Local angular velocity

The azimuthal component of the fluid velocity, v, is identical to vx and the local fluid angular velocity is co = v /q. This azimuthal velocity is y sin0oo(0) = qoo(0) and the local shear rate, y, is -sin0 which, for no slip boundary conditions, is Q/a for small a. Under uniform shear with no slip, it may be shown that dvx/dy 0 and dvx/dy y[2, 17]. 

Since historically the dissipation is evaluated using the local velocity at the boundary and the shear stress is evaluated as the product of the viscosity and the shear rate at the boundary, it follows that if the velocity is not frame indifferent then the dissipation will not be frame indifferent. As discussed previously in this chapter, rotation of the barrel at the same angular velocity as the screw are the conditions that produce the same theoretical flow rate as the rotating screw. Because the flow rate is the same and the dissipation is different, it follows that the temperature increase for barrel and screw rotation is different. This section will demonstrate this difference from both experimental data and a theoretical analysis. 

The velocity field between the cone and the plate is visualized as that of liquid cones described by 0-constant planes, rotating rigidly about the cone axis with an angular velocity that increases from zero at the stationary plate to 0 at the rotating cone surface (23). The resulting flow is a unidirectional shear flow. Moreover, because of the very small i//0 (about 1°—4°), locally (at fixed r) the flow can be considered to be like a torsional flow between parallel plates (i.e., the liquid cones become disks). Thus... 

Equation 7.2-22 indicates that the separating force is proportional to the local shear stress (fiy) in the liquid, it is a sensitive function of the Euler angles of orientation, and is proportional to the projection of the cross-sectional are (S = nc2). The angular velocities of rotation of the freely suspended spheroid particle were given by Zia, Cox, and Mason (46)... 

We believe that these results confirm that the external field has a deep influence on the relative orientation between the tagged dipole and its neighbors. We guess that a sudden removal of the external field will produce a fast local rearrangement (much faster than the microscopic depolarization) that will in turn excite the angular velocity to. In accordance with our remarks in Section V, we are then led naturally to study the non-Gaussian property of the angular velocity. 

When a particle is put at the origin it rotates following the fluid motion, but generally there exists still a velocity difference between the fluid and the particle. Let us consider the case of a spherical particle of radius a. As we d ussed it acquires a rotational motion of angular velocity qj2, but this motion is not enough to have an equi-velocity distribution, and a local disturbance of the flow appears. The effect is the same if a force is put at the origin, causing the perturbation flow. Clearly, the perturbation is due to the finite size of the sphere, but when an equivalent force is found one can replace the sphere by the force which does not, of course, require volume. 

Figure 13b shows the time evolutions of the three hyper-angles in the same dynamics as in Fig. 13a. It can be seen that the hyper-angles vary rapidly when the system is in the vicinity of the equilibrium structure, while they are almost locked to certain values during the period of collective isomerization motions. The origin of the rapid democratic rotations in the vicinity of the local equilibrium structure lies in the degeneracy among the three gyration radii there. Similarly to Eq. (30), the components of the democratic angular velocity yap (a, p = 1,2,3, a p) are expressed as...

The vorticity vector, w, is a measure of rotational effects, being equal to twice the local rate of rotation (angular velocity) of a fluid particle (i.e., uj = curl(v) = rot(v) = Vx V = 2Q) [168]. Many flows have negligible vorticity, uj curl(v) 0, and are appropriately called irrotational flows. 

Figure 10. The egg model of Stepanov and Shliomis a>, is angular velocity of yolk, to, is angular velocity of eggshell, ft is local angular velocity of the surrounding fluid, IX = magnetic viscosity represented by the viscosity of the white, r is the viscosity of the surrounding fluid, v is the volume of the yolk. V is the hydrodynamic volume.

Problem 2-11. Let Lbc a material volume moving with the fluid and let S be the material surface surrounding V. Assume the fluid is incompressible and inviscid, i.e., no viscosity. As usual, u(x, t) is the velocity field. The quantity [Pg.100]

Unless the contrary is explicitly stated, the following discussion of experimental and theoretical results is restricted to single, rigid, spherical particles freely suspended in a Poiseuille flow within a circular tube of effectively infinite length. Notation is as follows a = sphere radius 7 = tube radius (/ was used previously for this quantity) b = radial distance from tube axis to sphere center p = b/R = fractional distance from axis b = stable equilibrium distance of sphere from tube axis = b jR p = fluid density Pp = particle density p = viscosity v = pfp = kinematic viscosity. All velocities defined below are measured relative to the fixed cylinder walls V = mean velocity of flow vector (equal in magnitude to the superficial velocity and pointing parallel to tube axis in the direction of net flow) U = particle velocity vector—that is, the velocity of the sphere center (o = angular velocity of the sphere. The local velocity in the unperturbed Poiseuille flow is... 

The Reynolds number based upon the angular velocity of the particle is not an independent parameter for a freely supported particle rather according to Eq. (89b) a homogeneous spherical particle will, in the absence of wall and particle inertia effects, rotate with the local angular velocity of the undisturbed flow at its center—at least at sufficiently small Reynolds numbers. This velocity is... 

Use the force method to investigate the stability of a cylinder of A in B when both fluids rotate at an angular velocity co around the axis of the cylinder. Assume that interfacial tension exerts a local restoring force equal to -yd(2H), where H is the local mean curvature. If pg > Pa, how large must co be to overcome the basic capillary instability described in Section 4. 

Some experimental measurements of the local angular velocity have been carried out by Youcefi [9] with a hot film probe in polyacrylamide (PAA) solutions (Figure 12). Because of further experimental difficulties, it remains difficult to directly compare these results with the ones issued from the modeling work of Anne-Archard [14] because it is not possible to calculate in this case the value of the Deborah number. Nevertheless, we can consider these results compatible to the others. 

Given that the rotational inertia and the angular velocity are expressed in the local reference system, a GYS element appears similar to that in (9.14) with the following expression ... 

This way, the angular velocity of the fixed local frame to the body can be expressed as a function of the three angular velocities of the Cardan angles ... 

The second term of the above equation is due to the flow of the Uquid crystal. As shown in Figure 5.15, the local angular velocity, Q, of the director is related to dii jdt by Q.dtx n =dn. Therefore... 

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