Boundary layer thickness stagnation point

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

Since (he boundary layer thickness vanishes at the stagnation point (i.e., or = 0), the preceding equation is integrated to produce the following generalized expression for 8 ... 

Note that the boundary layer thickness continues to grow in the stagnation point region (x = 0.82). Previous analytic solutions assumed that the boundary layer thickness was zero at the moving stagnation point. Figure 16 shows that this is clearly incorrect. 

Figure 8.19 (a) Photograph of formation of a hydraulic jump upon liquid jet impingement on a flat surface. The schematic below indicates the stagnation point and the hydraulic jump radius in polar coordinates, (b) Estimated non-dimensional hydraulic jump radius as a function of impact pressure and tUt angle for the experimental conditions studied. Note that the values are calculated for 9 = n. (c) Estimated liquid film velocity just before the hydraulic jump, (d) Estimated boundary layer thickness as a function of jet impact pressure and substrate tUt angle. 

Hydrodynamic theory shows that the thickness, 8, of the boundary layer is not constant but increases with increasing distance y from the flow s stagnation point at the surface (Fig. 4.4) it also depends on the flow velocity ... 

In addition, a concentration boundary layer develops over the surface of the cylinder with its thinnest portion near the forward stagnation point. When the thickness of the concentration boundary layer is much less than the radius of the cylinder, the equation of convective diffusion simplifies to the familiar form for rectangular coordinates (Schlichling, 1979, Chapter XII) ... 

The thickness of the boundary layer at 90 from the forward stagnation point is given approximately by... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

A) Stages of the dip coating process (a-e) batch (f) continuous. (B) Detail of the liquid flow patterns in area 3 of the continuous process. U is the withdrawal speed, S is the stagnation point, S is the boundary layer, and h is the thickness of the fluid film. From Scriven [11. 

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