On the role of (weak) compressibility for fluid-structure interaction solvers

In the present study, a weakly compressible formulation of the Navier-Stokes equations is developed and examined for the solution of fluid-structure interaction (FSI) problems. Newtonian viscous fluids under isothermal conditions are considered, and the Murnaghan-Tait equation of state is employed for the evaluation of mass density changes with pressure. A pressure-based approach is adopted to handle the low Mach number regime, ie, the pressure is chosen as primary variable, and the divergence-free condition of the velocity field for incompressible flows is replaced by the continuity equation for compressible flows. The approach is then embedded into a partitioned FSI solver based on a Dirichlet-Neumann coupling scheme. It is analytically demonstrated how this formulation alleviates the constraints of the instability condition of the artificial added mass effect, due to the reduction of the maximal eigenvalue of the so-called added mass operator. The numerical performance is examined on a selection of benchmark problems. In comparison to a fully incompressible solver, a significant reduction of the coupling iterations and the computational time and a notable increase in the relaxation parameter evaluated according to Aitken's Δ² method are observed.     

Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

Equation in complex variable of axisymmetrical deformation problems for a general shell of revolution

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tölke(1938)^[3], R.A.Clark(1950)^[4]and V.V.Novozhilov(1951)^[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)^[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.     

Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]     

Fluid flow in a helical pipe

Without simplifying the N-S equations of Germano's^[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-order effects of curvature κ and torsion τ on the secondary flow and axial velocity are discussed in detail. The first-order effect of curvature is to form two counter-rotating cells of the secondary flow and to push the maximum axial velocity to the outer bend. The two cells are pushed to the outer bend by the pure second-order effect of curvature. The combined higher-order (second-, third-) effects of curvature and torsion, are found to be an enlargement of the lower vortex of the secondary flow at expense of the upper one and a clockwise shift of the centers of the secondary vortices and the location of maximum axial velocity. When the axial pressure gradient is small enough or the torsion is sufficiently larger than the curvature, the location of the maximal axial velocity is near the inner bend. The equation of the volume flux is obtained from integrating the perturbation solutions of axial velocity. From the equation the validity range of the perturbation solutions in this paper can be obtained and the conclusion that the three terms of torsion have no effect on the volume flux can easily be drawn. When the axial pressure gradient is less than 22.67, the volume flux in a helical pipe is larger than that in a straight pipe.     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

We study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t ?? ??, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy??s law. In this paper, we prove that any L ^?? weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural L ¹ topology with decay rates to the Barenblatt profile of the porous medium equation. The density function tends to the Barenblatt solution of the porous medium equation while the momentum is described by Darcy??s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.     

分层流体中gKdV型孤立波的迎撞

本文采用约化摄动法和PLK方法并通过双参数摄动展开,讨论了分层流体中以推广的Korteweg-de vries方程(gKdV方程)描述的孤立波的迎撞问题,求得了二阶近似解。分析结果表明,gKdV型孤立波碰撞后保持原来的形状不变,在碰撞时最大波幅为两个来碰孤立波的最大波幅的线性叠加。     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis

This paper is concerned with the time fractional Sharma–Tasso–Olver (FSTO) equation, Lie point symmetries of the FSTO equation with the Riemann–Liouville derivatives are considered. By using the Lie group analysis method, the invariance properties of the FSTO equation are investigated. In the sense of point symmetry, the vector fields of the FSTO equation are presented. And then, the symmetry reductions are provided. By making use of the obtained Lie point symmetries, it is shown that this equation can transform into a nonlinear ordinary differential equation of fractional order with the new independent variable ξ=xt ^?α/3. The derivative is an Erdélyi–Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions to the FSTO equation are given.     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.     

L 2-Stability Theory of the Boltzmann Equation near a Global Maxwellian

We present three a priori L ²-stability estimates for classical solutions to the Boltzmann equation with a cut-off inverse power law potential, when initial datum is a perturbation of a global Maxwellian. We show that L ²-stability estimates of classical solutions depend on Strichartz type estimates of perturbations and the non-positive definiteness of the linearized collision operator. Several well known classical solutions to the Boltzmann equation fit our L ²-stability framework.     

Numerical simulation of the flow around a porous covering square cylinder in a channel via lattice Boltzmann method

In this paper, a detailed investigation on the flow past a porous covering cylinder is presented through the lattice Boltzmann method. The Brinkman‐Forchheimer‐extended Darcy model is adopted for the entire flow field with the solid, fluid, and porous medium. The effects of several parameters, such as porous layer thickness, Darcy number, porosity, and Reynolds number on flow field are discussed. Compared with the case of a solid cylinder, the present work shows that the porous layer may play an important role on the flow, the lift and drag force exerted on the cylinder. The numerical results indicate that the maximal drag coefficient C_d and maximal amplitude of lift coefficient C_l exist at certain Darcy number which is in the range of 10^?6–10^?2. Copyright © 2010 John Wiley & Sons, Ltd.     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999     

A weighted least squares method for first-order hyperbolic systems

The paper presents a generalization of the classical L²-norm weighted least squares method for the numerical solution of a first-order hyperbolic system. This alternative least squares method consists of the minimization of the weighted sum of the L² residuals for each equation of the system. The order of accuracy of global conservation of each equation of the system is shown to be inversely proportional to the weight associated with the equation. The optimal relative weights between the equations are then determined in order to satisfy global conservation of the energy of the physical system. As an application of the algorithm, the shallow water equations on an irregular domain are first discretized in time and then solved using Laplace modification and the proposed least squares method.     

Setting and Analysis of the Multi-configuration Time-dependent Hartree–Fock Equations

In this paper, we formulate and analyze the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations for molecular systems with pairwise interaction. This set of coupled nonlinear PDEs and ODEs is an approximation of the N-particle time-dependent Schrödinger equation based on (time-dependent) linear combinations of (time-dependent) Slater determinants. The “one-electron” wave-functions satisfy nonlinear Schrödinger-type equations coupled to a linear system of ordinary differential equations for the expansion coefficients. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial rôle in the analysis. Under the full-rank assumption a fiber bundle structure emerges and produces unitary equivalence between different useful representations of the MCTDHF approximation. For a large class of interactions (including Coulomb potential), we establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular. A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix violates energy conservation. However, global well-posedness for this system in L ² is obtained with Strichartz estimates. Eventually, solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.     

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

A robust integration algorithm for implementing rate dependent crystal plasticity into explicit finite element method

Severe numerical instability in the integration of rate dependent crystal plasticity (RDCP) model is one of the main problems for implementing RDCP into finite element method (FEM), especially for simulating dynamic/transient forming process containing complicated contact conditions under large step length, large strain and high strain rate. In order to overcome the problem, an implicit model is deduced with the primary unknowns of shear strain increments of slip systems under the corotational coordinate system in the paper. The homotopy auto-changing continuation method combined with the Newton–Raphson (N–R) iteration is adopted. The subroutine VUMAT is developed for implementing RDCP model in ABAQUS/Explicit. Simulation results show that the algorithm is stable and accurate in 3D FE simulations on both dynamic simple loading and complicated loading process containing nonlinear contacts under the conditions of the maximal step length of 3.5 × 10⁻⁶ s, the maximal strain of 1.05, the maximal loading speed of 120 mm s⁻¹, and the minimal material rate sensitivity coefficient of 0.01. The predictions of the model on crystal behaviors of anisotropy, rate sensitivity and elasticity, as well as ear profiles in deep cup drawing are in agreement with experiments.     

Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity

By treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity, we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg-deVries equation as the evolution equation when the viscosity is of order of ε^3/2. We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma, the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter τ.