Acrivos number

Lighthill-Acrivos Generalized Expression Acrivos Number.312... 

The use of this theory has a real advantage as we do not need Newtonian fluids, it is applicable for large viscosity liquids, but the requisite is a laminar regime. Particularly, in the case of free convection, an Acrivos number (Ac) is defined. 

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... 

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. 

Writing of this chapter was facilitated by grants to Howard Brenner from the Office of Basic Energy Sciences of the Department of Energy and the National Science Foundation. The authors also wish to thank the referees, Professors Andreas Acrivos and William B. Russel, for bringing a number of relevant papers to our attention. 

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

T. D. Taylor and A. Acrivos, On the deformation and drag of a falling viscous drop at low Reynolds number, J. Fluid Mech. 18, 466-76 (1964). 

Acrivos and Taylor actually evaluated two additional terms in the inner expansion. Their expression for the correlation between Nusselt number and Peclet number is... 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

A general study of the streamlines for a circular cylinder in simple shear flow can he found in the following papers C. R. Robertson and A. Acrivos, Low Reynolds number shear flow past a rotating circular cylinder, Part I, Momentum Transfer, J. Fluid Mech. 40, 685-704 (1970) R. G. Cox, I. Y. Z. Zia, and S. G. Mason, Particle motions in sheared suspensions, 15. Streamlines around cylinders and spheres, J. Colloid Interface Sci. 27, 7-18 (1968). 

A. Acrivos, Heat transfer at high Peclet number from a small sphere freely rotating in a simple shear flow, J. Fluid Mech. 46, 233-40 (1971). 

Poe, G. G. and Acrivos, A., Closed streamline flows past small rotating particles heat transfer at high Peclet numbers, Int. J. Mult. Flow, Vol. 2, No. 4, pp. 365-377, 1976. 

Robertson, C. R. and Acrivos, A., Low Reynolds number shearflow past a rotating circular cylinder. Part 2. Heat transfer, J. Fluid Mech., Vol. 40, No. 4, pp. 705-718, 1970. 

More recently a number of detailed laminar dynamic analyses have been made of the flow in inclined channels with sedimentation. In these studies the flow is assumed to remain stable. Reference to them, along with historical background on settling in inclined channels, may be found in Davis Acrivos (1985). The volumetric flow rate b/sin 9)(dH/dt), as given from Eq. (5.4.28), is found to be generally satisfactory, though when the clarified layer occupies an appreciable portion of the channel the simple kinematically derived relation overestimates the fall speed dH/dt. 

DAVIS, R.H. c ACRIVOS, A. 1985. Sedimentation of noncolloidal particles at low Reynolds numbers. Ann. Rev. Fluid Mech. 17, 91-118. 

The observed stabilizing effect of surfactants toward convection induced by surface tension has been confirmed theoretically in a recent paper by Berg and Acrivos (B13), in which the stability analysis technique and the physical model were the same as Pearson s except that the free-surface boundary condition [(iii) of Table III] took into account the presence of surface active agents. Critical values for the Thompson number were computed as functions of two dimensionless parameters, one embodying the surface viscosity and the other the surface elasticity. ... 

A theoretical analysis of the Stokes flow problem for a noimeutraUy buoyant droplet is clearly called for. Germane to this problem is the theoretical analysis of Haberman (H3), dealing with axially symmetric Stokes flow relative to a liquid droplet at the axis of a circular tube, and Taylor and Acrivos (T2c) extension of the classical Hadamard-Rybczynski liquid droplet problem to the case of nonzero Reynolds numbers. In particular, Haberman shows that the assumption of a spherical shape for the droplet in a tube is incompatible with the differential equations and boundary conditions. Taylor and Acrivos (T2c) point out that, though Hadamard (H3a) and Rybczynski (RIO) were able to solve the Stokes flow problem by assuming a spherical shape for a liquid droplet, irrespective of the magnitude of the interfacial tension, the correctness of their a priori assumption was, to some extent, fortuitous. These remarks are undoubtedly pertinent to the resolution of Haberman s paradox and, ultimately, to the solution of the nonaxially symmetric droplet problem. 

O Brien (Ola) applied the Acrivos-Taylor analysis to the case where the Reynolds number, though small, is not identically zero as in the Stokes flow case. The analysis is vastly more complicated because, to any order in the Reynolds number, the velocity field v appearing in Eq. (298) is now expressed in terms of two, locally valid expansions [the inner and outer expansions of v given by the Proudman-Pearson analysis (PI 1)], rather than the single Stokes velocity field. For a solid spherical particle she obtains for small Pe and Re... 

An extension of the Acrivos-Taylor (Ala) heat transfer analysis to slip flow past the sphere is given by Taylor (T2a). Hartunian and Liu (H9a) and Taylor (T2b) include the elfect of surface chemical reactions in their singular perturbation treatment of Stokes-flow mass transfer from spheres at small Peclet numbers. 

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