Particle Viscous force

The behavior of colloidal suspensions is controlled by iaterparticle forces, the range of which rarely extends more than a particle diameter (see Colloids). Consequentiy suspensions tend to behave like viscous Hquids except at very high particle concentrations when the particles are forced iato close proximity. Because many coating solutions consist of complex mixtures of polymer and coUoidal material, a thorough characterization of the bulk rheology requires a number of different measurements. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

This corresponds to a Hamiltonian system which is characterized by a weak oscillatory perturbation of the SHV streamfunction T r, ) —> Tfr, Q + HP, (r, ( ) x sin(fEt). The equations of fluid motion (4.4.4) are used to compute the inertial and viscous forces on particles placed in the flow. Newton s law of motion is then... 

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... 

Archimedes number /VAr a, PtfifApd3 A/Ar - Pf = fluid density Ap = solid density — fluid density (Buoyant x inertial)/ (viscous) forces Settling particles, fluidization... 

In this form, u2JgL, the Froude number, can be viewed as a ratio of inertial to gravity forces Ps jPf is a ratio of particle to fluid inertial forces A UodpjP is the Reynolds number or ratio of particle inertial to fluid viscous forces and Pf u0 L[ p, a Reynolds number based on the bed dimensions and fluid density, is a ratio of fluid inertial to viscous forces. 

It is important to note that the viscous force is singular in the separation distance h and hence will predict infinite large forces at contact. Since this is physically impossible, a certain surface roughness has to be assigned to the granular surface as is shown in Fig. 21 where this parameter is assigned the value ha this prevents particles from touching. ... 

Consider a circulating spherical bubble (k 1, y 1) for which Re 1, and compare this with a rigid sphere at Re 1. For the latter case, the boundary layer is perceived as a thin layer at the particle surface where viscous forces... 

Equation (7-30) gives the natural frequency of the fundamental mode for stationary fluid particles undergoing small oscillations with viscous forces neglected. It has been modified to account for viscous effects (L4, MIO, SIO), surface impurities (MIO), finite amplitudes (S5, Yl), and translation (SIO). Observed oscillation frequencies are generally less than those given by Eq. (7-30), typically by 10-20% for drops in free motion in impure systems (S4) and by 20-40% for pure systems (El, E3, W8, Yl). The amplitude tends to be larger for pure systems (E3) and this explains the reduction in frequency. 

In the presence of fluids, i.e. in suspensions, and also in the polymer melt during homogenization, further forces act between the particles. Adam and Edmondson [22] specify two attractive forces, depending on the extent of wetting of the particles. In the case of complete wetting, viscous force (Fy) acts between the particles, which are separated from each other at a constant rate. Fy can be expressed as ... 

The fu st term is a modified Archimedes number, while the second one is the Froude number based on particle size. Alternatively, the first term can be substituted by the Reynolds number. To attain complete similar behavior between a hot bed and a model at ambient conditions, the value of each nondimensional parameter must be the same for the two beds. When all the independent nondimensional parameters are set, the dependent parameters of the bed are fixed. The dependent parameters include the fluid and particle velocities throughout the bed, pressure distribution, voidage distribution of the bed, and the bubble size and distribution (Glicksman, 1984). In the region of low Reynolds number, where viscous forces dominate over inertial forces, the ratio of gas-to-solid density does not need to be matched, except for beds operating near the slugging regime. 

The solid phase must be large enough to have at its interface a double layer. Then, equating the viscous force created by the movement of the particle to the electrical force due to the interaction of the field with the charge on the particle, one obtains... 

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,... 

As the net velocity of the particle is increased, the viscous force Fv opposing its motion also increases. Soon this force, shown in Figure 2.2b, equals the net driving force responsible for the motion. Once the forces acting on the particle balance, the particle experiences no further acceleration and a stationary state velocity is reached. It may be shown that, under stationary state conditions and for small velocities, the force of resistance is proportional to the stationary state velocity v ... 

The product of the viscous force Fv exerted by the particle on the fluid and the velocity vz of the fluid also equals the rate at which work is done on the fluid by the particle. Remember, a force times a distance equals energy, so a force times a velocity equals the rate of energy dissipation. Thus, we have two different ways of expressing the rate of energy dissipation associated with the presence of a spherical particle in the flowing liquid. Equating these two expressions gives... 

For small, stationary-state velocities, the viscous force on a particle is directly proportional to the velocity. 

In these equations v is the relative velocity between the particle and the surrounding medium. The difference between Equations (28) and (29) therefore equals the net viscous force on the volume element ... 

Archimedes number Ar gL3(pp-p)p P2 inertial forces x buoyancy forces (viscous forces)2 Particle settling... 

The term fa) is the volume-averaged dissipation function for the energy dissipated by the viscous force, which is irreversible dissipation of mechanical work into thermal energy or heat. For the solid-particle phase, the kinetic energy loss by attrition or inelastic collision may be included in this term. 

Impaction Collision of particles on an obstacle as a result of the action of inertial and viscous forces acting on the particle. 

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