Related papers: On Einstein's effective viscosity formula
This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we…
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…
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…
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…
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…
In this paper, we derive the closed form analytical solutions for the effective viscosity of the suspensions of solid spheres that take into account the size effects. This result is obtained using the solution for the effective shear…
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 $o(\lambda^2)$ approximation of the exact…
We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $\mu^\prime=\mu (1+\frac 5 2\phi+o(\phi))$, where $\phi$ is the volume fraction of the suspended particles. Up to now rigorous…
Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula…
We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\phi$. Our objective is to go beyond the Einstein's approximation…
We compute the first order correction of the effective viscosity for a suspension containing solid particles with arbitrary shapes. We rewrite the computation as an homogenization problem for the Stokes equations in a perforated domain.…
This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard…
This work is devoted to the definition and the analysis of the effective viscosity associated with a random suspension of small rigid particles in a steady Stokes fluid. While previous works on the topic have been conveniently assuming that…
When particulate suspensions are sheared, perturbations in the shear flows around the rigid particles increase the local energy dissipation, so that the viscosity of the suspension is effectively higher than that of the solvent. For bulk…
We propose an improved effective-medium theory to obtain the concentration dependence of the viscosity of particle suspensions at arbitrary volume fractions. Our methodology can be applied, in principle, to any particle shape as long as the…
We provide a rigorous derivation of an effective rheology for dilute viscous suspensions of self-propelled particles. We derive at the limit an effective Stokes system with two additional terms. The first term corresponds to an effective…
We give the first correction to the suspension viscosity due to fluid elasticity for a dilute suspension of spheres in a viscoelastic medium. Our perturbation theory is valid to $O(\phi\mathrm{Wi}^2)$ in the Weissenberg number…
We consider reversible diffusions in random environment and prove the Einstein relation for this model. It says that the derivative of the effective velocity under an additional local drift equals the diffusivity of the model without drift.…
We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point…
We solve the Einstein constraint equations for a first-order causal viscous relativistic hydrodynamic theory in the case of a conformal fluid. For such a theory, a direct application of the conformal method does not lead to a decoupling of…