Mechanical Modeling of Particle-Droplet Interaction Motivated by the Study of Self-Cleaning Mechanisms

Motivated by the study of self-cleaning surfaces, the interaction between a spherical solid particle and a water droplet is studied on the microsale level, in terms of the balance forces and the surface properties. To define the forces acting on the solid-liquid interface, the meniscus depression is computed from the Young-Laplace equation, using a non-linear finite element formulation. The equilibrium force is derived in terms of the contact angle, particle size, and the penetration. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closed Hypersurfaces Driven by Mean Curvature and Inner Pressure

We introduce a hyperbolic equation that describes the motion of closed hypersurfaces in a Riemannian manifold with surface tension and inner pressure as driving forces. In the case of spherical surfaces this equation can be considered as an idealized mathematical model for a moving soap bubble. The equation is derived as an Euler‐Lagrange equation from a suitable action integral. It is a quasi‐linear degenerate hyperbolic PDE of second order that describes the motion of the surfaces extrinsically. Our main results are the solution of the Cauchy problem by means of the Nash‐Moser inverse function theorem, a continuation criterion, and stability estimates. © 2012 Wiley Periodicals, Inc.     

The transition between the Stokes equations and the Reynolds equation: A mathematical proof

The Reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Using the asymptotic expansion in the Stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. We prove that the Reynolds equation is an approximation of the Stokes equations and that the kind of convergence is strongly related with the boundary conditions on the velocity field.     

Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the variational Lagrange equation, is described. The vectors of the forces reduced to the mesh points are constructed for the known stiffness matrices of the elements using the displacements at the mesh points found from the solution of the problem. On the other hand, these mesh point forces are determined in terms of the unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The stresses at the mesh points are determined for the values of these forces found on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations, which relate the forces, stresses and the normal to the surface. The special features of the use of the stress recovery procedure are demonstrated for a plane problem in the linear theory of elasticity.     

On non-Newtonian lubrication with the upper convected Maxwell model

A long-standing theoretical and practical problem, whether the viscoelasticity can have a measurable and beneficial effect on lubrication performance characteristics, is readdressed in this paper. The upper convected Maxwell model is chosen to study the influence of viscoelasticity on lubricant thin film flows. By employing characteristic lubricant relaxation times in an order of magnitude analysis, a perturbation method is developed for analysing the flow of a Maxwell lubricant between two narrow surfaces. The effect of viscoelasticity on the lubricant velocity and pressure is examined, and the influence of minimum film thickness on lubrication characteristics is investigated. An order of magnitude analysis reveals that the pressure distribution is significantly affected by the presence of fluid viscoelasticity when the minimum film thickness is sufficiently small. This mechanism suggests that viscoelasticity does indeed enhance the lubricant pressure field and produce a beneficial effect on lubrication performance, which is consistent with some experimental observations.     

On a variational inequality on elasto-hydrodynamic lubrication

We consider the problem of a deformable surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The mathematical model consists of the Reynolds variational inequality with nonlocal coefficients given by an integral operator which depends on the fluid pressure. The nonlocal operator represents the deformation of the lubricated surfaces. The problem considers the vertical displacement of the elastic surface from its reference configuration. The goal of the paper is to obtain the range of these admissible displacements. We present general results for nonlocal coefficients with applications to particular problems in elasto-hydrodynamic lubrication.     

Dynamic Reaction of a Composite Anisotropic Space with Tunnel Holes to the Action of Concentrated Shear Forces

An antiplane stationary dynamic problem of elasticity theory for a two-component anisotropic (orthotropic) space is considered, where one component of the pair is weakened by tunnel holes of arbitrary cross section. The space is subjected to the action of time-dependent harmonic shear forces concentrated along some line and shear stresses applied to the surfaces of the holes. To solve the problem, the fundamental solution for a composite space without holes is preliminarily constructed. Using integral representations for displacements in the composite space with holes, the boundary problem of elasticity theory is reduced to an integral equation of second kind, which is solved numerically by the method of quadratures.     

润滑理论中的广义雷诺方程和不等变分问题

本文运用张量分析工具和S-坐标系,推导了润滑理论中的广义雷诺方程及相应的不等变分问题,它计及了润滑流动中的弯曲效应,轴和轴瓦曲面内蕴性质对流动的影响.     

