Vortex kinematics

In order to interpret the local solutions we have just obtained in terms of the liquid distribution in the complete vortex it is required to transform our results back to physical variables. The local length scale may be recovered by inverting Eq. 3.2 and expressing the local strain rate in terms of the vortex kinematics using Eq. 2.10. This gives... 

The power required for a given stirrer type and associated vessel configuration depends on the speed of rotation N, the stirrer diameter du the density p, and the kinematic viscosity v of the medium. In vessels without baffles, the liquid vortex, and therefore the acceleration due to gravity, g, is immaterial, as long as no gas is entrained in the liquid. Thus, P = f(N, dt,p, v), and in the dimensionless form, Ne = /(Re), a relationship generally known as the power characteristics of the stirrer. Here, Ne = P/(pN3df) is the Newton or Power number, and Re s Ndf/v the Reynolds number. This relationship was described in Sections II and III for gas-liquid and gas-liquid-solid systems. 

Accompanying the impeded particle rotation is the (kinematical) existence of an internal spin field 12 within the suspension, which is different from one-half the vorticity to = ( )V x v of the suspension. The disparity to — 2 between the latter two fields serves as a reference-frame invariant pseudovector in the constitutive relation T = ((to — 12), which defines the so-called vortex viscosity ( of the suspension. Expressions for (( ) as a function of the volume of suspended spheres are available (Brenner, 1984) over the entire particle concentration range and are derived from the prior calculations of Zuzovsky et ai (1983) for cubic, spatially-periodic suspension models. 

The vortex depth h on the stirrer shaft depends, for a given stirrer type and for geometric installation conditions D/d, h/d) upon the stirrer diameter d, the stirrer speed n, the liquid height H above the stirrer at rest, the physical properties of the liquid (density p, kinematic viscosity v) and the acceleration due to gravity, g ... 

The left-hand side represents the advection (or convection) of vorticity by the velocity u, and the second term on the right-hand side represents the transport of vorticity by diffusion (with diffusivity = the kinematic viscosity v). These two terms are familiar in the sense that they resemble the convection and diflusion terms appearing in the transport equation for any passive scalar. A counterpart to the second term does not appear in these transport equations, however. Known as the production term, it is associated with the intensification of vorticity that is due to stretching of vortex lines. It is not a true production term, however, because it cannot produce vorticity where none exists. Indeed, because (10-5) contains to linearly in every term, it is clear that vorticity can be neither created nor destroyed in the interior of an isothermal, incompressible fluid It can only be convected, diffused, or changed in magnitude once it is already present.6... 

On the other hand, the flow in the region downstream of the cylinder and in the wake, where vortex formation occurs has only been scetchily investigated. Measurement of the detailed flow structure for this region is extremely difficult since the diameter of the cylinder should be very small (0.1-0.5 mm). Even measurements of Strouhal number, St=fd/V were accomplished in a small number of works [4,8,9], and in very limited range of Reynolds numbers, Re=dV/v, where f is the vortex shedding frequency, d-cylinder, diameter, V-velocity of the undisturbed flow, v-kinematic viscosity. The... 

The kinematics of moving fronts and interfaces has been studied in different physical contexts for over two hundred years. Most notable are the studies of free surfaces in ocean hydrodynamics and vortex sheets in free space (e.g., see Lamb, 1945), and more recently, flame propagation dynamics in combustion analyses. The following derivation, which applies to fluid fronts in porous media, is given in Chin (1993a). Let us consider a moving front or interface located anywhere within a three-dimensional Darcy flow (e.g., any surface marked by red dye), and let (()(x,y,z) denote the porosity. Furthermore, denote by u, v, and w the Eulerian speed components, and describe our interface by the surface locus of points... 

B.3 Striation Thickness from Kinematical Arguments. Rework Example 6.5 for a line oriented at an angle f with respect to axis x. Also, rework the same example for a two-dimensional point vortex flow with = 0 and Vg = olr. 

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