Acrivos

John C. Berg, Andreas Acrivos, and Michel Boudart, Evaporation Convection H. M. Tsuchiya, A. G, Fredrickson, and R. Aiis, Dynamics of Microbial Cell Populations Samuel Sideman, Direct Contact Heat Transfer between Immiscible Liquids Howard Brenner, Hydrodynamic Resistance of Particles at Small Reynolds Numbers... 

The necking mechanism has also been investigated using theoretical and numerical techniques. The theoretical approach, based on small deformation analysis (Barthes-Biesel and Acrivos, 1973) for the case of low Ca or high p shows the formation of lobes on the drop for Ca > Cacrit - Numerical techniques (Rallison, 1981) for p = 1 give similar results. The general conclusion is confirmation of the experimentally determined curve for Cacrit the drops in this case may break up rather than extend indefinitely. 

Acrivos, A., The breakup of small drops and bubbles in shear flows. 4th International Conference on Physicochemical Hydrodynamics, Ann. N. Y. Acad. Sci., 404, 1-11 (1983). 

Barth s-Biesel, D., and Acrivos, A., Deformation and burst of a liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61,1-21 (1973). 

John C. Berg, Andreas Acrivos. and Michel Boudart Dynamics of Microbial Cell Populations... 

ANDREAS Acrivos, Stanford University JOHN Dahler, University of Minnesota H. SCOTT FOGLER, University of Michigan Thomas J. Hanratty, University of Illinois JOHN M. PrausnITZ, University of California L. E. SCRIVEN, University of Minnesota... 

Background on Spin Casting. As early as 1958, Emslie, et al. (A) proposed a theoretical treatment of spin casting for nonvolatile Newtonian fluids. This theory predicted that films formed on a flat rotating disc would have radial thickness uniformity. They predicted that the final film thickness would depend on spin speed (w) and viscosity (ij) as well as other variables such as liquid density and initial film thickness. The dependence of thickness on u> and ij was also recognized by many of the other authors reviewed in this paper, and their proposed relationships are compared in Table I. Acrivos, et al. (5) extended the Emslie treatment to the general case of non-Newtonian fluids, a category into which most polymers fall. Acrivos predicted that non-Newtonian fluids would yield films with non-uniform radial thickness. 

Frankel and Acrivos11 have obtained models with well-defined hydrodynamics for very high concentrations of rigid and elastic particles. Here the solvent forms thin films and we enter the region of lubrication theory. The expressions describing the flow do bear some similarities to the semi-empirical expressions developed at lower concentrations. For example Frankel and Acrivos give... 

ACRIVOS, A. Ind. Eng. Chem. 45 (1956) 703. Method of characteristics technique. Application to heat and mass transfer problems. 

Using a somewhat more sophisticated singular perturbation method, Acrivos and Taylor (1962) obtained... 

Acrivos and Goddard (1965) developed an asymptotic expansion for the large-Pe problem and obtained... 

E.J. Hinch and A. Acrivos Long Slender Drops in a Simple Shear Flow. J. Fluid. Mech. 98, 305 (1980). 

Acrivos, A., On the combined effect of longitudinal diffusion and external mass transfer resistance in fixed bed operations. Chem. Eng. Sci. 13, 1 (1960). 

Fig. 3.10 External Sh for spheres in Stokes flow (1) Exact numerical solution for rigid and circulating spheres (2) Brenner (B6) rigid sphere, Pe 0, Eq. (3-45) (3) Levich (L3) rigid sphere, Pe 00, Eq. (3-47) (4) Acrivos and Goddard (A 1) rigid sphere, Pe oo, Eq. (3-48) (5) Approximate values fluid spheres.

Equation (3-45) is analogous to the Oseen correction to the Stokes drag, and is accurate to 0[Pe]." It applies for any rigid or fluid sphere at any Re, provided that Pe - 0 and the velocity remote from the particle is uniform. Figure 3.10 shows that Eq. (3-45) is accurate for Pe < 0.5. Acrivos and Taylor (A2) extended the solution to higher terms, but, as for drag, the additional terms only yield slight improvement at Pe < 1. 

Figure 3.10 shows that Eq. (3-47) gives Sh approximately 10% too low for Pe = 10, while the deviation becomes worse at lower Re. Acrivos and Goddard (Al) used a perturbation method to obtain the first-order correction to Eq. (3-47) ... 

Gupalo and Ryazantsev (GIO) followed the analysis of Acrivos and Taylor (A2) with the Proudman-Pearson stream function rather than Stokes flow. For Sc > 10, the two predictions for Sh agree within 1%, while for Sc = 1 they differ by at most 8% for Pe < 1. The results of Gupalo and Ryazantsev, although valid to higher Re, are still restricted to Pe 0, so that this extension is of little practical value. 

As noted in Chapters 2 and 3, deformation of fluid particles is due to inertia effects. For low Re and small deformations, Taylor and Acrivos (T3) used a matched asymptotic expansion to obtain, to terms of order We /Re,... 

From flow visualization and angular velocity measurements, Poe and Acrivos (P12) concluded that the analysis leading to Eqs. (10-37) and (10-38) is valid only for Roq <0.1, while for Re > 6 a sphere rotates unsteadily and the wake is oscillatory. Theoretical or numerical treatments appear to be lacking beyond the near-Stokesian range until much higher Reynolds numbers. 

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