Fore-and-aft symmetry

Since the pressure field depends only on the magnitude of the velocity (see Eq. (1-22)) and since the flow field has fore-and-aft symmetry, the modified pressure field forward from the equator of the sphere is the mirror image of that to the rear. This leads to d Alembert s paradox that the net force acting on the sphere is predicted to be zero. This paradox can only be resolved, and nonzero drag obtained, by accounting for the viscosity of the fluid. For in viscid flow, the surface velocity and pressure follow as... 

A body has a plane of symmetry if the shape is unchanged by reflection in the plane. Orthotropic particles have three mutually perpendicular planes of symmetry. An axisymmetric particle is symmetric with respect to all planes containing its axis, so that it is orthotropic if it has a plane of symmetry normal to the axis, i.e., if it has fore-and-aft symmetry. 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

The internal motion given by Eq. (3-8) is that of Hill s spherical vortex (H6). Streamlines are plotted in Figs. 3.1 and 3.2 for k = 0 and k = 2, and show the fore-and-aft symmetry required by the creeping flow equation. It may also be noted in Fig. 3.2 that the streamlines are not closed for any value of k, the solution predicts that outer fluid is entrained along with the moving sphere. This entrainment, sometimes known as drift, is infinite in creeping flow. This problem is discussed further in Chapter 4. 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

For a particle without fore-and-aft symmetry, condition (ii) is generally met only when the axis is vertical hence such particles fall with a tumbling motion. However, if the particle has fore-and-aft symmetry of shape and density, both F ) and immersed weight must act through the point where the plane of symmetry cuts the axis condition (ii) is automatically satisfied, and the particle falls without rotation. Condition (i) then determines the direction of motion. The angle (p becomes the inclination of the axis from the vertical, so that the... 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

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). 

Taking detectable departure from fore-and-aft symmetry as the upper limit of Stokes flow, Coutanceau found that increasing I increased the range of validity of the creeping flow approximation." The upper limit of Stokes flow was proposed as ... 

The first factor is that the agglomerates are not necessarily spherical in shape. A more general representation would be to assume that they are spheroids in shape with fore and aft symmetry. This case was treated in detail by Manas-Zloczower et al. (97). These particles enter the high shear zone in random orientation, and therefore some may rupture and others will pass without rupturing. The fraction of particles that rupture in a given set of condition can be calculated. 

Homogeneous symmetrical particles can take up any orientation as they settle slowly in a fluid of infinite extent. Spin-free terminal states are attainable in all orientations for ellipsoids of uniform density and bodies of revolution with fore and aft symmetry, but the terminal velocities will depend on their orientation. A set of identical particles will, therefore, have a range of settling velocities according to their orientation. This... 

Figure 4.5 Impuciiun ofa smail spherical p uiiclc on a cylinder placed normal to the How. The particle is unable to follow the fluid sireamline because of Us inertia. The drag on the particle is calculated by assuming that it is located in a unironn flow with velocity at infinity equal to the local fluid velocity (see detail). Fore-and-aft symmetry of the streamline exists for low-Reynolds-number flows and for an in viscid How. but the shapes of the strcamlinc.s differ.

Numerous available numerical solutions the Navier-Stokes equations, as well as experimental data (see a review in [94]), provide a detailed analysis of the flow pattern for increasing Reynolds numbers. For 0.5 < Re < 10, there is no flow separation, although the fore-and-aft symmetry typical of inertia-free Stokes flow past a sphere is more and more distorted. Finally, at Re = 10, flow separation occurs at the rear of the particle. 

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