Time-dependent Stokes Equations Research Articles

A postprocessing finite volume element method for time-dependent Stokes equations

In this paper, we utilize the P 1 - P 0 macro-element to develop a postprocessing finite volume element procedure for time-dependent Stokes equations. The method raises by one order the H 1 norm rate of convergence of the velocity and correspondingly the L 2 norm of the pressure. Numerical examples are presented to illustrate the theoretical analysis.

Convergence Analysis of High Order Algebraic Fractional Step Schemes for Time-Dependent Stokes Equations

In this paper we analyze the family of Yosida algebraic fractional step schemes proposed in [A. Quarteroni, F. Saleri, and A. Veneziani, Comput. Methods Appl. Mech. Engrg., 188 (2000), pp. 505-526], [F. Saleri and A. Veneziani, SIAM J. Numer. Anal., 43 (2005), pp. 174-194], and [P. Gervasio, F. Saleri, and A. Veneziani, J. Comput. Phys., 214 (2006), pp. 347-365] when applied to time-dependent Stokes equations. Under suitable regularity assumptions on the data, splitting error estimates both for velocity and pressure are established. In particular we analyze the first three methods of this family, providing, respectively, convergence (of the fractional step solution towards the numerical solution achieved without any operator splitting) of orders 3/2, 5/2, 7/2 for the velocity and 1, 2, 3 for the pressure. Moreover a general way to set up higher-order schemes is proposed. The present analysis is carried out when spectral element methods are employed for space discretization.

Simulation-based analysis of flow due to traveling-plane-wave deformations on elastic thin-film actuators in micropumps

One of the propulsion mechanisms of microorganisms is based on propagation of bending deformations on an elastic tail. In principle, an elastic thin-film can be placed in a channel and actuated for pumping of the fluid by means of introducing a series of traveling-wave deformations on the film. Here, we present a simulation-based analysis of transient, two-dimensional Stokes flow induced by propagation of sinusoidal deformations on an elastic thin-film submerged in a fluid between parallel plates. Simulations are based on a numerical model that solves time-dependent Stokes equations on deforming finite-element mesh, which is due to the motion of the thin-film boundary and obtained by means of the arbitrary Lagrangian Eulerian method. Effects of the wavelength, frequency, amplitude and channel’s height on the time-averaged flow rate and the rate-of-work done on the fluid by the thin-film are demonstrated and grouped together as the flow-rate and power parameters to manifest a combined parametric dependence.

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the time-dependent Stokes equations with a source term in L p (0, T; L q (Ω)) and prove uniform estimates on the time derivative and discrete Laplacian of the discrete velocity that are similar to those in Sohr and von Wahl [20].

A posteriori estimates for approximations of time-dependent Stokes equations

In this paper, we derive a posteriori error estimates for space-discrete approximations of the time-dependent Stokes equations. By using an appropriate Stokes reconstruction operator, we are able to write an auxiliary error equation, in pointwise form, that satisfies the exact divergence-free condition. Thus, standard energy estimates from partial differential equation theory can be applied directly, and yield a posteriori estimates that rely on available corresponding estimates for the stationary Stokes equation. Estimates of optimal order in L ∞ (L 2 ) and L ∞ (H 1 ) for the velocity are derived for finite-element and finite-volume approximations.

The Fundamental Solution of the Linearized Navier–Stokes Equations for Spinning Bodies in Three Spatial Dimensions – Time Dependent Case

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces Lp(R3), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.

Error Analysis of Pressure-Correction Schemes for the Time-Dependent Stokes Equations with Open Boundary Conditions

The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on part of the boundary are considered. It is shown that the standard pressure-correction method is not suitable for approximating the Stokes equations with open boundary conditions, whereas the rotational pressure-correction method yields reasonably good error estimates. These results appear to be the first ever published for splitting schemes with open boundary conditions. Numerical results in agreement with the error estimates are presented.

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

Network model for deep bed filtration

We study deep bed filtration, where particles suspended in a fluid are trapped while passing through a porous medium, using numerical simulations in various network models for flow in the bed. We first consider cellular automata models, where filtrate particles move in a fixed background flow field, with either no-mixing or complete-mixing rules for motion at a flow junction. The steady-state and time-dependent properties of the trapped particle density and filter efficiency are studied. The complete mixing version displays a phase transition from open to clogged states as a function of the mean particle size, while such a transition is absent in the (more relevant) no-mixing version. The concept of a trapping zone is found to be useful in understanding the time-dependent properties. We next consider a more realistic hydrodynamic network model, where the motion of the fluid and suspended particles is determined from approximate solutions of the time-dependent Stokes equation, so that the pressure field constantly changes with particle movement. We find that the steady-state and time-dependent behavior of the network model is similar to that of the corresponding cellular automata model, but the long computation times necessary for the simulations make a quantitative comparison difficult. Furthermore, the detailed behavior is extremely sensitive to the shape of the pore size distribution, making experimental comparisons subtle.

