Stokes Fundamental Solution Research Articles

Asymptotic profile for the interaction of a rigid ball and an incompressible viscous fluid

In this paper, we study the large time behavior of the solutions of the Stokes fluid-solid system. We compute the asymptotic expansion of the solution for Lq integrable initial data U0. We show that the asymptotic profile of the solution is a linear combination of the Stokes fundamental solution for the data with extra condition |x|U0∈L1. We also show that L1 integrability of the solution is strongly related to the net force exerted by the fluid on the boundary of the solid.

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

AbstractWe study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution of the Navier–Stokes equation such that and uniformly in as . Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall Blake (1971) [8]. However, to further impose periodic boundary conditions on directions parallel to the wall usually involves tedious derivations because single or double periodicity in Stokes flow may require the periodic unit to have no net force, which is not satisfied by the well-known image system. In this work we present a force-neutral image system. This neutrality allows us to represent the Stokes image system in a universal formulation for non-periodic, singly periodic and doubly periodic geometries. This formulation enables the black-box style usage of fast kernel summation methods. We demonstrate the efficiency and accuracy of this new image method with the periodic kernel independent fast multipole method in both non-periodic and periodic geometries. We then extend this new image system to other widely used Stokes fundamental solutions, including the Laplacian of the Stokeslet and the Rotne–Prager–Yamakawa tensor.

Note on the fast decay property of steady Navier–Stokes flows in the whole space

Abstract We investigate the pointwise asymptotic behavior of solutions to the stationary Navier–Stokes equation in {\mathbb{R}^{n}} ( {n\geq 3} ). We show the existence of a unique solution {\{u,p\}} such that {|\nabla^{j}u(x)|=O(|x|^{1-n-j})} and {|\nabla^{k}p(x)|=O(|x|^{-n-k})} ( {j,k=0,1,\ldots} ) as {|x|\rightarrow\infty} , assuming the smallness of the external force and the rapid decay of its derivatives. The solution {\{u,p\}} decays more rapidly than the Stokes fundamental solution.

On time-periodic Navier–Stokes flows with fast spatial decay in the whole space

We investigate the pointwise behavior of time-periodic Navier–Stokes flows in the whole space. We show that if the time-periodic external force is sufficiently small in an appropriate sense, then there exists a unique time-periodic solution $$\{ u,p \}$$ of the Navier–Stokes equation such that $$|u(t,x)|=O(|x|^{1-n})$$ , $$|\nabla u(t,x)|=O(|x|^{-n})$$ and $$|p(t,x)|=O(|x|^{-n})$$ uniformly in $$t \in {\mathbb {R}}$$ as $$|x| \rightarrow \infty$$ . Our solution decays more rapidly than the time-periodic Stokes fundamental solution. The proof is based on the representation formula of a solution via the time-periodic Stokes fundamental solution and its properties.

Non-singular time domain BEM with applications to 3D inertial soil–structure interaction

A direct time domain boundary element method is presented based on the Stokes fundamental solutions, discretized in both time and space, and an efficient time step-by-step solution that minimizes the accumulation of errors. A non-singular numerical integration procedure, in the Cauchy sense, is proposed for the generation of the associated influence matrices. This methodology is shown to be efficient for the solution of a number of computationally intensive problems in the area of soil–structure interaction. In addition, an algorithm for the direct calculation of the response of massive foundations to externally applied forces and/or obliquely incident seismic waves is introduced. The accuracy and computational efficiency of the proposed methodologies is established through a number of comparison studies.

Source near-field effects and pulse tests in soil samples

Shear modulus measurement using bender elements in laboratory samples has become very popular. However, the test results are hard to interpret. Simple plane wave test models are too coarse and result in substantial measurement uncertainty: near 100% in G0. A possible refinement of the test model is based on Stokes's fundamental solution for an isolated source. Near-field effects appear, and they have been regularly quoted as a major source of uncertainty in tests. There are some criteria in use for avoidance of near-field distortions, but they are signal dependent and not always successful. After some consideration of Stokes's fundamental solution the authors propose a new frequency domain signal-independent criterion to avoid near-field effects. Time-domain criteria are also given, but they are shown to be signal dependent. Applying this criterion to some experimental results the authors then also show how near-field induced errors are not responsible for much of the observed signal distortion. Uncertainty in signal arrival lasts well beyond the end of the Stokes source near field.

