Arbitrary Accelerated Motion

Apart from the specific classes of motion discussed above, understanding of unsteady fluid-particle interaction is not well advanced. Torobin and Gauvin 

The reason for writing the added mass term in this form is not clearly explained. 

The second general approach is based on extension of the creeping flow result, as in earlier sections. Corrsin and Lumley s modification (Cll) of the equation proposed by Tchen (Tl) allows Eq. (11-43) to be generalized as 

The pressure gradient term has been extended to its full form from the Navier-Stokes equation. Equation (11-70) has been discussed by Corrsin and Lumley (Cll), Hinze (H5), and Soo (S7). It is applicable only if a particle is small compared to the scale of velocity variations in the fluid (L8), i.e., if 

Effects such as lift due to particle rotation or fluid velocity gradients can readily be included in Eq. (11-70) if appropriate. The resulting equation of motion is 

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Remark 3.5. A mechanical system of rigid parts in accelerated motion will be in equilibrium if the virtual work of the actual impressed and inertia loads for arbitrary admissible virtual displacements vanishes. 

In view of the differential form of the equation of motion, Eq. (2-31), and the fact that (7 ) is arbitrary, we see that the angular acceleration principle, (2-33), requires that... 

For motions of a single fluid involving sohd boundaries, we have already noted that the no-slip and kinematic boundary conditions are sufficient to determine completely a solution of the equations of motion, provided the motion of the boundaries is specified. In problems involving two fluids separated by an interface, however, these conditions are not sufficient because they provide relationships only between the velocity components in the fluids and the interface shape, all of which are unknowns. The additional conditions necessary to completely determine the velocity fields and the interface shape come from a force equilibrium condition on the interface. In particular, because the interface is viewed as a surface of zero thickness, the volume associated with any arbitrary segment of the interface is zero, and the sum of all forces acting on this interface segment must be identically zero (to avoid infinite acceleration). 

In the steady, unidirectional flow problems considered in this section, the acceleration of a fluid element is identically equal to zero. Both the time derivative du/dt and the nonlinear inertial terms are zero so that Du/Dt = 0. This means that the equation of motion reduces locally to a simple balance between forces associated with the pressure gradient and viscous forces due to the velocity gradient. Because this simple force balance holds at every point in the fluid, it must also hold for the fluid system as a whole. To illustrate this, we use the Poiseuille flow solution. Let us consider the forces acting on a body of fluid in an arbitrary section of the tube, between z = 0, say, and a downstream point z = L, as illustrated in Fig. 3-4. At the walls of the tube, the only nonzero shear-stress component is xrz. The normal-stress components at the walls are all just equal to the pressure and produce no net contribution to the overall forces that act on the body of fluid that we consider here. The viscous shear stress at the walls is evaluated by use of (3 44),... 

Fig. 2.4. Analysis of acceleration of synchronous planetary coil-CCC from Schemes I and IV, Fig. 2.3. a Diagram of synchronous planetary motion b coordinate system for analysis c orbits of arbitrary points d distribution of centrifugal force vectors (170)...

For a sinusoidal motion of arbitrary frequency / = ojI2ti, the modulus of this complex function represents the amplitude relation between the mass relative motion z t) and the ground acceleration u t), and its phase is the relative phase between them. Both are plotted in Fig. 3 as functions of frequency. The amplitude response is flat for acceleration up to the comer frequency/o and decays at higher frequencies as/ The frequencies of interest are usually in the flat zone, so the instmmental correction for amplitude is simply a constant factor. Observe, nevertheless, that the phase response deviates from flatness even for frequencies well under the natural frequency. This response function is formally like the response of a second-order low-pass filter with cutoff frequency/o = (0(J2n. 

A body in which the distance between two arbitrary points remains at constant temperature, unchanged by any motions or interactions, is referred to as an ideally rigid body (IRB). During the translational motion of the IRB, any segment inside of it remains parallel to itself at any time. With such motion the displacement, velocity and acceleration of any point of the IRB are the same at any time. Therefore, many characteristics of the IRB s translational movement can be described by the motion of a single body s point with a mass eqnal to the mass of the whole body moving with velocity (acceleration) in any point of the body. The best point to choose is the centre of mass (CM) (see below). 

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