Upper and lower bounds on the drag coefficient of a sphere in a power‐model fluid

Abstract

The equation of continuity and the stress equations of motion are not sufficient in themselves to describe the motion of matter under given boundary conditions. In addition one must describe the behavior of the particular material to be considered by stating the relation between stress and deformation, the constitutive equation, A number of properly invariant constitutive equations have been proposed. Reiner (17) and Ridin (18, 19; see also 21) present the most general isotropic relation between stress and rate-of-deformation for homogeneous material. Oldroyd (13, 1 4 ) gives an excellent discussion of the formulation of properly invariant constitutive equations; he examines in detail the simplest constitutive equations which reduce to experimentally observed viscoelastic behavior at small rates of strain. Rivlin and Ericksen (20) assume the stress to be a function of the gradients of velocity, acceleration, second acceleration, Noll (12, 5 ) and Green and RivIin (6, 7) consider memory effects; that is, the state of stress in the fluid is assumed to depend upon its past history as well as its current kinematic state. All of these proposals appear to be qualitatively consistent with observed viscometric flows (for example, flow in a cone-plate viscometer, flow between rotating concentric cylinders, flow through a circular cylindrical tube) for some materials with the exception that the Stokesian fluid model proposed by Reiner and by Rivlin does not properly predict observed normal stresses (1 1 ) .The Stokesian fluid model for an incompressible fluid" ti = -pai, + J&+~ + yd",:is by far the simplest of all the constitutive equations proposed, It includes as special cases the simple empirical models long used in engineering work to represent the results of viscometric experiments such as the power model Ti = ti + F3 = gm(2II) (n--1)f2 & While the power model cannot fully represent the behavior of any real fluid presently known (an immediate objection is that it does not predict nonzero and finite limiting viscosities at very low and a t very high rates of deformation), over a limited range of stress it often gives a satisfactory approximation for the stress vs. rate-ofdefomation curve from a viscometric study [or in terms of Noll's theory for simple fluids, it gives a satisfactory approximation for the first material function in a viscometric flow ( 5 ) 1. In practice engineers have been using the power model [also the Ellis model ( 2 2 ) , the Sisko model ( 2 3 ) , and many others] as an approximate description for the behavior of real fluids which on the basis of viscometer studies appear to be approximately powermodel fluids over a limited range of stress, but which probably exhibit normal stress effects (1 1 ) (and according to popular usage would be called v8coelastic"). It seems reasonable to ask about the error incurred in using this model to represent the behavior of such a fluid in a more complicated geometry, where according to Noll's theory it is entirely possible that a description based upon viscometri...

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

“…The present results for bubbles with immobile interface show good agreement with the solutions derived using variational principle when the parameter in the Carreau model t/is close to 1 but the predictions of this work lie somewhat above the solutions from the variational principle in the highly shear-thinning region. It is to be noted, however, that this disagreement is equivalent to that observed between the linearization solution [19,20] and the variational principle solution [26] for solid spheres in a power-law fluid. Although the comparison between the present results and those based on variational principles shows that the present model may be applicable to 2* > 10, a more accurate assessment of the applicability of the present model must await the availability of more experimental data.…”

Cold Simulation of momentum transfer of a metal droplet falling through slag systems possessing net-work structured fluids

“…Wasserman and Slattery (1965) expressed as (Eq. (12)) by defining a parameter, X, that has an upper and a lower bound from which the average value of X is obtained.…”

Hydrodynamics of the motion of spherical particles, droplets, and bubbles in a non-Newtonian liquid: Analytical methods of investigation

“…The upper and lower boundaries of the drag coeffi cient for a sphere in non Newtonian liquid were esti mated using the variational method [2]. It is noted that, although the power law model does not represent the exact behavior of the real non Newtonian liquid over the entire range of shear rates (the model does not show the nonzero and final viscosities, respectively, at very small and large shear rates γ), it still provides a sat isfactory estimats for the limited variation of .…”