Reynolds number creep flows conditions

Problem 7-3. Linearity of Creeping Flow. In the laboratory you are looking at the sedimentation of rods under creeping-flow conditions (zero Reynolds number). These may be regarded as bodies of revolution characterized by an orientation vector (director) pL. In a simple experiment you measure the sedimentation velocity when the director is parallel to gravity (e.g., point down) to be 0.03 cm/s and when it is perpendicular to gravity to... 

Thus, the greater the Bingham number, the higher is the Reynolds number up to which the creeping flow conditions apply for spheres moving in Bingham plastic fluids. 

The Basset force accounts for the effect of past acceleration. The original formulation of the Basset force is derived from the creeping flow condition. For a particle moving in a liquid with a finite Reynolds number, the modified Basset force is given as (Mei and Adrian, 1992)... 

Sy et al (S8, S9) and Morrison and Stewart (M12) analyzed the initial motion of fluid spheres with creeping flow in both phases. For bubbles (y = 0, k = 0), the condition that internal and external Reynolds numbers remain small is sufficient to ensure a spherical shape. However, for other k and y, the Weber number must also be small to prevent significant distortion (S9). For k = 0, the equation governing the particle velocity may be transformed to an ordinary differential equation (Kl), to give a result corresponding to Eq. (11-16), i.e.,... 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

Under conditions of low Reynolds number (slow viscous flow), the Navier-Stokes equations are reduced to the Stokes or creeping-flow equations given by ... 

In reviewing our analysis of (9-152) leading to (9-141), we may note that we have used the conditions (9-143) and (9-144) only on the velocity field. Thus, as stated earlier, the result (9-141) or (9-142) is valid in streaming flow for any heated body with a uniform surface temperature provided that these conditions are satisfied and that /V Higher-order terms in (9-142) will depend on the details of the flow, and thus on the Reynolds number Re, as well as the shape and orientation of the body relative to the free stream. However, in the creeping-flow limit, Brenner was able to extend (9-142) to one additional term for a particle of arbitrary shape,... 

To specify the velocity field u, we must solve the Navier-Stokes equations subject to the boundary condition (9-160) at infinity. For present purposes, we follow the example of Section C and assume that the Reynolds number, defined here as Re = a2yp/ii, is very small so that the creeping-flow solution for a sphere in shear flow (obtained in Chap. 8, Section B) can be applied throughout the domain in which 6 differs significantly from unity. Hence, from (8-51) and (8-57), we have... 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

When the radius of the particle is very small, or the fluid viscosity is very large, or the relative velocity is very small, or the fluid density is very low, the Reynolds number becomes very small and the flow satisfies the conditions of the Stokes or creeping flow. In this limit, the inertial forces near the particle are small and can be neglected in the Navier-Stokes equations. The pressure distribution rm the drop in this limit takes the form... 

Taylor and Acrivos (1964) found an approximate expression for applicable for small valnes of the Reynolds nnmbCT (= l/ap /fi ) and capillary number (= U]4gly). They first obtained the creeping flow solution that satisfied all boundary conditions, those at the drop-flnid intaface being satisfied at r = a. The normal stress balance (Equation 7.19), which was not used in this initial proce-dnre, was then applied, with evaluated at r = a to obtain a first approximation... 

Our starting point for the lubrication approximation was intuitive, where we assumed that flow in a nearly parallel channel would approximate flow in a channel with a slowly varying cross section to within terms of order a. It can be shown that the approximate is valid provided apVH/r < 1, where His a characteristic gap spacing. This is a less restrictive condition than creeping flow, which requires pVH/r] since a < 1, the Reynolds number need not be small. The development of the ordering analysis can be found in many fluid mechanics books, including... 

Processing with mechanical mixers occurs under either laminar or turbulent flow conditions, depending on the impeller Reynolds number, defined as Re = pND / x. For Reynolds numbers below about 10, the process is laminar, also called creeping flow. Fully turbulent conditions are achieved at Reynolds numbers higher than about 10", and the flow is considered transitional between these two regimes. 

A rigorous solution exists for fj for the limiting condition of very low fluid flow rates around a sphere - in which the fluid streamlines follow the contours of the sphere, with no separation at the upper surface (the so called creeping flow regime). This may be regarded to occur at particle Reynolds numbers Re below about 0.1 ... 

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