Stokes equation, calculation

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

Centrifugal Sedimentation Methods These methods extend sedimentation methods into the submicron size range. Sizes are calculated from a modified version of Stokes equation ... 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

Recall that in the continuum case we derived the Navier-Stokes equations (9.16) by allowing for the possibility of having nonzero viscous terms in the form for the momentum tensor appearing in Euler s equation (9.9). While their LG analogs may be derived in essentially the same manner, however, the lack of Galilean invariance tend to make the calculations more involved. We will outline the procedure below. 

The method is strictly accurate for solid, spherical particles of a limited size range and becomes progressively less accurate with increasing particle diameter, with increasing departure from the spherical shape, or with increasing porosity The rate of air flow may be determined directly from charts supplied by the manufacturers or calculated from Stokes equation as follows ... 

When a ball falls (or ascends) in a melt under gravity, the velocity follows the Stokes equation, from which the viscosity can be calculated ... 

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

The more incisive calculation of Springett, et al., (1968) allows the trapped electron wave function to penetrate into the liquid a little, which results in a somewhat modified criterion often quoted as 47r/)y/V02< 0.047 for the stability of the trapped electron. It should be noted that this criterion is also approximate. It predicts correctly the stability of quasi-free electrons in LRGs and the stability of trapped electrons in liquid 3He, 4He, H2, and D2, but not so correctly the stability of delocalized electrons in liquid hydrocarbons (Jortner, 1970). The computed cavity radii are 1.7 nm in 4He at 3 K, 1.1 nm in H2 at 19 K, and 0.75 nm in Ne at 25 K (Davis and Brown, 1975). The calculated cavity radius in liquid He agrees well with the experimental value obtained from mobility measurements using the Stokes equation p = eMriRr], with perfect slip condition, where TJ is liquid viscosity (see Jortner, 1970). Stokes equation is based on fluid dynamics. It predicts the constancy of the product Jit rj, which apparently holds for liquid He but is not expected to be true in general. 

However, one difference exists with classical theory in this latter case, the Navier-Stokes equation (443) and the incompressibility condition (444) are assumed to be valid for all distances rict. In this case, it is an easy matter to calculate explicitly the higher-order terms in Eq. (445), and the boundary condition at the B-particle (assumed to be spherical) imposes the condition... 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

Since in hydrodynamic lubrication the friction force is completely determined by the viscous friction of the lubricant, the coefficient of friction can be calculated from hydrodynamics using the Navier-Stokes equations. This had already been done in 1886 when Reynolds published his classical theory of hydrodynamic lubrication [494], The friction force Fp between two parallel plates of area A separated by the distance d is given by ... 

In contrast to the pseudo 3-D models, tmly multi-dimensional models use, in general, finite element or finite volume CFD (Computational Fluid Dynamics) techniques to solve full 3-D Navier-Stokes equations with appropriate modifications to account for electrochemistry and current distribution. The details of electrochemistry may vary from code to code, but the current density is calculated almost exclusively from Laplace equation for the electric potential (see Equation (5.24)). Inside the electrolyte, the same equation represents the migration of ions (e g. 0= in SOFC), elsewhere it represents the electron/charge transfer. In what follows, we briefly summarize a commonly used multi-dimensional model for PEM fuel cells because of its completeness and of the fact that it also addresses most essential features of SOFC modeling. 

In the microfluid dynamics approaches the continuity and Navier-Stokes equation coupled with methodologies for tracking the disperse/continuous interface are used to describe the droplet formation in quiescent and crossflow continuous conditions. Ohta et al. [54] used a computational fluid dynamics (CFD) approach to analyze the single-droplet-formation process at an orifice under pressure pulse conditions (pulsed sieve-plate column). Abrahamse et al. [55] simulated the process of the droplet break-up in crossflow membrane emulsification using an equal computational fluid dynamics procedure. They calculated the minimum distance between two membrane pores as a function of crossflow velocity and pore size. This minimum distance is important to optimize the space between two pores on the membrane... 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

The proper procedure should be quite the opposite. Rq has to be found by fitting the non-Markovian asymptote (3.59) to the experimental data, provided D has been measured or calculated from the Stokes equation. Only then can the true Markovian rate constant k be found as AkRqD. The pure exponential decay with this rate constant is shown by the dashed line in Figure 3.7 for comparison with the true nonstationary kinetics. 

Particle radii (R) are calculated from the sedimentation times (t) by means of the Stokes equation. For a spin fluid with density and viscosity gradients... 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

Because the Reynolds number is much smaller than 1 and a Newtonian flow behavior is being observed in the first place, the Navier-Stokes equations convert to Stokes equations, and we obtain a system of linear equations for the flow calculations. It therefore follows that there must be a linear relationship both between the flow rate and the pressure and between the flow rate and the power. This is demonstrated in Figs. 8.10 and 8.11 in which the dimensionless conveying and power characteristic are illustrated, respectively. The red lines reflect the Newtonian cases. As expected, a linear relationship is revealed. 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

The exact solution of the convection-diffusion equations is very complicated, since the theoretical treatments involve solving a hydrodynamic problem, i.e., the determination of the solution flow velocity profile by using the continuity equation or -> Navier-Stokes equation. For the calculation of a velocity profile the solution viscosity, densities, rotation rate or stirring rate, as well as the shape of the electrode should be considered. 

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