Stokes frictional force

Charged compounds undergo the influence of the electric force (Egl) nd the Stokes frictional force (Es) in an electric field. When these two forces (in kg cm s ) are in equilibrium at steady-state conditions, the following equations can be written ... 

This force makes the ion move at velocity v. Since this is equal to the Stokes frictional force, 6ntjr0v, the flux, J = -cv, is given by... 

This equation can also be used approximately for microscopic particles like the molecules and ions mentioned above. We expect that the diffusion coefficient >b of a substance B will be the smaller, the more viscous the medium is in which it migrates. Let us consider a simple example of a rigid, spherical particle with the radius r. The diffusion force [expressed by Eq. (20.1)] is counteracted by the Stokes frictional force [Eq. (20.18)] ... 

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... 

A charged compound in an electric field undergoes the influence of two important forces (Fig. 17.1), namely, the electric force (Fe ) and the frictional force (Fs) approximated by Stokes law. At steady state, these two forces (in kg cm s 2) are in equilibrium ... 

In free solution, the frictional force obeys Stokes law so that... 

In the case of an immobile macromolecule in solution it is possible to estimate a value of the frictional force (F) developed between the solvent and polymer molecules by assuming the macromolecule to consist of a number (n) of solid spherical entities and applying the Stokes formula in a modified form (Eq. 5.8). 

This force is counterbalanced by the frictional force F, which in the case of a spherical ionic species is given by Stokes law ... 

This is different from Stokes s law we discussed in Equation (2.7), which results from the solution of Equation (29b) for flow over spheres (Bird et al. 1960). Equation (2.7) is for the frictional force on a sphere and is also known as the Stokes equation. 

The physical basis for the frictional force is not specified here but it could arise, e.g., from Stokes law friction on the bead. 

The droplet simultaneously experiences a frictional force due to the dynamics of the surrounding fluid that opposes its movement. Under laminar flow conditions, the frictional force is given by Ff = 67tr orv, where ri0 is the shear viscosity of the medium and vthe velocity with which the droplet moves. Under steady-state conditions, the so-called Stokes velocity (v) emerges from the force balance ... 

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 ... 

This relation is restricted, however, to frictionless media. If frictional forces are included, and acceleration is assumed to be damped out, an equally linear relationship between velocity v and force/can be established as in Stokes law, which clearly falls in the above category. 

In the diffusion region the reorientational motion of the molecules is impeded by a frictional force exerted by a medium considered structureless (continuum). For a spherical molecule, the rotational diffusion coefficient, D, is given by the Stokes-Einstein-Debye equation42... 

Modeling a disk by solving the full three-dimensional Navier-Stokes equations is a complicated task. Moreover, it is still not fully understood what is the cause of frictional forces in the disk. Molecular viscosity is by orders of magnitude too small to cause any appreciable accretion. Instead, the most widely accepted view is that instabilities within the disk drive turbulence that increases the effective viscosity of the gas (see Section 3.2.5). A powerful simplification of the problem is (a) to assume a parameterization of the viscosity, the so-called a-viscosity (Shakura Syunyaev 1973) ((3-viscosity in the case of shear instabilities, Richard Zahn 1999) and (b) to split the disk into annuli, each of which constitutes an independent one-dimensional (ID) vertical disk structure problem. This then constitutes a 1+1D model a series of ID vertical models glued together in radial direction. Many models go even one step further in the simplification by considering only the vertically integrated or representative quantities such as the surface density X(r) = p(r, z.)dz... 

The mobility is determined by the equilibrium of the driving electrical force and the frictional force approximated by Stokes law ... 

Description of the hydrodynamics in the cylindrical capillary experimental design is fairly simple. Considering only electrostatic and fluid frictional forces acting upon the suspended particles, apparent particle mobility at a given location r across the diameter of the capillary may be represented by a solution to the Navier-Stokes equation in the scheme of a coordinate system with the origin in the center of the capillary by... 

The coefficient k(h/a) has been introduced by H. Brenner 2[ and represents the modification of the Stokes friction due to the bottom wall located at a distance /( from the particle. If hja — oc, then k h/a) = I and we recover the usual Stokes law for an isolated particle. On the other hand, if the particle comes very close to the plane (h/a — I), then k(h/a) — oc and the particle will never reach the plane, owing to the presence of lubrication forces. The sedimentation velocity follows from Hq. (2) ... 

The polymer is then treated as a string of beads where the frictional force on each bead is described by Stokes law. However, this results in a relationship between [77] and molecular weight, M, of the form [77] AF+Jl, where A is a small fraction. Experimentally, we have seen that [77] AP, where a usually has values between 0.5 and 0.8. Obviously, the assumption that the polymer beads do not perturb the solvent is not a good one. 

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... 

The frictional force caused by the viscous resistance that a moving particle experiences from the fluid in which It moves. For an uncharged sphere, according to Stokes... 

Biological particles. Stokes law applies to spherical particles, which are large in comparison with the molecules that comprise the liquid medium, and are present at a concentration low enough to avoid modification of the liquid viscosity. Most biological particles are not spherical, and Strokes law must be modified to take this into account. One approach to this problem is to consider that the biological particles shapes could be approximated by ellipsoids of revolution , or spheroids with one major and one minor axis. Calculations show that the frictional force over these ellipsoids is greater than that expected for spherical particles of the same volume.3... 

As we have seen in the previous sections, the friction term in the Navier-Stokes equation (3.98) may not be neglected for large Reynolds numbers Re — oo, if we wish to correctly describe the flow close to the wall, and satisfy the no-slip condition. The region in which the friction forces may not be neglected compared to the inertia forces is generally bounded by a very thin zone close to the wall, as... 

In the calculation of the velocity profile of a hydrodynamic, fully developed, laminar flow we will presume the flow to be incompressible and all properties to be constant. The velocity profile of a fully developed, tubular flow is only dependent on the radius r, wx = wx(r) and wy = 0. Therefore the acceleration term gdw /dt in the Navier-Stokes equation (3.59) disappears body forces are not present, so fc = 0. An equilibrium develops between the pressure and friction forces. Balancing the forces on an annular fluid element, Fig. 3.32, gives... 

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

Solvent friction is measured by the Stokes friction coefficient = 6 r)is H- The interparticle forces = — d/dr, U ( rj ) derive from potential interactions of particle i with all other colloidal particles U is the total potential energy. The solvent shear-flow is given by v ° (r) = yyx, and the Gaussian white noise force satisfies (with a,j8 denoting directions)... 

The frictional term x(l) is assumed to be governed by Stokes s Law, which states that the frictional force decelerating a spherical particle of radius a and mass m is... 

Sedimentation or creaming, depending on the relative densities of the oil and water phases, results from the action of gravity on the droplets. Under a gravitational field a spherical droplet will accelerate until it reaches a velocity for which the friction force balances the gravitational force. At this point the particle will move at a constant velocity v predicted by Stokes law ... 

Stokes Law was used to estimate the frictional forces due to the movement of the ion in the solvent. 

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