Cross-flow velocity gradient

Cross-Flow Velocity Gradient Combined with a Field Action... 

A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is fiat The shear stress acting within die fluid Ry varies with velocity gradient du,/dy according to the relation ... 

The special flow conditions in circular (capillaries, tubes) or rectangular channels cause very different stresses depending on the position of the particles in the flow cross section. With laminar flow, for example the following applies to velocity gradient (see e.g. [37]) ... 

The membrane viscometer must use a membrane with a sufficiently well-defined pore so that the flow of isolated polymer molecules in solution can be analyzed as Poiseuille flow in a long capillary, whose length/diameter is j 10. As such the viscosity, T, of a Newtonian fluid can be determined by measuring the pressure drop across a single pore of the membrane, knowing in advance the thickness, L, and cross section. A, of the membrane, the radius of the pore, Rj., the flow rate per pore, Q,, and the number of pores per unit area. N. The viscosity, the maximum shear stress, cr. and the velocity gradient, y, can be calculated from laboratory measurements of the above instrumental parameters where Qj =... 

M 72] [M 73] [P 65] The analysis of cross-sectional velocity profiles (water as fluid Re =12) shows that the intersecting structures have intricate gradient fields near the bars of the internals, while the helical device displays entrance and exit effects over more than one-quarter of the flow field (see [155] e.g. for fluid flow through macroscopic helical static elements) [2],... 

To determine the pressure and velocity distributions, we must relate mean cross-sectional flow velocity (v) to the pressure gradient. This is done by means of Eq. 4.11 for capillary tubes and Eq. 4.22 for packed columns. To work with these equations, we must replace the overall pressure gradient Ap/L by the local pressure gradient - dpldx, where the negative sign arises because pressure decreases as one moves along the positive flow coordinate x. With this substitution, Eq. 4.11 can be rearranged to... 

Because in the porous media flow model being used, the effects of viscosity are assumed to be negligible, the velocity will be uniform across the duct, i.e., the velocity will be equal to um everywhere and the cross-stream velocity component, v, will, therefore, be zero everywhere. It will further be assumed that the temperature gradients across the flow, i.e., in the v-direction, will be much greater than those in the z-direction because W < L, L being the length of the duct. With these assumptions, the governing equation, i.e., Eq. (10.34), reduces to ... 

Assuming that the film is thin and that the pressure changes across the film are, therefore, negligible and that the velocity gradients in the cross-film direction (y-direction) are much greater than in the flow direction (x-direction), the flow in the film will be governed by the following set of equations ... 

While the engineer may frequently be interested in the heat-transfer characteristics of flow systems inside tubes or over flat plates, equal importance must be placed on the heat transfer which may be achieved by a cylinder in cross flow, as shown in Fig. 6-7. As would be expected, the boundary-layer development on the cylinder determines the heat-transfer characteristics. As long as the boundary layer remains laminar and well behaved, it is possible to compute the heat transfer by a method similar to the boundary-layer analysis of Chap. 5. It is necessary, however, to include the pressure gradient in the analysis because this influences the boundary-layer velocity profile to an appreciable extent. In fact, it is this pressure gradient which causes a separated-flow region to develop on the back side of the cylinder when the free-stream velocity is sufficiently large. 

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