Howard Brenner

Interfacial transport processes and rheology

Interfacial rheology and its applications basic properties of interfacial transport processes interfacial transport of momentum interfacial transport of species measurement of dynamic interfacial tension and dilatational elasticity measurement of interfacial shear viscosity measurement of the interfacial pilatational viscosity measurement of non-Newtonian interfacial theological behaviour interfacial stability foam rheology a surface-excess theory of interfacial transport processes surface-excess transport of momentum surface-excess transport of species a line-excess theory of equilibrium line tension.

Rheology of a dilute suspension of axisymmetric Brownian particles

Abstract Explicit results are presented for the complete rheological properties of dilute suspensions of rigid, axisymmetric Brownian particles possessing fore-aft symmetry, when suspended in a Newtonian liquid subjected to a general three-dimensional shearing flow, either steady or unsteady. It is demonstrated that these rheological properties can be expressed in terms of five fundamental material constants (exclusive of the solvent viscosity), which depend only upon the sizes and shapes of the suspended particles. Expressions are presented for these scalar constants for a number of solids of revolution, including spheroids, dumbbells of arbitrary aspect ratio and long slender bodies. These are employed to calculate rheological properties for a variety of different shear flows, including uniaxial and biaxial extensional flows, simple shear flows, and general two-dimensional shear flows. It is demonstrated that the rheological properties appropriate to a general two-dimensional shear flow can be deduced immediately from those for a simple shear flow. This observation greatly extends the utility of much of the prior Couette flow literature, especially the extensive numerical calculations of Scheraga et al. (1951, 1955). The commonality of many disparate results dispersed and diffused in earlier publications is emphasized, and presented from a unified hydrodynamic viewpoint.

The diffusion model of longitudinal mixing in beds of finite length. Numerical values

Abstract Accurate numerical values are provided for the solution of the diffusion model of miscible fluid displacement in beds of finite length. The calculations presented here apply to the case in which a solute, uniformly distributed throughout the medium at the outset of the experiment, is displaced by a solvent which may itself contain some of the solute. Results are given in dimensionless form for (i) the instantaneous concentration of solute leaving the bed, and (ii) the average solute concentration in the bed at any instant. Various limiting cases are discussed.

The Constrained Brownian Movement of Spherical Particles in Cylindrical Pores of Comparable Radius: Models of the Diffusive and Convective Transport of Solute Molecules in Membranes and Porous Media

Abstract Rigorous calculations are presented for the effect of a constraining circular cylindrical boundary upon the translational Brownian motion of an isolated spherical particle suspended in a Poiseuille flow. These results also apply to dilute, multiparticle systems. Results are presented in the format of modifications of the Taylor—Aris theory of dispersion, arising from nonzero values of the dimensionless parameter λ = sphere radius/tube radius. Differences between present, continuum-mechanical, results and those derived from global forms of the thermodynamics of irreversible processes are reconciled. An outline of a more general theoretical framework is presented, with a view toward eventual applications to nonspherical particles and noncylindrical boundaries, such as occur, for example, in problems of aerosol deposition.

The constrained brownian movement of spherical particles in cylindrical pores of comparable radius

Abstract Rigorous calculations are presented for the effect of a constraining circular cylindrical boundary upon the translational Brownian motion of an isolated spherical particle suspended in a Poiseuille flow. These results also apply to dilute, multiparticle systems. Results are presented in the format of modifications of the Taylor—Aris theory of dispersion, arising from nonzero values of the dimensionless parameter λ = sphere radius/tube radius. Differences between present, continuum-mechanical, results and those derived from global forms of the thermodynamics of irreversible processes are reconciled. An outline of a more general theoretical framework is presented, with a view toward eventual applications to nonspherical particles and noncylindrical boundaries, such as occur, for example, in problems of aerosol deposition.

The motion of a closely-fitting sphere in a fluid-filled tube

Abstract Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses. Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube. The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.

The slow motion of a sphere through a viscous fluid towards a plane surface. II - Small gap widths, including inertial effects.

Abstract Singular perturbation techniques are employed to calculate the hydrodynamic force experienced by a sphere moving, at small Reynolds numbers, perpendicular to a solid plane wall bounding a semi-infinite viscous fluid, for the limiting case where the gap width between the sphere and plane tends to zero. Two distinct, but related, analyses of the problem are presented. In the first analysis, the exact bipolar-coordinate expression for the force given independently by M aude [1] and B renner [2] for the quasistatic Stokes flow case is expanded via a novel asymptotic procedure. The second analysis, which is somewhat more general in scope, provides a perturbation solution of the unsteady Navier—Stokes equations for a more general axisymmetric particle than a sphere, taking accounts of the finiteness of the Reynolds number. When the Reynolds number is taken into account, the force on the particle differs, according as it moves towards or away from the wall. Using the techniques developed in the first portion of this paper, formulas are derived for the Stokes couple required to maintain the symmetrical rotation of a dumbbell in an unbounded fluid, as well as the couples required to maintain the symmetrical rotation of a sphere touching a rigid plane wall and touching a planar free surface.

The Stokes resistance of an arbitrary particle

Abstract At small particle Reynolds numbers it is demonstrated that the intrinsic hydrodynamic resistance of an arbitrary particle to translational motion through an incompressible, unbounded, viscous fluid can be represented by a symmetric second-rank tensor (dyadic), uniquely determined by the exterior geometry of the particle. Similar remarks apply to the intrinsic resistance of the body to rotation about an axis. Unlike the previous tensor, however, the rotation tensor is shown to vary with position, an explicit formula for its variation being derived. The existence of a unique geometrical point, through which the hydrodynamic force always acts, is established. It is pointed out that this point serves to differentiate translational and rotational particle motions. The ultimate, stable orientation attained by a particle settling under the influence of gravity is shown to devolve upon the relative positions of its centres of mass, buoyancy and “hydrodynamic stress”, the latter being the point referred to in the previous paragraph. General dynamical equations are derived for the steady and unsteady motion of a settling particle and their solutions given for a few simple cases. Extension of the fundamental formulae to the case of a fluid in net flow is discussed.

The lateral migration of solid particles in Poiseuille flow — I theory

Abstract The general time independent problem concerning the motion of solid particles of arbitrary shape at low Reynolds numbers is considered. The effects of a basic flow, of solid boundaries and of inertia are jointly included in a theory which is used to examine the behaviour of spheres in laminar tube flows. Thus formulae are obtained for the lateral migration of spheres in laminar flows through vertical tubes of circular and non-circular cross-section. Cases of neutrally and non-neutrally buoyant particles are considered as is also the case of zero flow along the tube. Also given is a general discussion of some of the qualitative aspects of the results.

The Stokes resistance of an arbitrary particle—IV Arbitrary fields of flow

Abstract A phenomenological scheme is formulated for calculating the quasistatic Stokes force and torque on a rigid particle of any shape immersed in a flow field which tends to an arbitrary Stokes flow at infinity. This generalizes a previous result (Part III) limited to a uniform shear flow at infinity. The phenomenological resistance coefficients are shown to be constant polyadics which are intrinsic properties of the particle, dependent only on its external shape. In particular they are independent of the density, viscosity, and state of motion of the fluid. It is demonstrated that these coefficients can be computed solely from a knowledge of the solutions of Stokes equations for translational and rotational motions of the particle, along any three non-coplanar axes, in a fluid at rest at infinity. Explicit formulae for the polyadic coefficients are given for ellipsoidal and slightly deformed spherical particles.