This paper is a follow-up of article [6], on the derivation of accurate effective models for viscous dilute suspensions. The goal is to identify an effective Stokes equation providing a $o(λ^2)$ approximation of the exact fluid-particle system, with $λ$ the solid volume fraction of the particles. This means that we look for an improvement of Einstein's formula for the effective viscosity in the form $μ_{eff}(x) = μ+ \frac{5}{2} μρ(x) λ+ μ_2(x) λ^2$. Under a separation assumption on the particles, we proved in [6] that if a $o(λ)^2$ Stokes effective approximation exists, the correction $μ_2$ is necessarily given by a mean field limit, that can then be studied and computed under further assumptions on the particle configurations. Roughly, we go here from the conditional result of [6] to an unconditional result: we show that such a $o(λ^2)$ Stokes approximation indeed exists, as soon as the mean field limit exists. This includes the case of periodic and random stationary particle configurations.
We present a mathematical proof of Einstein's formula for the effective viscosity of a dilute suspension of rigid neutrally--buoyant spheres when the spheres are centered on the vertices of a cubic lattice. We keep the size of the container finite in the dilute limit and consider boundary effects. Einstein's formula is recovered as a first-order asymptotic expansion of the effective viscosity in the volume fraction. To rigorously justify this expansion, we obtain an explicit upper and lower bound on the effective viscosity. A lower bound is found using energy methods reminiscent of the work of…
We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction $ϕ$. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $ϕ\to 0$, we show that the microscopic dynamics is approximated to order $ϕ^2 |\log ϕ|$ by a …
We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled …
We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $ϕ$. Our objective is to go beyond the Einstein's approximation $μ_{eff}=(1+\frac{5}{2}ϕ)μ$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O(ϕ^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(ϕ^2)$-correction, and p…
In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allo…
Some properties of characteristic curves in connection with viscosity solutions of Hamilton-Jacobi equations defined by Hopf-type formula are studied. We investigate the points where the Hopf-type formula $u(t,x)$ is differentiable, and the strip of the form $(0,t_0)\times \R^n$ of the domain $Ω$ where the viscosity solution $u(t,x)$ is continuously differentiable. Moreover, we present the propagation of singularity in forward of $u(t,x).$
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDE problem involving sublinear operators. This is done through a dynamic programming principle derived from [8]. The formula can be seen as a nonlinear extension of the Feynman--Kac formula and is based on the backward stochastic differential equations theory.
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDEs given as a sup--envelope function. This is done through a dynamic programming principle derived from Denis, Hu, Peng (2010). The formula can be seen as a nonlinear extension of the Feynman--Kac formula and is based on the backward stochastic differential equations theory.
We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $μ^\prime=μ(1+\frac 5 2ϕ+o(ϕ))$, where $ϕ$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $ϕ=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ es…
Some properties of characteristic curves in connection with viscosity solution of Hamilton-Jacobi equations $(H,σ)$ defined by Hopf formula $u(t,x)=\max_{q\in\R^n}\{ \langle x,q\rangle -σ^*(q)-tH(q)\}$ are studied. We are concerned with the points where the solution $u(t,x)$ is differentiable, and the strip of the form $\mathcal R=(0,t_0)\times \R^n$ of the domain $Ω$ where $u(t,x)$ is of class $C^1(\mathcal R).$ Moreover, we investigate the propagation of singularities in forward of this solution.
Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned} &\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in M\times(0,+\infty), \\ &u(x,0)=φ(x), \end{aligned} \right. \end{equation*} where $φ\in C(M)$ and the stationary one \begin{equation*} H(x,u(x),\partial_x u(x))=0, \end{equation*} where $H(x,u,p)$ is continuous, convex and coercive in $p$, uniformly Lipschitz in $u$. By introducing a solution semigroup, we provide a representation formula of the viscosity solution of the …
Based on a fixed point argument, we give a {\it dynamical representation} of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we obtain some regularity results related to the viscosity solution, including a partial extension of Lions' regularizing effect \cite{L} to the case of weakly coupled systems.
Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector eq…
We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation $(H,σ)$ on a given domain $Ω= (0,T)\times \R^n.$ It is known that, if the Hamiltonian $H = H(t,p)$ is not a convex (or concave) function in $p$, or $H(\cdot, p)$ may change its sign on $(0,T)$, then the Hopf-type formula does not define a viscosity solution on $Ω.$ Under some assumptions for $H(t,p)$ on the subdomains $(t_i, t_{i+1})\times \R^n\subset Ω$, we are able to arrange "partial solutions" given by the Hopf-type formula to get a viscosity solution on $Ω.$ Then we study …
We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense.
We introduce a level-set formulation for the mean curvature flow with obstacles and show existence and uniqueness of a viscosity solution. These results generalize a well known viscosity approach for the mean curvature flow without obstacle by Evans and Spruck and Chen, Giga and Goto in 1991. In addition, we show that this evolution is consistent with the variational scheme introduced by Almeida, Chambolle and Novaga (2012) and we study the long time behavior of our viscosity solutions.
We study some differential properties of viscosity solution for Hamilton - Jacobi equations defined by Hopf-Lax formula $u(t,x)=\min_{y\in \R^n} \big\{σ(y)+tH^*\big (\frac {x-y}{t}\big)\big \}.$ A generalized form of characteristics of the Cauchy problems is taken into account the context. Then we examine the strip of differentiability of the viscosity solution given by the function $u(t,x).$
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic…
This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path.
