Fluid flow, kinematics

The unit of viscosity, the poise, is defined as the force in dynes cm-2 required to maintain a relative velocity of 1 cm/sec between two parallel planes 1 cm apart. The unit commonly used for milk is the centi-poise (10 2 poise). A useful quantity in fluid flow calculations is the kinematic viscosity, or viscosity divided by density. 

Ya.B. assumes that 1) a fast dynamo based on this mechanism can be generated by a time-periodic three-dimensional fluid flow, but that 2) a fast kinematic dynamo (in three-dimensional space with steady flow and periodic boundary conditions) is impossible. 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

Assuming that the flow kinematics of CEF and Newtonian fluids are identical, the velocity profile in steady torsional disk flow is... 

Thus, we require a fluid with a kinematic viscosity (p/p) one tenth that of air at atmospheric pressure for our experiments. Water at moderate temperatures should be suitable. See Chapter 3, Concepts of Fluid Flow for information on kinematic viscosity. 

This dispersion number, (D/uL), for fluid flow in a cylinder can be obtained from a chemical engineering correlation by Levenspiel [9] noting that the intensity of dispersion D/udf, (where df is the diameter of the cylinder) is plotted as a function of Reynolds s number Re = up JlXg, pg is the gas density and p.g is the gas viscosity. (Please note that the Reynolds s number of the flow is altered by the presence of particles. Particles increase the gas density and reduce the effective kinematic viscosity. The net result is to accentuate turbulence and... 

In fluid flow and heat transfer studies, the ratio of dynamic viscosity to density appears frequently. For convenience, this ratio is given the name kinematic viscosity v and is expressed as v = filp. Two common units of kinematic viscosity are m /s and stoke (1 stoke = 1 cm /s = 0.0001 mVs). 

Nusselt provided a simple model for laminar liquid flow down an inclined plane. This assumed that the liquid had reached fully developed conditions in which drag due to viscous shear exactly balanced the weight of the film. Under these conditions, Nusselt showed that for a Newtonian fluid of kinematic viscosity, v, film thickness, /, could be written in terms of the liquid flow rate, Q, moving over a vertically inclined surface of width, w, under a gravitational acceleration, g, using the following relationship ... 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

Kinematic similarity is the similarity of fluid flow behavior in terms of time within the similar geometries. Kinematic similarity requires that the motion of fluids of both the scale model and prototype undergo similar rate of change (velocity, acceleration, etc.). This similarity criterion ensures that streamlines in both the scale model and prototype are geometrically similar and spatial distributions of velocity are also similar. 

The presence of droplets also introduces new kinematic and dynamic boundary conditions on the fluid flow. Since the immiscible fluids cannot cross the interface, boundary condition states that the local normal component of the velocities in each fluid must be equal to the interface velocity, the velocity tangents to the interface must be also equal inside and outside the droplet, and the tangential shear stresses must be balanced at the interface when it is clean of surfactants. 

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