Concentric cylinders working equations

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

In the case of the thermal-conductivity, there are three main techniques those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates, and in Paris with concentric cylinders, the effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK). 

Other geometries can readily be worked out. As a useful analogy, the concentration c in equation 1 can be viewed as the electrostatic potential around a conductor of potential co. The analog to the local evaporation rate is the electric field, evaluated at the surface of the conductor. By this analogy a fresh set of intuitive ideas can be brought to bear on evaporation problems. For example, it is not surprising that the vapor density around an infinite cylindrical source drops logarithmically with radial distance, and that the evaporation rate varies inversely with the radius of the cylinder. Similarly, the vapor concentration above an infinite sea drops linearly with distance, whereas the evaporation rate is constant everywhere on the surface. 

The common configurations of measuring geometries are capillary, cone-plate, plate-plate, and concentric cylinder. The following is a review of working equations associated with each geometry and its limitations. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

This result follows from the Richardson-Zaki equation. In their original work, Richardson and Zaki (1954) studied batch sedimentation, in particular the settling of coarse solid particles through a liquid in a vertical cylinder with a closed bottom. Richardson and Zaki found that the settling speed uc of the equal-sized particles in the concentrated suspension was related to the terminal settling speed u, of a single particle in a large expanse of liquid by the equation... 

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