Velocity profile developing flow

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

When a fluid flows over a surface, that part of the stream which is close to the surface suffers a significant retardation, and a velocity profile develops in the fluid. The velocity gradients are steepest close to the surface and become progressively smaller with distance from the surface. Although theoretically there is no outer limit at which the velocity gradient becomes zero, it is convenient to divide the flow into two parts for practical purposes. 

Here vq is the measured tangential velocity profile at time t and (ve,steady) is the value at steady-state. Both intensity indices have a value of unity at f = 0, and approach zero as t approaches infinity. Figure 4.5.15 shows the variation of the intensity indices with average strain, for an outer cylinder velocity of 0.05 cm s 1. These plots indicate that the mixing process occurs in two stages, where the velocity profile develops only after the droplet concentration profile is essentially uniform. It can be seen that 1 decays to zero at approximately 100 strain units, whereas Iv shows that the steady-state velocity profile is reached only when y ps 400. From Figure 4.5.14 it can be seen that when y = 115, flow is detected... 

As usual, we must first solve for the velocity profile. The flow can be taken as laminar for low Reynolds numbers, vR/i> < 2500, where v is the maximum velocity of the flow reached at the centre of the channel (r = 0) [52], For simplicity, we will consider the particular case where the Poiseuille velocity profile has been fully developed when the solution carrying the active species... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

If there is laminar flow in a pipe or tube, equation (E4.3.15) is used to compute the friction factor accurately after some velocity profile development length. 

Unfortunately, these equations cannot be modeled using the simple parallel-flow assumptions. In the entry region the radial velocity v and the pressure gradient will have an important influence on the axial-velocity profile development. Therefore we defer the detailed discussion and solution of this problem to Chapter 7 on boundary-layer approximations. 

The fully developed velocity profile for flow between two flat plates can be used,... 

The photo in Fig. 6-3 obtained from a video clip clearly shows the evolution of a velocity gradient as a result of the fluid sticking to the surface of a blunt nose. The layer that slicks lo the surface slows the adjacent fluid layer because of viscous forces between the fluid layers, which slows the next layer, and so on. Therefore, the no-slip condition is responsible for (he development of the velocity profile. The flow region adjacent lo the wall in which the viscous effects (and thus the velocity gradients) are significant is called the boundary layer. The fluid propeity responsible for the no-slip condition and the development of the boundary layer is viscosity and is discussed briefly in Section 6-2. 

Consider steady flow of a fluid through a circular pipe attached to a large tank. The fluid velocity everywhere on the pipe surface is zero because of the no-slip condition, and the flow is two-dimensional in the entrance region of the pipe since the velocity changes in both the r- and z-directions. The velocity profile develops fully and remains unchanged after some distance from the... 

The hydrodynamics of flow of solution past the electrode, which is essential to the cell design, was rigorously investigated. In the range of flow rates used (10 4to 10-1cm3s-1), the flow was laminar (Reynolds Number, Re < 10) and hence, beyond a lead-in section of length 0.1 Re b, a parabolic velocity profile developed across the narrow channel. Thus, the hydrodynamics of the coaxial cell were equivalent to those of the conventional channel electrode [59]. It was predicted that the diffusion-limited current would obey the Levich equation... 

When fluid flows through elbows, tees, and valves a swirling flow with an uneven velocity profile develops. For accurate flow measurement fhe flow upstream of a differential pressure meter should be free of these disturbances. In order to eliminate these disturbances, a long length of straight pipe is required. For example, swirl flow inside a pipe can require 50 to 100 diameters of sfraight pipe to eliminate the spin [16]. 

Answer Use the postulated form of the one-dimensional velocity profile developed in part (a) and neglect the entire left side of the equation of motion for creeping flow conditions at low rotational speeds of the solid sphere. The fact that does not depend on cp, via symmetry, is consistent with the equation of continuity for an incompressible fluid. The r and 9 components of the equation of motion for incompressible Newtonian fluids reveal that dynamic pressure is independent of r and 9, respectively, when centrifugal forces are negligible. Symmetry implies that does not depend on cp, and steady state suggests no time dependence. Hence, dynamic pressure is constant, similar to a hydrostatic situation. Fluid flow is induced by rotation of the solid and the fact that viscous shear is transmitted across the solid-liquid interface. As expected, the -component of the force balance yields useful information to calculate v. The only terms that survive in the (/ -component of the equation of motion are... 

Correction to the shear rate is necessitated by the fact that unlike in isothermal Newtonian flow where the velocity distribution from wall to wall in a tube is parabolic, nonparabolic velocity profile develops in non-Newtonian flow. The Rabinowitsch correction [21] is applied to shear rate to eliminate this error as follows ... 

Several variations of microflow visualization have been developed for microfluidic applications such as particle-based flow velocimetry and scalar-based flow velocimetry [5]. In terms of the zeta potential measurement, these visualization techniques such as micro-PlV are used to measure the velocity profile and flow rate under electroosmotic flow. Once the velocity is known, the zeta potential can be calculated from Eq. 6. The main advantage of using a flow visualization technique is that the electroosmotic velocity can be measured directly and in real time. In general, the small amount of particles or dye used has a negligible effect on the electroosmotic flow being measured. The oifly significant disadvantage of this technique is that the extent and cost of the hardware may be prohibitive. 

In general, axial dispersion decreases with increasing values for Re and Re Sc. An exception is the behavior of empty tubes under laminar flow conditions. For laminar flow a parabolic velocity profile develops. Under these conditions, molecular diffusion in axial and radial directions plays an important role in RTD. The diffusion in the radial direction tends to diminish the spreading effect of the parabohc velocity profile, while in the axial direction the molecular diffusion increases the dispersion. As a result the axial dispersion passes through a minimum (Pe passes through a maximum) as function oiRe- Sc = u- d /D at Re - Sc= (see Equation 3.59). 

Application Laminar flow was established using flow rates such that the Reynolds Number, Re < 10. Thus, after sufficient lead-in (specifically 0.1 x Re x h), which in this case is negligible, a parabolic velocity profile develops across the flow-path. In this manner, the hydrodynamics of this electrochemical setup are equivalent to that of the channel electrode flow system the mass transport-limiting current is therefore given by the Levich equation [86],... 

Provided that the flow is laminar, the velocity profile develops in a microchannel from the entrance to the position where a fully developed parabolic profile is established. The length of the entrance zone (Lz) can be estimated by Equation 9.10 [13] ... 

The formulation of Reynolds boundary conditions In terms of pressures p o and Ap = o Is not the primary physical one, this Is continuity within the total pressure development. Circumferential flow and the two different types of velocity profiles develop independently as long as the gap is filled. Dynamically loaded bearings are overfeed and the oil inlet Is chosen according to the state of the art. The point Is not a pure mathematical correct solution, but a physical one, concerned with pressure development, which Is given In Holland-Lang as well as In the mobility method or impulse-whirl angle method. 

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