Stokes viscous force

Stokes viscous force — The Stokes viscous force relevant to electrochemistry is the force acting on a sphere in the medium with viscosity, q. When the sphere in... 

One can therefore define the absolute mobility u bs for diffusing particles by dividing the drift velocity by either the diffusional driving force or the equal and opposite Stokes viscous force... 

This relation shows that, owing to the Stokes viscous force, the conventional mobility of an ion depends on the charge and radius of the solvated ion and the viscosity of the medium. The mobility given by Eq. (4.183) is often called the Stokes mobility. It will be seen later that the Stokes mobility is a highly simplified expression for mobility, and ion-ion interaction effects introduce a concentration dependence that is not seen inEq. (4.183). 

Hitherto, the drift velocity has been related to macroscopic forces (e.g., the Stokes viscous force F = 6nrrf or the electric force F = zc X) through the relation... 

The ionic atmosphere is accelerated by the externally applied electric force Z6qX but is retarded by a Stokes viscous force. When the cloud attains a steady-state electrophoretic velocity v, then the viscous force is exactly equal and opposite to the electric force driving the cloud... 

The general formula for Stokes viscous force is 6nnjv, where r and v are the radius and velocity of the moving sphere. In computing the viscous force on the cloud. 

The starting point to elucidate the way the proton moves in solution is to consider its movement through the solvent at a steady state—constant velocity—and at a concentration so low that there is no interionic interaction (zeroth approximation). This occurs when the electric driving force ze X balances the Stokes viscous force, 6nrr v. Thus, the Stokes mobility is... 

However, if the concentration gradient of ions can be balanced by the electric field, it can also be balanced by some other force, such as the Stokes viscous force Fs = 6 Trrrjv, where r is a radius of an ion, r] is the viscosity of the solution, and v is a velocity of an ion. This consideration shows the relationship between the diffusion coefficient and the viscosity of the medium ... 

Stokes viscous force — The Stokes viscous force relevant to electrochemistry is the force acting on a sphere in the medium with viscosity, tj. When the sphere in radius a moves at the constant speed v, the force is given by Gnrjav. This equation is valid for lavpjt] < 0.1 for the density p, of the medium [i]. The force is often supplied by diffusion and an electric field. 

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 Reynolds number Re = vl/v, where v and l are the characteristic velocity and length for the problem, respectively, gauges the relative importance of inertial and viscous forces in the system. Insight into the nature of the Reynolds number for a spherical particle with radius l in a flow with velocity v may be obtained by expressing it in terms of the Stokes time, t5 = i/v, and the kinematic time, xv = l2/v. We have Re = xv/xs. The Stokes time measures the time it takes a particle to move a distance equal to its radius while the kinematic time measures the time it takes momentum to diffuse over... 

In 1851, Stokes (well known for his pioneering work on luminescence see Chapter 1) showed that the relation linking the force exerted by a fluid on a sphere to the viscosity tj of the medium is F = Gitt/rv, where r is the radius of the sphere and v its constant velocity. In this relation, the quantity 6ra/r appears as a friction coefficient, i.e. the ratio of the viscous force to the velocity. 

Most drop situations in extraction are far above the upper limit of application of the preceding equations. A drop moving through a liquid at a velocity such that the viscous forces could be termed negligible can not exist. It will break up into two or more smaller droplets (HIO, K5). Most real situations involve both viscous and inertial terms, and the Navier-Stokes equations can not then be solved. 

When the particle attains a steady-state velocity, the electric and viscous forces are exactly equal and by utilizing Stokes law with tc as the radius,... 

Viscous Forces In the momentum equation (Navier-Stokes equation), forces F acting on the system result from viscous stresses. It is necessary to relate these stresses to the velocity field and the fluid s viscosity. This relationship follows from the stress and strain-rate tensors, using Stokes postulates. 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

If the ion is in a fluid medium, then the electrostatic force seeking to accelerate the ion is opposed by a viscous force trying to slow the ion down. Though it strictly applies only to spheres of macroscopic dimensions, Stoke s law can provide an approximate expression... 

Settling of particles less than 0.5 pm is slowed by Brownian motion (random motion of small particles from thermal effects) in the water. Conversely, large sand-sized particles are not affected by viscous forces and typically generate a frontal pressure or wake as they sink. Thus, Stokes law can only apply to particles with Reynolds numbers (Re) that are less than unity. The particle Reynolds number according to Allen (1985) is defined as follows ... 

These are the Navier-Stokes equations for steady constant fluid property flow. They are sometimes termed the x-, y-, and z-momentum equations, respectively. Basically these equations state that die net rate of change of momentum per unit mass in any direction (the left-hand side) is the sum of the net pressure force and the net viscous force in that direction. 

The most important relationships used are - Fick s laws and - Einsteins equation for diffusion, Newtons viscosity law and Stokes s law (- Stokes s viscous force)... 

The driving force F in Fl-FFF is the viscous force exerted on a particle by the cross-flow stream. Application of Stokes law gives for F [17] ... 

The Walden rule is interpreted in the same manner as the Stokes-Einstein relation. In each case it is supposed that the force impeding the motion of ions in the liquid is a viscous force due to the solvent through which the ions move. It is most appropriate for the case of large ions moving in a solvent of small molecules. However, we will see here that just as the Stokes-Einstein equation applies rather well to most pure nonviscous liquids [30], so does the Walden rule apply, rather well, to pure ionic liquids [15]. When the units for fluidity are chosen to be reciprocal poise and those for equivalent conductivity are Smol cm, this plot has the particularly simple form shown in Figure 2.6. 

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