Shimon Haber

On the low reynolds number motion of two droplets

Abstract Exact solutions are derived for the quasi steady-state creeping flow internal and external to two spherical droplets moving along their line-of-centers. Numerical results are presented, which include all previous solutions as special cases.

Low Reynolds number motion of a droplet in shear flow including wall effects

Abstract The hydrodynamic interaction between a droplet immersed in Couette flow and the containing walls is studied. The analysis is based on the assumptions that the disturbance flow induced by the droplet is without inertia, that the droplet maintains its nearly spherical shape and that the radius of the droplet is small compared with the distance between the walls. Based on Lorentzs reflection method, a first-order simple analytical solution is derived for the case of a droplet in close vicinity to one wall. An integral solution is given for the general configuration of a droplet interacting with two walls. First-order corrections for wall effects are obtained for the drag force and the droplets deviation from sphericity.

Shear flow over a self-similar expanding pulmonary alveolus during rhythmical breathing

Alternating shear flow over a self-similar, rhythmically expanding hemispherical depression is investigated. It provides a fluid-mechanical model for an alveolated respiratory unit, by means of which the eect of lung rhythmical expansion on gas mixing as well as aerosol dispersion and deposition can be studied. The flow is assumed to be very slow and governed by the quasi-steady linear Stokes equations. Consequently, superposition of the following two cases provides an easy route toward characterizing the aforementioned flow eld. The rst case treats the flow eld that is generated by a rhythmically expanding spherical cap (the alveolus). The cap is attached at its rim to a circular opening in an expanding unbounded plane bounding a semi-innite fluid region. The rate of expansion of the cap and the plane are chosen such as to maintain the system’s congurational self-similarity. The second case addresses the flow disturbance that is generated by an alternating shear flow encountering a rigid hemispherical cavity in a plane bounding a semi-innite fluid domain. For the rst case, a stream-function representation employing toroidal coordinates furnishes an analytical solution, whereas the second case was solved numerically by Pozrikidis (1994). Linear superposition of the two flow cases results in particularly rich streamline maps. In the symmetry plane (bisecting the cap and parallel to the mean shear flow), for a certain range of shear to expansion-rate ratios, the streamline maps are self-similar and display closed orbits and two internal stagnation points. One of the stagnation points is a ‘centre’ surrounded by closed streamlines whereas the other constitutes a ‘saddle point’. For other planes, no stagnation points exist in the eld, but the streamlines associated with the saddle point display complex looping patterns. These unique flow structures, when subjected to a small perturbation (e.g. a small asynchrony between ductal and alveolar entering flows) cause highly complex stochastic particle trajectories even in the quasi-static Stokes alveolar flow. The observed irreversible flow phenomena in a rhythmically expanding alveolus may be partially responsible for the ‘stretch-and-fold’ flow mixing patterns found in our recent flow visualization studies performed in excised animal lung acini.

The Validity of the Reynolds Equation in Modeling Hydrostatic Effects in Gas Lubricated Textured Parallel Surfaces

Microdimples generated by laser surface texturing (LST) can be used to enhance performance in hydrostatic gas-lubricated tribological components with parallel surfaces. The pressure distribution and load carrying capacity for a single three-dimensional dimple, representing the LST, were obtained via two different methods of analysis: a numerical solution of the exact full Navier-Stokes equations, and an approximate solution of the much simpler Reynolds equation. Comparison between the two solution methods illustrates that, despite potential large differences in local pressures, the differences in load carrying capacity, for realistic geometrical and physical parameters, are small. Even at large clearances of 5% of the dimple diameter and pressure ratios of 2.5 the error in the load carrying capacity is only about 15%. Thus, for a wide range of practical clearances and pressures, the -Simpler approximate Reynolds equation can safely be applied to yield reasonable predictions for the load carrying capacity of dimpled surfaces.

LOW REYNOLDS NUMBER MOTION OF A DROPLET BETWEEN TWO PARALLEL PLATES

The low Reynolds number hydrodynamics of a droplet moving in a quiescent fluid between two parallel flat plates is studied. The method of reflections is utilized to obtain approximate solutions for the pressure field, the drag force exerted on the droplet and the deformation of the droplet.

Boundary conditions for darcy's flow through porous media

Abstract A novel formulation for the boundary conditions to be applied at a porous surface is proposed. Interfaces between porous and clear media and porous and solid media are considered. The well known Beavers & Joseph boundary condition is applicable for interfaces between porous and clear media. An equivalent boundary condition is obtained for interfaces between porous and impermeable media, namely, v · n = √(κ)∇ t · v 1 where v is the velocity field inside the porous medium, n denotes a unit vector normal to the interface pointing towards the porous medium, κ stands for the permeability and the subscript t refers to components tangential to the interface. A sample problem is solved for the flow fields exterior to a porous spherical particle and interior to it, assuming that the particle has a rigid concentric spherical core and that the submerging flow field is Newtonian. Stokesian and uniform at infinity. Both Brinkmans equation and Darcys law are utilized to obtain general forms of the velocity and pressure fields. Comparison of the two solutions yields the desired boundary conditions applicable to the Darcy problem.

