Streamline thickness

The axial flow compressors in aero gas turbines are heavily loaded. The aspecl ratio of the blades, especially the first few stages, can be as high as 4.0, and the effecl of streamhne curvature is substantial. The streamline configuration is a function of the annular passage area, the camber and thickness distribution of the blade, and the flow angles at the inlet and outlet of the blades. The shafts on these units are supported on antifriction bearings (roller or ball bearings). 

The recommended radius not only reduces the brittleness effect but also provides a streamlined flow path for the plastic melt in the mold cavity. The radiused corner of the metal in the mold reduces the possibility of its breakdown and thus eliminates a potential repair need. Too large a radius is also undesirable because it wastes material, may cause sink marks, and may even contribute to stresses from having excessive variations in thickness. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Liquid is flowing at a volumetric flowrate Q per unit width down a vertical surface. Obtain from dimensional analysis tile form of the relationship between flowrate and film thickness. If the flow is streamline, show that the volumetric flowrate is directly proportional to the density of the liquid. 

For streamline flow it is found that the film thickness is proportional to the one third power of the volumetric-flowrate per unit width. Show that the heat transfer coefficient would be expected to be inversely proportional... 

Calculate the thickness of the boundary layer at a distance of 75 mm from the leading edge of a plane surface over which water is flowing at the rate of 3 m/s. Assume that the flow in the boundary layer is. streamline and that the velocity of the fluid at a distance y from the surface may be represented by the relation u = a + by + ey2 + dyi, where the coefficients a, h, c, and d are independent of v. Take the viscosity of water as 1 mN s/nr. 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

The streamlines of this flow are shown by Peters and Smith (12). In this case, the effective thickness of this layer appears to be about equal to the gap with the wall, indicating a pressure flow about equal to the drag flow. It can be calculated that this would increase the maximum shear rate on the fluid passing under the agitator blade by a factor of seven. 

Different species, belonging to the same sample, form exponential distributions or layers of different thickness I (see Figure 12.5c) the greater the thickness I, the higher the mean elevation above the accumulation wall and the further the penetration into the fast streamlines of the parabolic flow profile. The thickness is inversely proportional to the force exerted on the particle by the field (see Equation 12.8). Usually, this force increases with particle size and this defines the so-called normal mode of elution smaller particles migrate faster and elute earlier than larger particles (see Figure 12.4a). This sequence is referred to as the normal elution order. The above-described equilibrium-Brownian mode will behave as normal mode. However, Brownian, equilibrium, and normal concepts are strictly interrelated. 

The thickness of each successive layer in the fluid is the infinitesimal mathematical increment dr. Since the streamlines roughly follow the outline of the particle and are thin compared to the radius of the sphere R we can think of each flow layer as moving tangentially to the surface of the sphere. 

FIG. 12.1 Streamlines (which also represent the electric field) around spherical particles of radius Rs. The dashed lines are displaced from the surface of the spheres by the double-layer thickness k. In (a) kRs is small in (b) kRs is large. 

Figure 6.13 illustrates the streamline patterns and velocity profiles for two rotation rates. The outer flow for the rotating disk is seen to be quite different from the semi-infinite stagnation-flow situation. In the rotating-disk case, the inviscid flow outside the viscous boundary layer has only uniform axial velocity. In the stagnation flow, the axial velocity varies linearly with the distance from the stagnation surface z and the scaled radial velocity v/r is a constant (cf. Fig. 6.6). The rotating-disk solutions reveal that as the rotation rate increases, the axial velocity increases in the outer flow and the boundary-layer thickness decreases as fi1/2 and f2-1/2, respectively.

Machine direction product nonuniformities always accompany melt fracture, and this is why the phenomenon marks the throughput upper limit to die forming. These nonuniformities can be intense or mild, depending on the die streamlining, but they are always high frequency disturbances in the product thickness. Other causes for machine... 

If the upward motion of the suspension is very slow and the. radial motion of the particles is streamline, the equations governing the behavior of particles in the suspension can be derived by a ipethod due to Hauser and Lynn (1940). Since the particles are distributed over a thickness of i 2 — i i, our problem is somewhat complicated with regard to the initial location of any particle and its absolute path through the concentric ring of suspension. However, while sharp fractionation by size as determined by a measured position in the cylinder (all particles assumed of the same density) cannot be obtained, positions along the... 

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