Spheres neutrally buoyant

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Similar methods were employed by Schonberg et al. (1986) to investigate the multiparticle motions of a finite collection of neutrally buoyant spheres suspended in a Poiseuille flow. They were also used by Ansell and Dickinson (1986) to simulate the fragmentation of a large colloidal floe in a simple shear flow. 

Figure 3.50 Experimental set-up for measuring flow velocity, (a) Cross-section view of silicon-based flow cell with FPW transducer on left side of bottom plate and glass slide to cover top of cell, (b) Cell on stage of optical microscope that is fitted with video camera and VCR for recording, (c) Schematic cross section of flow cell showing marker spheres and light rays converging from lens having small depth of field and large focal distance. (Neutrally buoyant spheres are depicted here see text for details.) (Reprinted with permission. See Ref. [75]. 1994 IEEE.)...

Let us then consider a suspension of identical, neutrally buoyant solid spheres of radius a. We are interested in circumstances in which the length scale of the suspension at the particle scale (that is, the particle radius) is very small compared with the characteristic dimension L of the flow domain so that the suspension can be modeled as a continuum with properties that differ from the suspending fluid because of the presence of the particles. Our goal is to obtain an a priori prediction of the macroscopic rheological properties when the suspension is extremely dilute, a problem first considered by Einstein (1905) as part of... 

It follows that a neutrally buoyant sphere in any linear flow can be treated as though the origin of coordinates for the undisturbed flow is coincident with the center of the sphere, namely,... 

Again, suppose that there is no external force acting on the sphere (suppose it is neutrally buoyant). Then, according to (8-226),... 

A. Babiano, J.H.E. Cartwright, O. Piro, and A. Provenzale. Dynamics of a small neutrally buoyant sphere in a fluid and targeting in hamiltonian systems. Phys. Rev. Lett., 84 5764, 2000. 

The Navier-Stokes equation has been solved By Cox and Brenner (unpublished data) who computed the lateral force required to maintain a sphere at a fixed radial position (r). They used Stoke s law for the neutrally buoyant case ... 

Drop Deformability When a neutrally buoyant, initially spherical droplet is suspended in another liquid and subjected to shear or extensional stress, it deforms and then breaks up into smaller droplets. Taylor [1932,1934] extended the work of Einstein [1906, 1911] on dilute suspension of solid spheres in a Newtonian liquid to dispersion of single Newtonian liquid droplet in another Newtonian liquid, subjected to a well-defined deformational field. Taylor noted that at low deformation rates in both uniform shear and planar hyperbolic fields, the sphere deforms into a spheroid (Figure 7.9). 

If the sphere is neutrally buoyant the former relation requires that Ur = Ur, so that the center of the ellipsoid is carried along with the fluid. We note the identities x ( - ) x r = ( - ) ( r — Qr) and x (S r) = — X T. Thus, in terms of a system of Cartesian coordinates instantaneously coinciding with the principal axes of the ellipsoid, the following expressions are obtained for the components of the couple about the center of the ellipsoid ... 

With regard to wall effects in fluids undergoing net flow, Goldman, Cox, and Brenner (G5e), using bipolar coordinates, obtained an exact solution to the problem of a neutrally buoyant sphere near a single plane wall in a semiinfinite fluid undergoing simple shear. In an unbounded fluid the translational... 

The couple on the sphere vanishes unless it is restrained from rotating. If the sphere is also neutrally buoyant then F = 0, and only the last term in Eq. (147a) survives. By noting that the local rate of mechanical energy dissipation in the unperturbed flow is 2/iSjj Sjj, this ultimately leads to a simple proof of Einstein s law of suspension viscosity (Ela) for flow through cylinders (B17), provided that the spheres are randomly distributed over the duct cross section. 

Segre and Silberberg (S6) Dilute suspension of neutrally buoyant spheres 1.12 0.028-0.153... 

Goldsmith and Mason (G9) Neutrally buoyant sphere Karnis, Goldsmith, and Neutrally buoyant sphere 0.800 0.123-0.135... 

Oliver (02) Neutrally buoyant sphere (see also discussion for other cases) 0.940 0.245-0.361... 

Repetti and Leonard (R4a) Neutrally buoyant sphere Rectangular / 0.187... 

Intermediate case between a neutrally buoyant and nonneutrally buoyant sphere WJjV = 0(a/RJ ... 

At larger particle Reynolds numbers (Re x 10" ), where inertial effects are more significant, rigid, neutrally buoyant rods and disks behave in essentially the same manner as do rigid spheres. They migrate inward if introduced near the wall and outward if introduced near the axis, attaining a stable terminal position at about j8 = 0.5 (K5, K5b). 

In regard to nonsteady tube flows. Mason et al. have observed both inward and outward radial migration of rigid, neutrally buoyant spheres in oscillatory (S9b, G9b) and pulsatile (Tl) flows in circular tubes at frequencies up to 3 cps, at which frequencies inertial effects are likely to be important. We refer here to inertial effects arising from the local acceleration terms in the Navier-Stokes equations, rather than from the convective acceleration terms. In the oscillatory case the spheres (a/R 0.10) attained equilibrium positions at about P = 0.85. Important Reynolds numbers here are those based upon mean tube velocity for one-half cycle and upon frequency. Nonneutrally buoyant spheres in oscillatory flow migrate permanently to the tube axis, irrespective of whether they are denser or lighter than the fluid (K4a). 

Note that for hard spheres, surprisingly there is no influence of the particle size on the viscosity. The only concern about particle size in hard sphere suspensions is that if the particles are too big they will settle out. If the particles are neutrally buoyant sedimentation issues are not significant. 

Dimensional analysis thus reduces the task of characterizing the dependency by three orders of magnitude. Further reductions are possible in practical situations. For example, at moderate shear rates, the steady-state relative viscosity of a dispersion of neutrally buoyant spheres depends on only the dimensionless concentration (volume fraction) and the Peclet Number Pe, which is a dimensionless shear rate ... 

Einstein [63-65] was the pioneer in the study of the viscosity of dilute suspensions of neutrally buoyant rigid spheres without Brownian motion in a Newtonian hquid. He proposed the following relationship between the relative viscosity of the suspension and the volume fraction of the suspended particles... 

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