Kinematics of fluid flow

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. 

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. 

For a fluid, the axisymmetric kinematics of the flow (equation [7.24]) leads to a simple expression of the strain rate tensor (Table 1.2 of Chapter 1) ... 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Kinematic similarity exists in two geometrically similar units when the velocities at corresponding points have a constant ratio. Also, the paths of fluid motion (flow patterns) must be alike. 

In addition, the effects of pulsatile flow cannot be ignored. One measure of the impact of oscillary flow is the Wcmersley parameter (a) a= h/2tt f/v where r is the tube radius, f the frequency of oscillation and v is the kinematic viscosity of the fluid (Wcmersley, 1955). The degree of departure from parabolic flow increases with and frequency effects may become important in straight tubes when a > 1 (Ultman, 1985). For conditions of these experiments, a exceeds one to beyond the third generation. 

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

Relaxing the restriction of low Reynolds number, Rimmer (1968,1969) used a matched asymptotic expansion technique to develop a solution in terms of Pe and the Schmidt number Sc (or Prandtl number Pr for heat transfer), where Sc = v/D.j and Pr = v/a in which v is the kinematic viscosity of the flowing fluid. His solution, valid for Pe < 1 and Sc = 0(1), is... 

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. 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

Both the normal and tangential components of stress must balance, and the kinematics of the surface and flow field must be consistent along a melt-fluid interface. For the meniscus shown in Figure 6 and described by the normal and tangent vectors (n, t), these conditions dictate that... 

The methods described in this book are primarily concerned with the measurement of the microstructure of complex fluids subject to the application of external, orienting fields. In the case of flow, it is also of interest to measure the kinematics of the fluid motion. This chapter describes two experimental techniques that can be used for this purpose laser Doppler velocimetry for the measurement of fluid velocities, and dynamic light scattering (or photon correlation spectroscopy) for the determination of velocity gradients. 

---