Dissipation per unit volume

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

What about the micro-scale phenomena These are dependent primarily on the energy dissipation per unit volume, although one must also be concerned about the energy spec tra. In general, the energy dissipation per unit volume around the impeller is approximately 100 times higher than in the rest of the tank. Tnis results in an rms velocity fluc tuation ratio to the average velocity on the order of 10 I between the impeller zone and the rest of the tank. 

The energy dissipation per unit volume to fracture a network consisting of a bond density Nev = bonds per unit volume is... 

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

Of course, the energy dissipated per unit volume is not uniform and the nonuniformity depends on the details of the stirring device. Nonetheless, in what follows we will consider, for the sake of simplicity, that the energy dissipated per unit volume is uniform. The following problems are of interest ... 

The motion of the liquid is in this case a result of the dissipation of the mechanical energy supplied by the stirring. The energy dissipated per unit volume is assumed to be uniform and related to the mechanical energy P... 

As in Section IV,N, the state of motion near the interface can be characterized by the energy dissipated per unit volume, the viscosity t], the surface tension [Pg.92]

Since the force under consideration measures viscous resistance to flow, the quantity dE/dVmeasures the energy dissipated per unit volume. 

Einstein was able to derive another equation describing the rate of energy dissipation per unit volume for a dispersion of spheres ... 

Notice that the left-hand side is now energy dissipation per unit volume. Comparing this result with Equation (3) shows that the increment in viscosity caused by the free-draining chain is given by the coefficient of (dv/dy)2 and may be written... 

The total power (energy/time) dissipated per unit volume of fluid in this system is the sum of the power dissipated due to fluid viscosity and the power dissipated by friction between the pore fluid the counterions pinned in the fluid7. For two-dimensional flow the power dissipated due to fluid viscosity is given by ([5]),... 

With the shear rate at hand, we can calculate the local viscous dissipation per unit volume. From Table 2.3 we note that the only nonvanishing shear-stress component is zyz — which is given by... 

From Eq. E2.5-17 we can calculate the total viscous dissipation between the parallel plates. The second invariant of the rate of strain tensor multiplied by the viscosity gives the viscous dissipation per unit volume. From Table 2.3 we find that, for the case at hand, the second invariant reduces to y2z therefore, the total viscous energy dissipation (VED) between the plates will be given by... 

What about the microscale phenomena These are dependent primarily on the energy dissipation per unit volume, although they must also be concerned about the... 

The temperature rise caused by internal damping may be estimated in the following manner. The energy dissipated per unit volume per unit time, q, is approximately given by... 

IfP = Ta is the local rate of energy dissipation (per unit volume), then, obviously. 

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