equations are identical to the 14-moment approximation albeit with different values of these coefficients. We also derive the anisotropic transport coefficients in the Navier-Stokes limit.