Fluid flow velocity gradient

The natural process of bringing particles and polyelectrolytes together by Brownian motion, ie, perikinetic flocculation, often is assisted by orthokinetic flocculation which increases particle coUisions through the motion of the fluid and velocity gradients in the flow. This is the idea behind the use of in-line mixers or paddle-type flocculators in front of some separation equipment like gravity clarifiers. The rate of flocculation in clarifiers is also increased by recycling the floes to increase the rate of particle—particle coUisions through the increase in soUds concentration. 

When a fluid flows over a surface, that part of the stream which is close to the surface suffers a significant retardation, and a velocity profile develops in the fluid. The velocity gradients are steepest close to the surface and become progressively smaller with distance from the surface. Although theoretically there is no outer limit at which the velocity gradient becomes zero, it is convenient to divide the flow into two parts for practical purposes. 

In words, the prediction is that the current per unit of pressure gradient should be equal to the fluid flow velocity per unit of electric field. Experiments prove that this is indeed the case. 

As a first approximation to the motion of two spheres in a solvent (which can be regarded as a continuum), the spheres can be presumed to move about the solvent sufficiently slowly that the very much simplified Navier—Stokes equation of fluid flow is applicable. The application of a pressure gradient VP(r) in the fluid develops velocity gradients within the fluid, Vv(r). If another force F(r) is included in the fluid, this can generate a pressure gradient and further affect the velocity gradients. The Navier— Stokes equations [476] becomes... 

Figure 4.2 Fluid flow velocity through the channel of a membrane module is nonuniform, being fastest in the middle and essentially zero adjacent to the membrane. In the film model of concentration polarization, concentration gradients formed due to transport through the membrane are assumed to be confined to the laminar boundary layer...

The dynamic viscosity can be illustrated with the Couette flow (Figure 3.17). A fluid is located between two plates at the distance H. If a shearing force F is applied, a velocity gradient in the fluid is built up. The maximum velocity will occur at the point where the stress is applied, whereas the velocity is zero at the opposite side due to the wall adherence. For a Newtonian fluid, the velocity gradient is constant across the distance between the two plates. With A as the surface area of the upper plate, the shear stress r is defined as... 

Couette flow A type of flow in which a fluid is sandwiched between two parallel plates, one of which is stationary and the other is moving at some constant velocity. For a Newtonian fluid, the velocity gradient is linear between the plates. It is named after French physicist Maurice Marie Alfred Couette (1858-1943). 

Normal stress difference Complex fluid Rheometers Rheometry Viscometric flows Velocity gradient Rate of deformation Weissenberg effect Extrudate Die swell Melt fracture Hydrostatic pressure Shear direction Anisotropic... 

For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direc tiou perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is caUed a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most uoupolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of uou-Newtouiau fluids. 

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Number of vanes. The greater the number of vanes, the lower the vane loading, and the closer the fluid follows the vanes. With higher vane loadings, the flow tends to group up on the pressure surfaces and introduces a velocity gradient at the exit. 

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