Drag, form

When we consider many particles settling, the density of the fluid phase effectively becomes the bulk density of the slurry, i.e., the ratio of the total mass of fluid plus solids divided by the total volume. The viscosity of the slurry is considerably higher than that of the fluid alone because of the interference of boundary layers around interacting solid particles and the increase of form drag caused by particles. The viscosity of a slurry is often a function of the rate of shear of its previous history as it affects clustering of particles, and of the shape and roughness of the particles. Each of these factors contributes to a thicker boundary layer. 

There are strict limitations to the application of the analogy between momentum transfer on the one hand, and heat and mass transfer on the other. Firstly, it must be borne in mind that momentum is a vector quantity, whereas heat and mass are scalar quantities. Secondly, the quantitative relations apply only to that part of the momentum transfer which arises from skin friction. If form drag is increased there is little corresponding increase in the rates at which heat transfer and mass transfer will take place. 

Rao et al. (R5) and Raju et al. (R2) also investigated mass transfer at vibrating electrodes for low vibration frequencies (higher frequencies would cause cavitation). Mass transfer follows a laminar-type correlation both for a transverse vibration of a vertical cylinder and for a vertical plate vibrating parallel to the face. In the case of the plate, the Reynolds number is based on width, indicating the predominance of form drag. When vibrations take place perpendicular to the thickness, skin friction predominates and the Reynolds number is then preferably based on the equivalent diameter (total surface area divided by transverse perimeter). 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

Figure 8.5 shows a Venturi meter. The theory is the same as for the orifice meter but a much higher proportion of the pressure drop is recoverable than is the case with orifice meters. The gradual approach to and the gradual exit from the orifice substantially eliminates boundary layer separation. Thus, form drag and eddy formation are reduced to a minimum. 

For the flow of a viscous fluid past the cylinder, the pressure decreases from A to B and from A to C so that the boundary layer is thin and the flow is similar to that obtained with a non-viscous fluid. From B to D and from C to D the pressure is rising and therefore the boundary layer rapidly thickens with the result that it tends to separate from the surface. If separation occurs, eddies are formed in the wake of the cylinder and energy is thereby dissipated and an additional force, known as form drag, is set up. In this way, on the forward surface of the cylinder, the pressure distribution is similar to that obtained with the ideal fluid of zero viscosity, although on the rear surface, the boundary layer is thickening rapidly and pressure variations are very different in the two cases. 

Turbulence may arise either from an increased fluid velocity or from artificial roughening of the forward face of the immersed body. Prandtl roughened the forward face of a sphere by fixing a hoop to it, with the result that the drag was considerably reduced. Further experiments have been carried out in which sand particles have been stuck to the front face, as shown in Figure 3.3. The tendency for separation, and hence the magnitude of the form drag, are also dependent on the shape of the body. 

If form drag were neglected for all velocities less than 0.083 m/s, the distance moved by the particle would be given by ... 

Utilization of Eq. (16) and the sum of the resistance forces over the surface of the sphere (B4) leads to an expression for the form drag... 

The pressure distribution given by Eq. (3-9) is an odd function of 0, so that the particle experiences a net pressure force or form drag. Integration of the pressure over the surface of the particle leads to a drag component given by... 

It is an interesting semantic question whether C 2 should be regarded as a component of form drag or of skin friction. 

Two thirds of this drag arises from skin friction, one third from form drag, and the component due to deviatoric normal stress is zero. The corresponding terminal velocity follows from Eq. (3-15) as ... 

Equations (3-32) and (3-33) differ from the Stokes solution only in the Re terms. The contribution to p is symmetrical about the equator, so that the form drag is the same for the two solutions. 

Form drag and skin friction drag coefficients are obtained from the numerical results by integrating the distributions of surface pressure and vorticity ... 

Predicted and observed wake lengths and wake volumes agree closely for Re = 100 (Figs. 5.7 and 5.8). For Re > 100, the excess pressure over the leading surface of the sphere approaches that for an ideal fluid, but there is little recovery in the wake. As Re increases, the importance of skin friction decreases relative to form drag. 

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