The Stokes Equation

In 1851, Stokes derived Eq. (4.1) from the model of solid spherical particles falling independently through a homogeneous liquid without Brownian motion, slippage, and wall effects. Slippage is an inconstant rate of fall wall effects refer to axial orientation in the outermost planes of fluid in contact with a surface, and the differential velocity of flow in the outermost and innermost planes of a fluid in a confining tube  

The spheres with radius rt and density ds fall through a solvent with density d and viscosity T)0 at a rate dx/dt g is gravity. F, counterbalancing gravity, is equal to the product of fc and dx/dt, i.e., 

The variables in Eqs. (4.1) and (4.2) are conventionally expressed in cgs13 units. For a spherical geometry (Hiemenz, 1986), 

Equations (4.2) and (4.3) show that molecularly homogeneous solute providing a larger fc settles more slowly than does solute providing smaller fc. Alternatively stated, larger particles settle more slowly than smaller particles with the same density, barring hydration. Other empirical offshoots from the Stokes law were attempted, but complications arose from an initial lack of awareness of the contributions of hydration to particle factors (Mehl el al., 1940). 

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Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

We shall mate further use of the Stokes equation later in this chapter for the... 

Falling ball viscometers are based on Stokes law, which relates the viscosity of a Newtonian fluid to the velocity of the falling sphere. If a sphere is allowed to fall freely through a fluid, it accelerates until the viscous force is exactly the same as the gravitational force. The Stokes equation relating viscosity to the fall of a soHd body through a Hquid may be written as equation 34, where ris the radius of the sphere and d are the density of the sphere and the hquid, respectively g is the gravitational force and p is the velocity of the sphere. 

The speed at which a sphere roUs down a cylindrical tube filled with a fluid or down an angled plate covered with a film of the fluid also gives a measure of viscosity. For the cylindrical tube geometry, equation 35, a generalized form of the Stokes equation is used for any given instmment, where p is the translational velocity of the rolling sphere and k is the instmment constant determined by caUbration with standard fluids. 

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... 

Cunningham correction factor A factor used as a refinement to the Stokes equation for falling particles of small diameter. These tend to slip between the air molecules and, as a result, fall faster. Cup anemometer A device used by meteorologists for the measurement of wind speed. 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

A wide range of particle terminal velocities for various Reynolds numbers have been investigated by Kunii and Levenspiel.43 They suggested that if the particles were assumed to be spherical and operated at low particle Reynolds number (Rep < 0.4), the Stokes equation was found to be acceptable (see Figure 17.4). Therefore, the terminal velocity Ut can be expressed as ... 

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

Qian and Bau [144] have analyzed such electroosmotic flow cells with embedded electrodes on the basis of the Stokes equation with Helmholtz-Smoluchowski boimdary conditions on the channel walls. They considered electrode arrays with a certain periodicity, i.e. after k electrodes the imposed pattern of electric potentials repeats itself An analytic solution of the Stokes equation was obtained in the form of a Eourier series. Specifically, they analyzed the electroosmotic flow patterns with regard to mixing applications. A simple recirculating flow pattern such as the one... 

The problem of following the interface for Newtonian fluids can be described by the Stokes equations,... 

The slip correction factors are important for particles smaller than 1 pm in diameter, which is rarely the case for pharmaceutical aerosols. Slip correction is required for the Stokes equation to remain predictive of particle behavior for these small particles. Therefore, assuming the absence of shape effects for particles in the Stokes regime of flow, Eq. (1) collapses into the following expression ... 

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... 

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. 

This equation, with or without the body-force term pg, is called the Stokes equation ... 

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. 

What is the Navier-Stokes equation What is the physical significance of each of the terms appearing in the Navier-Stokes equation How does the Navier-Stokes equation differ from the Stokes equation Can you use the Navier-Stokes equation for a non-Newtonian fluid ... 

We assume that the flow geometry, as well as the fluid density and viscosity (/x > 0, given) are not affected by the reactions and that the flow is described by the Stokes equations relating the fluid velocity q and fluid pressure p ... 

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 sedimentation of isometric particles, such as cubes and octa-hedra, deviates only slightly from the Stokes equation. Significantly anisometric particles that can be approximated by ellipsoids of revolution are amenable to rigorous sizing by centrifugation if the axial ratios q are known. If the major semi-axis is a and the minor semi-axis is c, the sedimentation velocity dx/dt can be written as... 

Insofar as liquid solvents are concerned, the most important factor governing / is viscosity 77, as the Stokes equation clearly demonstrates. Therefore any systematic effort to increase separation speed requires close attention to viscosity, with an emphasis on finding solvents and conditions for which viscosities are minimal. The reduction of viscosity can be pursued systematically in place of a hit or miss search for low viscosity solvents. The approach below was developed by the author and his colleagues [32] for use in optimizing size-exclusion chromatography, but the conclusions are generally applicable to separations. 

The state of a pure liquid without any viscoelastic film coverage is designated as I when e = 0 and /< = 0. Al this point, the frequency a>0 agrees with Eq. 10 while the damping coefficient a is given by the Stokes equation [64] with a correction similar to the one used for the Kelvin equation ... 

In electrophoresis of colloidal particles, the inertial effect is typically much smaller than the viscous effect for the fluid motion so that the fluid flow is governed by the Stokes equations incorporating an electric body force... 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

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