Quiescent Viscous Fluid Research Articles

Long-time molecular diffusion, sedimentation and Taylor dispersion of a fluctuating cluster of interacting Brownian particles

Generalized Taylor dispersion theory, incorporating so-called coupling effects, is used to calculate the transport properties of a single deformable ‘chain’ composed of hydrodynamically interacting rigid Brownian particles bound together by internal potentials and moving through an unbounded quiescent viscous fluid. The individual rigid particles comprising the flexible chain or cluster may each be of arbitrary shape, size and density, and are supposed ‘joined’ together to form the chain by a configuration-dependent internal potentialV. Each particle separately undergoes translational and rotational Brownian motions; together, their relative motions give rise to a conformational or vibrational Brownian motion of the chain (in addition to a translational motion of the chain as a whole). Sufficient time is allowed for all accessible chain configurations to be sampled many times in consequence of this internal Brownian motion. As a result, an internal equilibrium Boltzmann probabilistic distribution of conformations derived fromVeffectively obtains.In contrast with prior analyses of such chain transport phenomena, noad hocpreaveraging hypotheses are invoked to effect the averaging of the input conformation-specific hydrodynamic mobility data. Rather, the calculation is effected rigorously within the usual (quasi-static) context of configuration-specific Stokes-Einstein equations.Explicit numerical calculations serving to illustrate the general scheme are performed only for the simplest case, namely dumb-bells composed of identically sized spheres connected by a slack tether. In this context it is pointed out that prior calculations of flexible-body transport phenomena have failed to explicitly recognize the existence of a Taylor dispersion contribution to the long-time diffusivity of sedimenting deformable bodies. This fluctuation phenomenon is compounded of shape-sedimentation dispersion (arising as a consequence of the intrinsic geometrical anisotropy of the object) and size-sedimentation dispersion (arising from fluctuations in the instantaneous ‘size’ of the object). Whereas shape dispersion exists even for rigid objects, size dispersion is manifested only by flexible bodies. These two Taylor dispersion mechanisms are relevant to interpreting the non-equilibrium sedimentation-diffusion properties of monodisperse polymer molecules in solutions or suspensions.

On the deformation of two droplets in a quasisteady stokes flow

A theoretical method is given for the determination of the shape of two drops (bubbles) moving with constant velocities parallel to their line of centres in a quiescent viscous fluid. The Reynolds numbers for the motions within the fluids are assumed to be sufficiently small that the equations governing these motions are quasisteady Stokes' equations. It is also assumed that the maximum deviation of the interfaces from spherical form is small when compared with the radius of the “equivalent” spherical drop. The paper deduces the first-order pressure distribution exterior and interior to the droplets. Effects of fluid viscosities, capillary numbers and distances between the droplets are taken into account. Special attention is paid to the influence of a solid plane or solid sphere on the shape of a drop (bubble) approaching or receding away from the solid boundary. The obtained solutions may serve as a first iteration of an iterative procedure for determining more accurate flow fields, taking into account the deviation from sphericity of the deformed particles.

Taylor dispersion in systems of sedimenting nonspherical Brownian particles: III. Time-periodic forces

An Aris-type moment scheme is employed to calculate the time-average mean velocity vector and Taylor-Aris dispersion tensor for the Stokes sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles which simultaneously undergo translational and rotational Brownian motions while settling in an otherwise quiescent viscous fluid under the influence of a time-periodic external force. This generalizes to time-periodic forces prior dispersion results previously restricted to time-independent forces. Each harmonic component of the force independently gives rise to its own Taylor dispersion contribution, in which appears a dimensionless coefficient quantifying the effects of frequency and particle geometry—namely size and shape. This parameter vanishes identically for spherical and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.), whose translational hydrodynamic resistances are orientation independent. Advantages attaching to an experimental “wet” particle-shape characterization device based upon the principles developed herein are pointed out. In this context a novel experimental device is proposed based upon preventing any net (i.e., time-average) gravitational sedimentation of the nonneutrally buoyant particles by continuous rotation of the suspension about a horizontal axis.

Taylor dispersion in systems of sedimenting nonspherical Brownian particles: II. Homogeneous ellipsoidal particles

An Aris-type moment scheme is applied to calculate the Taylor-Aris dispersion tensor for the sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles settling under the influence of gravity in a quiescent viscous fluid and undergoing rotational and translational Brownian motions. This generalizes to triaxial particles a prior dispersion result for centrally symmetric bodies of revolution, such as spheroids. An independent Langevin-type dispersivity calculation is shown to yield results identical to those obtained by the moment scheme. The components D ̄ |∗ and D ̄ ⊥∗ of the transversely isotropic dispersion dyadic, parallel and perpendicular, respectively, to the direction of the gravitational field, are shown to be given by the Taylor—Aris-type formulas D ∥ ∗ = D+ γ U ∗2 15 d , D ⊥ ∗ = D+ γ U ∗2 20d . Here, U ̄ ∗ is the mean settling velocity of the particle, and D ̄ = ( 1 3 )(D 1 + D 2 + D 3) and d ̄ = ( 1 3 ) × (d 1 + d 2 + d 3) are respectively the mean translational and rotational diffusivities of the Brownian particle, with (D 1,D 2,D 3) and (d 1,d 2,d 3) respectively the appropriate diffusivity components along the principal axes of the particle. The dimensionless coefficient γ, which is of order unity, is given by the formula γ= d[( D 1 D−1 ) 2d 1+( D 2 D−1 ) 2d 2+( D 3 D−1 ) 2d 3] d 1d 2+d 2d 3+d 3d 1 , This anisometric parameter vanishes identically for spherical particles and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.) whose translational and rotational hydrodynamic resistances are independent of the orientation of the particle relative to the directions of its linear and angular velocity vectors. Upon utilizing the translational and rotational Stokes-Einstein equations, explicit numerical values of D ̄ |∗ and D ̄ ⊥∗ are furnished for ellipsoids of revolution of various aspect ratios and sizes when settling in water. Physical restrictions pertaining to sedimentation-vessel apparatus size and the requirement of reasonable sedimentation times greatly restrict the range of particle sizes whose anisometric properties may be experimentally investigated by this new particle-shape characterization technique.

Asymmetric stokes flows generated by the motion of a closed torus

Exact solutions of the Stokes equations are given for asymmetric flows produced by a closed torus which steadily translates along or rotates about a direction perpendicular to its axis of rotational symmetry in a quiescent viscous fluid. The force and couple acting on the body are calculated and the general motion of a closed torus is considered.

The hydrodynamic interaction of two unequal spheres moving under gravity through quiescent viscous fluid

The hydrodynamic forces and couples that act on two spherical particles in slow motion through a quiescent fluid are determined as functions of the relative configuration of the particles from the solution of the Stokes equation for the motion of the fluid in the vicinity of the particles. General formulae that relate the translational and rotational velocities of the particles to the ratios of their radii a = a2/a1 and net densities I are obtained, as are asymptotic forms for the velocities in the limiting cases of very large and very small interparticle separation. Relative trajectories of the particles when they move solely under gravity and their own interaction are calculated for several values of I and a. A particularly interesting feature of the results is that, for certain ranges of values of I and a, trajectories of finite length and trajectories having the form of closed periodic orbits may occur.