A steady state, one dimensional, model for boiling two phase flow in triangular micro-channel

Abstract The potential advantages of triangular micro-channels incorporated into heat generating small devices are discussed. A simple one dimensional model of boiling two-phase flow and heat transfer in a single triangular micro-channel is investigated. The flow of the liquid phase inside the micro-channels is driven by surface tension and friction forces that exist at the interface between the fast moving vapor and liquid. The flow of the vapor phase is controlled by the heat flux generated and removed from the device. As the liquid flows through the channel it evaporates, its cross-section diminishes and the radius of curvature at the liquid vapor interface decreases. Thus, according to Young–Laplace equation, the liquid–vapor pressure difference increases along the channel. Consequently, a large decrease in the liquid pressure along the channel is obtained if the vapor pressure remains almost uniform. That pressure drop in the liquid phase is responsible for the onset of liquid flow. Along the micro-channel the increasing amount of generated vapor causes vapor velocity to increase and friction forces exerted on the liquid phase become significant until dry-out occurs. Since in the dry-out zone the heat transfer is drastically diminished, dry-out length estimates are of major concern in micro-channel design. A solution of a first order non-linear differentiated equation is required to predict dry-out lengths and their dependence on the dimensionless parameters governing the flow. A numerical simulation was carried out for the case of water flowing in a vertical channel of equilateral triangular cross-section. Hydraulic diameters from 0.1 to 1 mm, heat fluxes from 10 to 600 W/cm 2 and contact angles of 5° to 40° were assumed. The results validate the basic assumption that vapor pressure along the micro-channel is almost uniform. In many practical applications the differential equation can be simplified and solved analytically and the dry-out length are determined via a solution of an algebraic equation. Finally, it was demonstrated that the dry-out lengths seem to fit the dimensions of microelectronic devices.

Rheological properties of dilute suspensions of centrally symmetric Brownian particles at small shear rates

Abstract A constitutive equation is derived for the steady-state rheological properties of a dilute monodisperse suspension of centrosymmetric particles subject to a general homogeneous isochoric shearing flow for cases where disorienting rotary Brownian motion effects dominate over the orienting effects of the shear (small rotary Peclet numbers). Explicit results for the suspension viscosity and normal stresses are given for general triaxial ellipsoidal particles, supplementing those already available for ellipsoids of revolution. Calculations of the orientational distribution function to the second order in the rotary Peclet number yield complete dynamical rheological properties correct to the zeroth and first orders in the nominal shear rate. Alternative energy dissipation rate calculations of rheological properties are shown to be consistent with those of the (more complete) dynamical scheme. Present calculations supplement and are concordant with the numerical results of J. M. Rallison ( J. Fluid Mech. 84 , 237 (1978))—concemed, inter alia , with the same theme of nonaxisymmetric particle suspension rheology. However, use of the infinitesimal rotation pseudovector in place of Rallisons (finite) rotation dyadic as the invariant representation of rigid-body rotations simplifies the requisite algebraic manipulations sufficiently to enable our theological calculations of the material constants to be brought to fruition in an explicit analytical form.

Levitation force induced by pressure radiation in gas squeeze films

An analytical and numerical study on the levitation force induced by pressure radiation in gas squeeze films is investigated. The levitation phenomenon is known to occur when a planar object is placed at close proximity to a vibrating piston. The existing analytical approaches are based on either conventional acoustic radiation, where the fluid is assumed inviscid or on a variant of the Reynolds equation that incorporates viscous effects. Alas, these solutions are often in poor agreement with accurate numerical results and, at best, describe appropriately cases that include a limited range of object weights and vibration frequencies. In this work, two cases are addressed: the flow induced by vibrations perpendicular to a flat surface and that by flexural wave propagation parallel to the surface. For the first case, numerical and second-order analytical perturbation solutions are obtained and compared, proving them to be in good agreement. In addition, a novel, analytical expression for the levitation forc...

Lagrangian approach to time-dependent laminar dispersion in rectangular conduits. Part 1. Two-dimensional flows

Les vitesses moyennes et les coefficients de dispersion sont evalues pour un ecoulement laminaire bidimensionnel. On adopte une methode lagrangienne par laquelle une particule brownienne est tracee dans un champ de vitesse artificiellement restructure. Des expressions asymptotiques on obtient des expressions asymptotiques pour des periodes de temps courtes, moyennes et longues pour un ecoulement de Couette, un ecoulement de Poiseuille plan et un ecoulement en conduite decouverte sur une surface plane inclinee