Relative angular velocity

We can identify four pairs of thermodynamic forces and fluxes, the symmetric traceless strain rate (Vu) and the symmetric traceless pressure tensor, the director angular velocity relative to the background, (l/2)Vxu-I2 and the torque density X, the streaming angular velocity relative to the background (l/2)Vxu- and the torque density and the trace of the strain rate V-u and difference between the trace of the pressure tensor and the equilibrium... 

Here Wj are components of angular velocity relative to coordinate axes, are components of rotational inertia tensor. The dimension of Sly is cube of length, therefore the tensor is interpreted as equivalent volume. Note, that the relation (8.14) can be represented similar to (8.5), namely, as... 

Here is a generalized (6N dimensional) force-torque vector, U -u (6N dimensional) is the particle translational-angular velocity relative to the bulk fluid flow evaluated at the particle centre, (3x3 matrix) is the traceless symmetric rate of the strain tensor (supposed to be constant in space). The resistance matrices Rfu (6N x6N) and Rfe (6N x 3 x 3) which depend only on the instantaneous relative particle configurations (position and orientation) relate the force-torque exerted by the suspending fluid on the particles to their motion relative to the fluid and to the imposed shear flow, respectively. Note that in ER (MR) fluids torques can be neglected. 

A being the familiar rate of strain tensor or matrix, and o the angular velocity relative to the background rotation of the continuum. While it is possible to derive the viscous stress from the above assumption [46], we opt here for a more direct approach, and replace (Eq. 91) by... 

Invmiance to supoposed rigid body motions requires instead that T is a function of n, A and co, who-e A is the familiar rate of strain matrix and o> is the angular velocity relative to the background rotation of die continuum dius... 

Fig. 27. Concentric cylinder viscometer. R and R are the radii of the inner and outer cylinder, respectively, and Q is the relative angular velocity.

The relationship between viscosity, angular velocity, and torque for a Newtonian fluid in a concentric cylinder viscometer is given by the Margules equation (eq. 26) (21,146), where M is the torque on the inner cylinder, h the length of the inner cylinder, Q the relative angular velocity of the cylinder in radians per second, T the radius of the inner cylinder wall, the radius of the outer cylinder wall, and an instmment constant. 

The equation of motion as given in terms of angular momentum can be transformed into other forms that are more convenient to understanding some of the basic design components. To understand the flow in a turbomachine, the concepts of aboslute and relative velocity must be grasped. Absolute velocity (V) is gas velocity with respect to a stationary coordinate system. Relative velocity (IV) is the velocity relative to the rotor. In turbomachinery. 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

Fortunately, the earth rotates with a relatively small angular velocity, when the force of attraction plays the dominant role. It is interesting to raise the following... 

As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an influence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth s surface. With this purpose in mind let us choose a non-inertial frame of reference, shown in Fig. 3.5a its origin 0 is located near the earth s surface and it rotates together with the earth with angular velocity a>. The unit vectors i, j, and k of this system are fixed relative to the earth and directed as follows i is horizontal, that is, tangential to the earth s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth s axis, drawn from south to north, I is the unit vector along OiO, and K is a unit vector parallel to SN. 

Equations (3.75 and 3.76) describe the motion of a free particle in a rotating frame of reference, when the angular velocity is relatively small and the z-axis is directed along the plumb line. Certainly, a presence of terms with a> is related to the rotation of the earth. Besides, the gravitational field also contains a term with this frequency. It may be proper to emphasize again that in deriving these formulas we assumed... 

More modern systems (diffractometers) follow the same principles but the diffracted X-rays are detected with a solid state detector, as described earlier. Typically, the X-ray source is static and the sample and detector are rotated, with the detector moving at twice the angular velocity of the sample to maintain the equivalent angle. Such instruments typically make use of relatively large samples compressed into the window of a 35 mm sample holder. However, where the sample size is restricted, as is common with archaeological applications, a smaller sample (a few mg) can be attached to a silica wafer. In all cases the sample needs to be hnely ground to ensure a uniform diffracted beam. 

---