Stokes resistance

The model for inviscid liquids is equally well applicable to viscous liquids also, provided that the resistance due to viscous drag is included in the analysis. As a first approximation, the viscous drag may be evaluated by a Stokes resistance term, since the bubble is not followed by a wake. Thus we proceed as before, first evaluating the force-balance bubble volume Kr and then the total bubble volume by reference to the detachment stage. 

If all forces are balanced, that is, m dvldt = 0, the particle is not accelerating and moves with a uniform velocity if it moves at all. When the Stokes resistance is equal to zero, the particle velocity with respect to the airstream is zero. 

When the viscosities of the medium and the particle are the same, then the correction factor has a value of %, and the Stokes resistance is... 

To determine thermophoretic velocity, the Stokes resisting force is equated with the thermal force. Then... 

Here Eq. (21.10) is the usual Navier-Stokes equation. The term yu on the left-hand side of Eq. (21.11) represents the frictional forces exerted on the liquid flow by the polymer segments in the polyelectrolyte layer, and y is the frictional coefficient. If it is assumed that each resistance center corresponds to a polymer segment, which in turn is regarded as a sphere of radius and the polymer segments are distributed at a uniform volume density of in the polyelectrolyte layer, then each polymer segment exerts the Stokes resistance 6nriapU on the liquid flow in the polyelectrolyte layer so that... 

G5c. Goldman, A. J., Cox, R. G., and Brenner, H., The Stokes resistance of an arbitrary particle—Part VI Terminal motion of a settling particle (to be published) see also Goldman, A. J., Investigations in low Reynolds number fluid-particle dynamics. Ph.D. Dissertation, New York University, New York, 1966. 

The term (-ur]lk) in Eq. 3.23 is the Darcy resistance term, and the term (rjW u) is the viscous resistance term the driving force is still considered to be the pressure gradient. When the permeability k is low, the Darcy resistance dominates the Navier-Stokes resistance, andEq. 3.23 reduces to Darcy s law. Therefore, the Brinkman equation has the advantage of considering both viscous drag along the walls and Darcy effects within the porous medium itself. In addition, because Brinkman s equation has second-order derivatives of u, it can satisfy no-slip conditions at solid surfaces bounding the porous material (e.g. the walls of a packed bed reactor), whereas Darcy s law cannot. In that sense, Brinkman s equation is more exact than Darcy s law. 

A force of viscous resistance is proportional to the stationary-stage velocity Vj according to Stokes law ... 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

From the standpoint of collector design and performance, the most important size-related property of a dust particfe is its dynamic behavior. Particles larger than 100 [Lm are readily collectible by simple inertial or gravitational methods. For particles under 100 Im, the range of principal difficulty in dust collection, the resistance to motion in a gas is viscous (see Sec. 6, Thud and Particle Mechanics ), and for such particles, the most useful size specification is commonly the Stokes settling diameter, which is the diameter of the spherical particle of the same density that has the same terminal velocity in viscous flow as the particle in question. It is yet more convenient in many circumstances to use the aerodynamic diameter, which is the diameter of the particle of unit density (1 g/cm ) that has the same terminal settling velocity. Use of the aerodynamic diameter permits direct comparisons of the dynamic behavior of particles that are actually of different sizes, shapes, and densities [Raabe, J. Air Pollut. Control As.soc., 26, 856 (1976)]. 

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... 

Viscosity is a measurement of resistance to flow. Although the unit of absolute viscosity is poise, its measurement is difficult. Instead, kinematic (flowing) viscosity is determined by measuring the time for a given flow through a capillary tube of specific diameter and length. The unit of kinematic viscosity is the stoke. However, in general practice, centistoke is used. Poise is related to stoke by the equation ... 

In this equation, % is a proportionality factor known as the bead-solvent friction coefficient which purports to account in some kind of average way for the complex molecular interactions as the polymer segments (schematized by the bead) move about in the solvent. Following Stokes law of drag resistance, this friction coefficient is usually given as = 67trisa, with a equal to the bead radius. 

The fluid resistance force acting on the droplet should be taken as that given by Stokes law, that is 3ntidu where /< is the viscosity of the continuous phase, velocity relative to the continuous phase. 

TTiere are several particle sizing methods, all based upon sedimentation and Stokes Law. If a particle is suspended in a fluid (which may be gas, or any liquid), the force of resistance to movement by the particle will be proportional to the particle s velocity, v, and its radius, r, vis-... 

If the electric field E is applied to a system of colloidal particles in a closed cuvette where no streaming of the liquid can occur, the particles will move with velocity v. This phenomenon is termed electrophoresis. The force acting on a spherical colloidal particle with radius r in the electric field E is 4jrerE02 (for simplicity, the potential in the diffuse electric layer is identified with the electrokinetic potential). The resistance of the medium is given by the Stokes equation (2.6.2) and equals 6jtr]r. At a steady state of motion these two forces are equal and, to a first approximation, the electrophoretic mobility v/E is... 

It may be assumed that the resistance force may be calculated from Stokes Law and is equal to 3np,du, where u is the velocity of the particle relative to the liquid. 

According to Stokes Law, the resistance force F acting on a particle of diameter d, settling at a velocity u in a fluid of viscosity p is given by ... 

The velocity times the Stoke s friction factor gives the viscous resistive force ... 

The more common procedure is to perform a linear extrapolation of resistance vs. f1 or/-1/2, the value at infinite frequency being taken as the true resistance. For Nal solutions in acetonitrile it has been found that f l plots yield straight lines at low concentrations, but/-1/2 plots must be used to achieve linearity at higher concentrations 28>. Robinson and Stokes 2> discuss the causes of these variations. 

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