Reynolds number highly viscous flows

FIGURE 1.3 Fluid flow characteristics and profiles of fluid flow in pipes (a) At low Reynolds numbers, where streamline flow is obtained throughout the cross section, (b) At high Reynolds numbers, where turbulent flow is obtained for most of pipe volume. Streamline flow is only obtained in a thin boundary layer adjacent to the pipe wall where the influence of the wall and viscous forces control turbulence. 

Let us investigate steady-state convective diffusion on the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers (the Blasius flow). We assume that mass transfer is accompanied by a volume reaction. In the diffusion boundary layer approximation, the concentration distribution is described by the equation... 

When immiscible fluid streams are contacted at the inlet section of a microchannel network, the ultimate flow regime depends on the geometry of the microchannel, the flow rates and instabilities that occur at the fluid-fluid interface. In microfluidic systems, flow instabilities provide a passive means for co-flowing fluid streams to increase the interfacial area between them and form, e.g. by an unstable fluid interface that disintegrates into droplets or bubbles. Because of the low Reynolds numbers involved, viscous instabilities are very important At very high flow rates, however, inertial forces become influential as well. In the following, we discuss different instabilities that either lead to drop/bubble breakup or at least deform an initially flat fluid-fluid interface. Many important phenomena relate to classical work on the stability of unbounded viscous flows (see e.g. the textbooks by Drazin and Reid[56]and Chandrasekhar [57]). We will see, however, that flow confinement provides a number of new effects that are not yet fully understood and remain active research topics. 

At very high Reynolds numbers the viscous forces are quite small compared to the inertia forces and the viscosity can be assumed as zero. These equations are useful in calculating pressure distribution at the outer edge of the thin boundary layer in flow past immersed bodies. Away from the surface outside the boundary layer this assumption of an ideal fluid is often valid. 

Since viscous forces are a manifestation of intermolecnlar attractive forces, they are stabilising, whereas inertial forces tend to pull the fluid elements apart and are destabilising. It is thns qnite logical that stable (laminar) flow should occur at low Reynolds numbers where viscous forces dominate, whereas unstable (turbulent) flow occurs at high Reynolds numbers where inertial forces dominate. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

High-Viscosity Systems A axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in... 

At low values of the Reynolds number, less than about 10, a laminar or viscous zone exists and the slope of the power curve on logarithmic coordinates is — 1, which is typical of most viscous flows. This region, which is characterised by slow mixing at both macro-arid micro-levels, is where the majority of the highly viscous (Newtonian as well as non-Newtonian) liquids are processed. 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects. 

In view of the usually viscous nature of highly non-Newtonian materials it is not likely that Reynolds numbers appreciably greater than 70,000 will be very common, at least for some time to come. This fact places great importance on the region below NRe = 70,000, and its detailed study would appear to be of primary importance. In well-developed turbulent flow, which apparently may be delayed to... 

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