Front stagnation point

In region III near the tube center, viscous stresses scale by the tube radius and for small capillary numbers do not significantly distort the bubble shape from a spherical segment. Thus, even though surfactant collects near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension, aQ. Region... 

Two particularly useful equations can be derived by applying the thin concentration boundary layer approximation to steady-state transfer from an axisymmetric particle (L2). The particle and the appropriate boundary layer coordinates are sketched in Fig. 1.1. The x coordinate is parallel to the surface x == 0 at the front stagnation point), while the y coordinate is normal to the surface. The distance from the axis of symmetry to the surface is R. Equation (1-38), subject to the thin boundary layer approximation, then becomes... 

The dimensions of the attached wake are shown in Figs. 5.6, 5.7, and 5.8. The various numerical solutions agree closely with flow visualization results of Taneda (T2), although other workers (K2) report separation slightly closer to the rear. The separation angle, measured in degrees from the front stagnation point, is well approximated by... 

Fig. 5.6 Angle from front stagnation point to separation from the surface of a rigid sphere.

Figure 5.15 shows streamlines and concentration contours calculated by Masliyah and Epstein (M6). Even in creeping flow, Fig. 5.15a, the concentration contours are not symmetrical. The concentration gradient at the surface, and thus Shjoc is largest at the front stagnation point and decreases with polar angle see also Fig. 3.11. The diffusing species is convected downstream forming a region of high concentration at the rear (often referred to as a concentration wake ) which becomes narrower at higher Peclet number.

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

Marchildon et al (M3) related oscillation of a falling cylinder to movement of the front stagnation point, and obtained an expression for the frequency ... 

Previous workers have also made use of potential flow pressure distributions about spheroids, but no allowance was made for lack of fore-and-aft symmetry, while the constant pressure condition was satisfied only near the front stagnation point (SI) or at the equator and poles (H6, Mil). 

Note that the polar angle is again defined as 0 = n at the front stagnation point and 0 = 0 at the back. We thus see that either the interface velocity is zero or that T = 0 and the tangential-stress must be continuous across the interface [see, e g., (7-264)]. This physical picture has been called the spherical-cap model. It is sketched in Fig. 7 20. The problem then is to determine the velocity fields in the two fluids and determine the critical angle 0C. 

Figure 10-8. A sketch showing streaming flow past a circular cylinder. On the right is the flow as seen with a course level of resolution. On the left is the flow in the immediate vicinity of the front-stagnation point, as seen from a much finer resolution. It is evident that the local flow examined in a region close enough to the stagnation point reduces to a classic stagnation point flow as described by the Falkner-Skan equation for, 8 = 1.

Figure 10-9. The dimensionless shear stress as a function of position on the surface of a circular cylinder as calculated with the approximate Blasius series solution. Note that x is measured in radians from the front-stagnation point. The predicted point of boundary-layer separation corresponds to the second zero of du/dY 0. and is predicted to occur just beyond the minimum pressure point atx = jt/2.

The boundary layer equations are valid only in the region between the front stagnation point and the separation point. Behind the separation point there is a wake region with absolutely different hydrodynamic laws. The position of the separation point can be determined either experimentally or by using numerical or approximate analytical methods. 

One can see that the local diffusion flux attains its maximum at the front stagnation point on the surface of the sphere (at 0 = 7r) and monotonically decreases with the angular coordinate to the minimum value, which is equal to zero and is attained at 0 = 0. 

At high surface activity due to the residual surface mobility, the zone of low surface concentration can appear near the front stagnation point. In other words the angle )/ characterising the dimension of the stagnant cup can be slightly less than n and... 

If Re increases beyond 20 the separation ring moves forward so that the attached re-circulating wake widens and lengthens. The separation angle measured in degrees from the front stagnation point is well approximated by 0 = 180-42.5 (ln(Re/20)) at 20 < Re < 400. The steady wake region appears at 20wake instability corresponds to 130 < Re < 400. 

The maximum possible continuous-phase heat transfer coefficient obtainable for nonoscillating drops was suggested by Elzinga and Banchero (E2). Their equation is based on the maximum heat transfer to a solid sphere, calculated in the vicinity of the front stagnation point. Applying it to drops with internal circulation they obtained... 

The motion is assumed to be axisymmetric and a spherical polar coordinate system r, 0 is used. All variables are independent of 0 therefore, and 9 = 0 is taken to represent the front stagnation point which is the first point the oncoming stream meets as it approaches the bubble. The assumption of axisymmetry (i.e. a spherical bubble) is a reasonable one as long as inertial and viscous forces are small relative to surface tension forces. This requires the Weber number, Wq = and the capillary number, Ca = to be small. Here a is a representative value for the surface tension coefficient. Both these conditions are usually met in the applications we are considering here because bubble sizes and rise velocities are small and surface tension is relatively large. 

The variation of the local Nu number along the circumference of a cylinder in cross flow of air (Pr= 0.7) for low and high Reynolds number is shown in Figures 3.2.10 and 3.2.11, respectively. The reason for the local variation of Nu is that the cross flow over a cylinder (and also over other bodies) exhibits complex flow characteristics. The fluid approaching the cylinder at the front stagnation point (angle y = 0) branches out and... 

Figure 3.2.10 Variation of local Nusselt number along the circumference of a cylinder for cross flow of air (Pr=0.7) for low Reynolds numbers Re = ud< /v [Sucker and Brauer, 1976, short-dash line values of Khan, Culham, and Yovanovich (2005) for0 (front stagnation point)

Figure 6.4.13 indicates that the size of the local boundary layer Siocai.heat changes along the circumference of the wire, for example, in the upstream direction we get 280 p-m and at the front stagnation point we have the lowest value of 65 pm. 

A submarine submerged in seawater travels at 10 km hr Estimate the pressure at the front stagnation point when it is 1.5 m below the sea surface. The density of seawater may be taken as constant and equal to 1026 kg m". ... 

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