Pressure gradient field

Besides the drag force, Basset force, and Saffman force, another force may act on the particle as a result of the existence of a pressure gradient in the fluid. Using the axisymmetric condition, the force on a differential element of a sphere in a pressure gradient field shown in Fig. 3.2 can be expressed by... 

Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(—/fflf) in a symmetrical electrolyte solution of viscosity r] and relative permittivity r in an applied oscillating pressure gradient field Vp exp(—imt) due to a sound wave propagating in the suspension, where m is the angular frequency 2n times frequency) and t is time. We treat the case in which m is low such that the dispersion of r can be neglected. We assume that the particle core of radius a is coated... 

FIGURE 26.3 A spherical soft particle in an applied pressure gradient field, a = radius of the particle core. = thickness of the polyelectrolyte layer covering the particle core. 

Figures 26.4 and 26.5 show the dependence of the magnitude (Fig. 26.4) and phase (Fig. 26.5) of each of CVI, IVl, and TVI on the frequency oj of the pressure gradient field due to the applied sound wave for the case where a=l /im for an aqueous KCl solution of concentration n = 0.01 M. It is seen that the m-dependence is negligibly small mUn < 10" Hz and becomes appreciable for mUn > 10" Hz. That is, CVI is essentially equal to its static value at (U = 0 for l2n < 10" Hz and drops sharply to zero for (oUn > 10" Hz, while the phase of CVI is zero for (oUn < 10" Hz and increases sharply for the frequency range (ol2n> 10" Hz. The magnitude of IVl, on the other hand, is constant independent of , while the phase of IVl is always zero, since in the present approximation IVl is a real quantity.

Figure 13.4 shows a snapshot of the pressure-gradient field at t =... 

Figure 13.4 Snapshot of pressure-gradient field at < = 0.80 ms Ucyde = 3 ms, Uciose = 2.1 ms, and Tpurge = 0.1 ms. (Refer color plate, p. XXIV.)...

Control algorithm part B Solve a least squares problem for the necessary voltage actuations to induce a pressure gradient field that will create a flow fleld that will carry the particles along the desired directions obtained in step 3. 

Step 4 of the algorithm requires more elaboration. Since the pressure field obeys Laplace s equation, which is linear, we can consider linear combinations of pressure boundary conditions due to voltage actuation at the electrodes (see Fig. 6). The problem of computing the necessary boundary conditions to create a pressure gradient field to move the particles in the directions we want leads to a least squares problem. 

The synoptic case during April 18-21, 1999 was characterized by an upper trough located in the northern part of the area of interest. There was a small pressure gradient field at lower levels. This synoptic case produces a light variable wind near the surface, and a much more intensive westerly wind at upper levels. Fronts were observed during April 18-20 followed by heavy rain, thunderstorms, and intense wind (17 m s ) from a westerly direction. 

There are some constraints on the shape functions in order to have a consistent finite element formulation. Equation 8.12 demands that the shape function Nn be at least linear in the spatial coordinates. The simplest and most widely used shape function in our case is perhaps the piecewise linear interpolation function as described in Kennedy (1995). With this shape function, one can achieve C° continuity in pressure, that is, the pressure field variable is continuous at element interface, but its gradients are not. The pressure gradient field is piecewise constant over the elements and is discontinuous across element interfaces. Consequently, the resulting velocity and shear rate fields are not continuous across element boundaries. An example of triangular elements with higher order shape functions can be found in Hieber and Shen (1980). 

The field unit for permeability is the Darcy (D) or millidarcy (mD). For clastic oil reservoirs, a good permeability would be greater than 0.1 D (100 mD), while a poor permeability would be less than 0.01 D (10 mD). For practical purposes, the millidarcy is commonly used (1 mD = 10" m ). For gas reservoirs 1 mD would be a reasonable permeability because the viscosity of gas is much lower than that of oil, this permeability would yield an acceptable flowrate for the same pressure gradient. Typical fluid velocities in the reservoir are less than one metre per day. 

In pressure diffusion, a pressure gradient is estabUshed by gravity or in a centrifugal field. The lighter components tend to concentrate in the low pressure (center) portion of the fluid. Countercurrent flow and cascading extend the separation effect. 

There is a field operation method by which the fracture pressure gradient can be experimentally verified. Such tests are known as leak-off tests. The leak-off test will be discussed in Chapter 4. 

The two-layer model is being progressively updated as fresh experimental results and correlations become available. The most satisfactory starting-point for anyone wishing to use the model to calculate pressure gradients for flow of solids-liquid mixtures in a pipeline is the text of SHOOK and Roc.o(52) which includes a worked example. However, there are many pitfalls to be avoided in this area, and there is no substitute for pracucal experience gained by working in the field. 

FIGURE 11.32 Flow profiles in microchannels, (a) A pressure gradient, - AP, along a channel generates a parabolic or Poiseuille flow profile in the channel. The velocity of the flow varies across the entire cross-sectional area of the channel. On the right is an experimental measurement of the distortion of a volume of fluid in a Poiseuille flow. The frames show the state of the volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule, (b) In electroosmotic flow in a channel, motion is induced by an applied electric field E. The flow speed only varies within the so-called Debye screening layer, of thickness D. On the right is an experimental measurement of the distortion of a volume of fluid in an electroosmotic flow. The frames show the state of the fluorescent volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule [165], Source http //www.niherst.gov.tt/scipop/sci-bits/microfluidics.htm (see Plate 12 for color version). 

The hydrodynamic equation of motion (Navier-Stokes equation) for the stationary axial velocity, vfr), of an incompressible fluid in a cylindrical pore under the influence of a pressure gradient, dP /dz, and an axial electric field, E is... 

The flux of liquids or gases through the membrane is in most cases driven by a pressure gradient and sometimes by an electric field gradient. Membranes can be used for ... 

Consider then a mixture of A and B which are being subjected to a uniform gravitational acceleration or centrifugal acceleration g in the — z direction. It should be noted that there is no mass flux due to forced diffusion according to Eq. (35) inasmuch as jt(F) vanishes when we replace t by —g. The effect of the gravitational field is to produce a pressure gradient, the latter being determined by Eq. (25), which for the case under consideration becomes ... 

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. 

The Navier-Stokes equations equations involve the pressure gradient, but the pressure itself does not appear explicitly. As a result a further simplification is often available and useful. Assuming nominal atmospheric pressure (patm 105 N/m2), pressure variations associated with the characteristic velocity scales are very often quite small. For air at standard atmospheric conditions, the sound speed is a0 350 m/s. The pressure variations for a low-speed atmospheric flow, say u0 = 10 m/s, are around p p0u20 100, which is three orders of magnitude lower than p0. Thus the pressure field can be usefully separated into two components [255,303] as... 

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