Long-time tails in the solid-body motion of a sphere immersed in a suspension

Long-time tails in the translational and rotational motion of a sphere immersed in a suspension of spherical particles are discussed on the basis of the linear, time-dependent Stokes equations of hydrodynamics. It is argued that the coefficient of the t(-3/2) long-time tail of translational motion depends only on the effective mass density and shear viscosity of the suspension. A similar expression holds for the coefficient of the t(-5/2) long-time tail of rotational motion. In particular, the long-time tails are independent of the sphere radius, and therefore the expressions hold also for a particle of the suspension. On account of the fluctuation-dissipation theorem the long-time tails of the velocity autocorrelation function and the angular velocity autocorrelation function of interacting Brownian particles are also given by these expressions.

Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

La limite pour la viscosité zéro des équations de Stokes dans un demi-espace

The time-dependent Stokes equations are considered. It is shown that the solution is the sum of an inviscid solution, a boundary layer solution, and a small (in the zero viscosity limit) correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data.

Iterative techniques for time dependent Stokes problems

In this paper, we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent Stokes equations using a mixed finite element spatial discretization. At each time step, a two by two block system corresponding to a perturbed Stokes problem must be solved. Although there are a number of techniques for iteratively solving this type of block system, to be effective, they require a good preconditioner for the resulting pressure operator (Schur complement). In contrast to the time independent Stokes equations where the pressure operator is well conditioned, the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced (and/or the Reynolds number is increased). In this paper, we shall describe and analyze preconditioners for the resulting pressure systems. These preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and Reynolds number parameter k.

The boundary layer analysis for stokes equations on a half space

The time dependent Stokes equations on a half space are considered. We decompose the solution of these equations in an inviscid solution: a boundary layer solution and a correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction is shown to be of the same order of magnitude of the square root of the viscosity.

Nonlinear drift interactions between fluctuating colloidal particles: oscillatory and stochastic motions

In this work we consider how nonlinear hydrodynamic effects can lead to a mean force of interaction between two spheres of equal radiusaundergoing translational fluctuations parallel or perpendicular to their line of centres. Motivated by amplitudes and Reynolds numbers characteristic of Brownian motion in colloidal systems, nonlinearities due to motion of the boundaries and to inertia throughout the fluid are treated as regular perturbations of the time-dependent Stokes equations. This formulation ultimately leads to a prescription for computing, at leading order, the time-average nonlinear force for the case of pure oscillatory modes – which represents the Fourier decomposition of more general motions. The associated hydrodynamic problems are solved numerically using a least-squares boundary singularity method. Frequency-dependent results over the whole spectrum are presented for a sphere-sphere gap equal to one radius; illustrative calculations are also carried out at other separations. Subsequently we extend the analysis of nonlinear drift to a Langevin equation formulation of the more complex problem of stochastic motion due to thermal fluctuations in the suspending fluid, i.e. Brownian motion. By integrating (numerically) over the spectrum of frequencies, we quantify how the mutual interactions of all translational disturbance modes give rise, on ensemble average, to a stochastic nonlinear force of interaction between the particles. It is particularly interesting that this net interaction – arising from a zero-mean random force – is ofO(1) on the Brownian scalekT/a, even though it represents a smallO(Re) correction at each frequency of pure oscillations. Finally, we discuss how the presence of stochastic nonlinear drift would lead to non-uniform equilibrium distributions of dilute colloidal suspensions, unless one adds to the random force in the Langevin equation a cancelling non-zero mean component.

Solutions for Steady and Nonsteady Entrance Flow in a Semi-Infinite Circular Tube at Very Low Reynolds Numbers

Analytic solutions are found to the time-dependent Stokes and continuity equations describing entrance flow of an incompressible fluid into a circular tube at very low Reynolds numbers. Fourier transform methods are used, and entrance-flow solutions are obtained in the form of normal-mode expansions. Complex dispersion relations are obtained for each mode, and analytic expressions are determined for the normal-mode coefficients in terms of entrance boundary conditions. Results for velocity and pressure near the entrance are given. The generalized method is applied to obtain solutions to specific nonsteady-flow problems with a sinusoidal time dependence. In the last section of the paper, the steady-flow solutions are obtained by taking the limits of the nonsteady-flow solutions as the frequency approaches zero.

A Stable Algorithm For A Fluid-structureInteraction Problem In 3D

A fluid-structure interaction problem is studied in the following hypotheses: the fluid is incompressible and it is governed by the time-dependent Stokes equations and the sructure is governed by the linear elasticity equations. The interaction beet wen the blood and the left ventricle of the heart is the physical system which is modeled. This paper presents a stable algorithm wich computs the velocity and the pressure of the fluid and the velocity of the structure, using the Lagrange multiplier method.