Local flow velocities, origin

Origin of the Pipe and Local Flow Velocities. For an understanding of the origin of the pipe effect, the influence of the chain extension on the local flow field, which initially created the chain extension, needs invoking. Local flow modification arises through a combination of two factors (1) the energy requirement for chain extension, and (2) the localized nature of the chain extension. It follows from factors 1 and 2 that the flow velocity is expected to become modified at strain rates when the chains start to... 

Because the role of convection is to transfer information in the direction of the flow, the derivative dV/dz must be approximated by a difference formula that uses information that is upstream of the flow. If the derivative is approximated as (Vj - Vj- )/dz when the velocity is negative, then the derivative communicates information ahead of the flow, which is physically unrealistic. Moreover, and importantly, such a downwind difference formula can cause severe numerical instabilities. From the point of view of a control volume, recall the origin of the convective terms in the substantial derivative. They represent the mass, momentum, or energy that is carried into or out of the control volume from the surrounding regions with the fluid flow. Thus the term must have a directional behavior that depends on the local fluid velocity. 

Figure 4 shows exemplarily simulated flow velocities for the gas and liquid phase. Figure 5 summarizes a comparison between simulated and predicted local values of the specific gas hold-up. For more detailed information and comparisons at different operation conditions the reader is referred to the original papers [43,44]. 

In the context of microfluidics and nanofluidics, visualization implies the determination of fluid velocity in small-scale internal flows. Visualization requires an optically detectable fluid marker that does not alter the local fluid velocity of interest. A variety of visualization methods have been developed for small-scale flows, many of which were derived from methods originally developed for macroscale flows. Molecular tagging describes one class of methods that involve molecules being rendered optically differentiable from the bulk fluid to serve as optically detectable fluid markers. The molecular dimensions of these fluid markers make them well suited to small-scale flows. Molecular tagging is commonly achieved by select exposure to light, although the specific photochemical mechanisms vary over the four techniques described here. 

The mobility tensor can be derived from Stokes-flow hydrodynamics. Consider a set of spherical particles, located at positions r, with radius a, surrounded by a fluid with shear viscosity rj. Each of the particles has a velocity v which, as a result of stick boundary conditions, is identical to the local fluid velocity on the particle surface. The resulting fluid motions generate hydrodynamic drag forces Ff, which at steady state are balanced by the conservative forces, Ff + F- = 0. The commonly used approximation scheme is a systematic multipole expansion, similar to the analogous expansion in electtostatics [17-21]. For details, we refer the reader to the original literature [17], where the contributions from rotational motion of the beads are also considered. As a result of the linearity of Stokes flow, the particle velocities and drag forces are linearly related,... 

The last term is the rate of viscous energy dissipation to internal energy, Ev = jv <5 dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <5 for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such... 

When a particle is put at the origin it rotates following the fluid motion, but generally there exists still a velocity difference between the fluid and the particle. Let us consider the case of a spherical particle of radius a. As we d ussed it acquires a rotational motion of angular velocity qj2, but this motion is not enough to have an equi-velocity distribution, and a local disturbance of the flow appears. The effect is the same if a force is put at the origin, causing the perturbation flow. Clearly, the perturbation is due to the finite size of the sphere, but when an equivalent force is found one can replace the sphere by the force which does not, of course, require volume. 

The definition (7.32) masks the local and non-local contributions from bodies to the flow. A more systematic approach to characterising the Eulerian mean velocity is to decompose the flow into (i) a far field flow contribution - far from each body but still within the group of bodies - and (ii) a near field flow contribution - local to each body. This concept, originally described qualitatively by Kowe et al. [353], is strictly valid for dilute arrays since it formally requires the bodies to be widely separated, so that there is a separation of lengthscales between the near and far field, scaling approximately as 0(a) and 0(LS) respectively. The decomposition is defined formally here for potential flows. The far field flow, u, is defined mathematically as the sum of the dipolar and source contributions from the bodies, by assuming the bodies shrink to zero, so that (from (7.31))... 

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