On the optimization of surface textures for lubricated contacts

The pressure field that develops inside a lubricated contact obeys an elliptic equation known as Reynolds equation, with coefficients that depend on the shape of the contacting surfaces. The load-carrying capacity of a contact, defined as the integral of the pressure field, is an important performance indicator that should be as high as possible to avoid wear and damage of the surfaces. In this article, the effect of arbitrary uniform periodic textures on the load-carrying capacity of lubricated devices known as thrust bearings is investigated theoretically by means of homogenization techniques and first-order perturbation analysis. It is shown that the untextured shape is a local optimum for the load-carrying capacity of the homogenized pressure field. This is proved for bearings of general shape and considering both incompressible and compressible models for the lubricant. The homogenization technique however implies an error. Suitable bounds for the effect of this error are provided in a simplified case.     

Elrod–Adams cavitation model for a new nonlinear Reynolds equation in piezoviscous hydrodynamic lubrication

In previous studies, different cavitation models have been incorporated into the classical Reynolds equation in piezoviscous regimes. The advantages of the Elrod–Adams cavitation model compared with the Reynolds model have been demonstrated in this classical framework. Recently, a new nonlinear Reynolds equation was rigorously justified [15] for lubricated line contact problems by introducing the piezoviscous Barus law into the departure Navier–Stokes equations before passing to the thin film limit. In addition, the corresponding nonlinear first order ordinary differential equation (ODE) has been proposed.In the present study, we incorporate the Elrod–Adams model for cavitation and we pose the free boundary problem associated with the nonlinear first order ODE, which involves a multivalued Heaviside operator for the relationship between the lubricant pressure and saturation. After analyzing the qualitative properties of the solution, we propose suitable numerical techniques for solving the problem as well as obtaining the lubricant pressure, saturation, and viscosity. Finally, we give some numerical results to illustrate the performance of the proposed numerical methods as well as comparisons with alternative models.     

Lubrication of porous solids in reference to human joints

A simple mechanical model which has some features in common with load bearing human joints is described. The normal approach of two plane surfaces, one of which is covered with porous material is analysed. The gap between the two surfaces is filled with micropolar fluid to represent a particulate suspension (i.e, synovial fluid) as lubricant. The poresize diameter is so small that only the suspending medium,i.e, the viscous fluid enters into the porous matrix due to the filtration action. The problem has been solved separately in two regions; flow of viscous fluid in the porous matrix and the squeeze film lubrication with micropolar fluid as lubricant in between the two approaching surfaces along with suitable matching conditions at the porous boundary. Several interesting results have been brought out. Agreement with available experimental results and the computational results presented, herein, is quite good.     

A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.     

Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies

A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.     

Optimal control problems for wave equations

The problem on minimizing a quadratic functional on trajectories of the wave equation is considered. We assume that the density of external forces is a control function. A control problem for a partial differential equation is reduced to a control problem for a countable system of ordinary differential equations by use of the Fourier method. The controllability problem for this countable system is considered. Conditions of the noncontrollability for some wave equations were obtained.     

A method for the experimental determination of the viscoelastic properties of a cylindrical sample

Forced oscillations of a cylinder, clamped to a vibrating base and loaded on top with a certain mass, is considered using a two-dimensional deformation model. The boundary conditions are specified over the whole cylinder surface, and the equation of the balance of forces on its outer surface is used. An algorithm for solving the problem using the collocation method is proposed. It is shown that the accuracy with which the modulus of elasticity and the loss factor are determined is much higher than when using methods based on the one-dimensional model, since the proposed method does not require a knowledge of the dynamic form factor.     

A weak solvability of a system of thermoviscoelasticity for the Jeffreys model

We establish a weak solvability of the initial-boundary value problem for a dynamic model of thermoviscoelasticity. The problem under consideration is an extension of the Jeffreys model obtained with the help of a consequence of the energy balance equation. We study the corresponding initial-boundary value problem by splitting the problem and reducing it to an operator equation in a suitable Banach space.     

Mathematical modeling of contact problems of plastic flow

The nonstationary space contact problems of plastic flow in the comparatively thin layer between surfaces of external bodies have been analysed in the present paper. Earlier encountered in mathematical modeling peculiarities characterized for considered processes (as the slipping of layer material along contact, high - comparatively with shears - contact pressures, normal elastic movements commensurable with thickness of the layer, the formation of hardened layers near contact) are noted. Other characterized peculiarities, such as the property of anisotropy of friction forces on contact, the possibility of preserving the intermediate lubricant volumes on contact and contact friction faces control, the influence of initial nonhomogeneity on the formation of plastic flow are also considered. In each caw the induced qualitative effects by them are represented.