arXiv subjects

Leonid Berlyand

Publications and source records attributed to Leonid Berlyand.

40 records · Page 3Linked to original sources

arXiv2006

A network model for granular statics with impenetrability constraints

We study quasi-static deformation of dense granular packings. The packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). We prove that under certain geometric conditions on the network, in the energy-minimizing configuration there are at least two non-stretched springs attached to each node, which means that every particle has at least two solid-like contacts. This eliminates micro-avalanches as a mechanism for structural weakening in small shear deformation.

math.OC↗

arXiv2006

Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate in a 2D Model of Concentrated Suspensions

We present a two-dimensional (2D) mathematical model of a highly concentrated suspension or a thin film of the rigid inclusions in an incompressible Newtonian fluid. Our objectives are two-fold: (i) to obtain all singular terms in the asymptotics of the overall viscous dissipation rate as the interparticle distance parameter $δ$ tends to zero, (ii) to obtain a qualitative description of a microflow between neighboring inclusions in the suspension.

math.AP↗

arXiv2006

Capacity of a multiply-connected domain and nonexistence of Ginzburg-Landau minimizers with prescribed degrees on the boundary

Suppose that $ω\subsetΩ\subset R^2$. In the annular domain $A=Ω\setminus\barω$ we consider the class $J$ of complex valued maps having degree 1 on $\partial Ω$ and on $\partialω$. It was conjectured by Berlyand and Mironescu ('04), that he existence of minimizers of the Ginzburg-Landau energy $E_κ$ in $J$ is completely determined by the value of the $H^1$-capacity $cap(A)$ of the domain and the value of the Ginzburg-Landau parameter $κ$. The existence of minimizers of $E_κ$ for all $κ$ when $cap(A)\geqπ$ (domain $A$ is ``thin'') and for small $κ$ when $cap(A)<π$ (domain $A$ is ``thick'') was established by Berlyand and Mironescu ('04). Here we provide the answer for the remaining case of large $κ$ when $cap(A)<π$. We prove that, when $cap(A)<π$, there exists a finite threshold value $κ_1$ of the Ginzburg-Landau parameter $κ$ such that the minimum of the Ginzburg-Landau energy $E_κ$ is not attained in $J$ when $κ>κ_1$ while it is attained when $κ<κ_1$.

math.AP↗

arXiv2004

Effective shear and extensional viscosities of cocentrated disordered suspensions of rigid particles

We study effective shear viscosity $μ^\star$ and effective extensional viscosity $λ^\star$ of concentrated non-colloidal suspensions of rigid spherical particles. The focus is on the spatially disordered arrays. We use recently developed discrete network approximation techniques to obtain asymptotic formulas for the viscosities as the typical inter-particle distance $δ$ tends to zero. For disordered arrays, the volume fraction alone does not determine the effective viscosity. Use of the network approximation allows us to study the dependence of the effective viscosities on variable distances between neighboring particles. Our analysis can be characterized as global because it goes beyond the local analysis of the flow between two particles. The principal conclusion in the paper is that, in general, asymptotic formulas obtained by global analysis are different from the formulas obtained from local analysis. In particular, the leading term in the asymptotics of $μ^\star$ is of lower order than suggested by the local analysis (weak blow up), while the order of the leading term in the asymptotics of $λ^\star$ depends on the geometry of the particle array (either weak or strong blow up). We obtain geometric conditions on a random particle array that lead to the strong blow up of $λ^\star$, and show that these conditions are generic. We also provide an example of a closely packed particle array for which the leading term in the asymptotics of $λ^\star$ degenerates (weak blow up).

physics.flu-dyn↗