Source near-field effects and pulse tests in soil samples

Shear modulus measurement using bender elements in laboratory samples has become very popular. However, the test results are hard to interpret. Simple plane wave test models are too coarse and result in substantial measurement uncertainty: near 100% in G0. A possible refinement of the test model is based on Stokes's fundamental solution for an isolated source. Near-field effects appear, and they have been regularly quoted as a major source of uncertainty in tests. There are some criteria in use for avoidance of near-field distortions, but they are signal dependent and not always successful. After some consideration of Stokes's fundamental solution the authors propose a new frequency domain signal-independent criterion to avoid near-field effects. Time-domain criteria are also given, but they are shown to be signal dependent. Applying this criterion to some experimental results the authors then also show how near-field induced errors are not responsible for much of the observed signal distortion. Uncertainty in signal arrival lasts well beyond the end of the Stokes source near field.

Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis

A coupled BEM–FEM methodology is presented for 3D wave propagation and soil–structure interaction analysis in the direct time domain. The employed boundary element method (BEM) uses a new generation of the Stokes fundamental solutions that utilize the B-Spline family of polynomials. A standard finite element methodology for dynamic analysis along with direct integration in time is coupled to the BEM through a staggered solution approach. Each method provides initial conditions to the other at the beginning of each time step. Formulation and computational aspects of the proposed coupling scheme are discussed. A number of numerical examples are presented for the validation and demonstration of the general nature of the proposed methodology.

On the Asymptotic Behavior of Physically Reasonable Solutions to the Stationary Navier—Stokes System in Three-dimensional Exterior Domains with Zero Velocity at Infinity

We consider physically reasonable solutions of the stationary Navier—Stokes system in a three-dimensional exterior domains with zero velocity at infinity. We show that when these solutions are asymptotically expanded near infinity, the leading term cannot be the product of a non-zero vector with the Stokes fundamental solution. This result should be contrasted with the case when the velocity at infinity is not zero. Then, as is well known, such an expansion is possible, with the leading term being the product of a suitable constant vector with the fundamental Oseen solution.

A time domain BEM for 3-D elastodynamic analysis using the B -spline fundamental solutions

A Boundary Element for 3-D elastodynamic analysis is introduced in detail. The method uses a new generation of the Stokes fundamental solutions that utilize the B-spline family of polynomials. The integration techniques of the boundary element kernels are also discussed for both the singular and non-singular cases. A number of numerical examples are presented for the validation of the proposed methodology.

BOUNDARY INTEGRAL EQUATION FOR NAVIER-STOKES EQUATIONS IN A NON-SMOOTH DOMAIN

Boundary integral equations corresponding to the differential equations describing a transient flow of incompressible viscous fluid in three dimensions are considered. Emphasis is put on the treatment of edges and corners. The boundary Γ is assumed piecewise Lyapunov surface and the interior solid angle Θ(x) at the non-smooth boundary point x must satisfy the inequality lim⁡δ→0 sup⁡x∈Γ12π{ ∫0<|y−x|≤δ|dΘx(y)|+|2π−Θ(x)| }<1. Corresponding to the Dirichlet problem of the Navier-Stokes equations, the following series of Volterra integral equations of the first kind for unknown tractions σj(n)(j=1,2,3:n=0,1,2,…) is derived. Gσj(n)(x,t)=∫t0∫Γσi(n)(y,τ)Uij*(y,τ;x,t)dS(y)dτ=bj(n)(x,t), where Uij* are components of the Stokes fundamental solution tensor and bj(n) can be regarded as given functions. The integral Gσj(n) is the single layer potential. The integral involved in the definition of bj(n) (see the text) is the double layer potential. Those integrals are shown to be weakly singular for the non-smooth domain under consideration. It is proved that, with ∑​=Γ×[0,T], the operator G:H−12,−14(∑)→H12,14(∑) is coercive; ((Gσ,σ))L2(∑)≥β|||σ|||H−12,−14(∑)2 with a constant β>0, σ=(σ1,σ2,σ3).

Viscoelastic formulations of BEM in time and frequency domain

Formulations of the boundary element method (BEM) currently include conventional viscoelastic constitutive equations in the frequency domain. The aim of the present paper is to implement viscoelastic behaviour in a time domain approach as well. The elastic Stokes fundamental solution is converted to a viscoelastic one by adopting a correspondence principle. A novel viscoelastic fundamental solution is obtained analytically by inverse Laplace transformation. A frequency domain BE approach is generalized by taking viscoelastic constitutive equations with fractional order time derivatives into account. It is shown that the boundary matrix ϵ ij for non smooth boundaries in a dynamic formulation equals the elastostatic matrix. The transfer behaviour of a mounting system has been calculated by adopting the developed BE formulation to a viscoelastic resilient support mount.