Determining velocity fields

Finally, we conclude this chapter with notes on calculating velocity fields from integral expressions for pressure. To keep the discussion general, we consider the arbitrary fluid of Example 2-2 with nonzero m, taking +1 

The strength f(x) and the constant of integration H are assumed to be known. The horizontal velocity parallel to the fracture is obtained by first differentiating Equation 2-44 with respect to x. This leads to 

The integral in Equation 2-118 is a Cauchy principal value integral and can be evaluated as discussed in Example 2-5. If the fracture pressure pf is constant, 

This derivative is used to calculate the normal Darcy velocity at the slit. The preeeding examples demonstrate the power and elegance of integral equation methods. In Chapter 3, similar methods are used to analyze flows about shales. Chapter 4 introduces modem issues in streamline traeing and the fundamentals of complex variables this background is helpful to understanding Chapter 5, where more eomplicated shapes are considered. 

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In order to determine velocity field one uses the Stokes equations... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

The results of computations of T o for an isolated fiber are dhistrated in Figs. 17-62 and 17-63. The target efficiency T t of an individual fiber in a filter differs from T o for two main reasons (Pich, op. cit.) (1) the average gas velocity is higher in the filter, and (2) the velocity field around the individual fibers is influenced by the proximity of neighboring fibers. The interference effect is difficult to determine on a purely theoretical basis and is usually evaluated experimentally. Chen (op. cit.) expressed the effecd with an empirical equation ... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

Flow velocity field determined by PIV. Lean limit flames propagating upward in a standard cylindrical tube in methane/air and propane/ air mixtures, (a) Methane/air—laboratory coordinates, (b) propane/air—laboratory coordinates, (c) methane/air—flame coordinates, and (d) propane/air—flame coordinates. 

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

For laminar flow in channels of rectangular cross-section, the velocity profile can be determined analytically. For this purpose, incompressible flow as described by Fq. (16) is assumed. The flow profile can be expressed in form of a series expansion (see [100] and references therein), which, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross-section is needed. Purday [101] suggested an approximate solution of the form... 

The Navier-Stokes equation and the enthalpy equation are coupled in a complex way even in the case of incompressible fluids, since in general the viscosity is a function of temperature. There are, however, many situations in which such interdependencies can be neglected. As an example, the temperature variation in a microfluidic system might be so small that the viscosity can be assumed to be constant. In such cases the velocity field can be determined independently from the temperature field. When inserting the computed velocity field into Eq. (77) and expressing the energy density e by the temperature T, a linear equahon in T is... 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

The Navier-Stokes equations are solved first to determine the velocity field throughout the reactor, as described by Armaou and Christofides [4], and subsequently by Brass and Lee [5] using FEMLAB. Then, the species mass balances are solved to determine the concentrations of SiFL (1), SiH2 (2), SiH3 (3), and H (4), throughout the reactor. Finally, the deposition rate of silicon is ... 

Now, consider a natural river, illustrated in Figure 9.3. There are many sources of vorticity in a natural river that are not related to bottom shear. Free-surface vortices are formed in front of and behind islands and at channel contractions and expansions. These could have a direct influence on reaeration coefficient, without the dampening effect of stream depth. The measurement of p and surface vorticity in a field stream remains a challenge that has not been adequately addressed. The mean values that are determined with field measurements are not appropriate. Most predictive equations for reaeration coefficient use an arithmetic mean velocity, depth, and slope over the entire reach of the measurement (Moog and Jirka, 1998). The process of measuring reaeration coefficient dictates that these reaches be long to insure the accuracy of K2. Flume measurements, however, have generally shown that K2 u /hor K2 (Thackston and Krenkel, 1969 ... 

The flow characteristics can be determined using the model shown in Figure 7.79. During the time interval nnder consideration, it is assumed that the polymer has been heated to a uniform temperatnre, T , that is equivalent to the mold wall temperature. As long as the preform radius, R, is less than the radius of the outer wall of the mold cavity, Ro, we can treat the problem as an isothermal radial flow of an incompressible power law flnid flowing between two disks that approach each other at a constant rate, h. In this way, the velocity field, Vr z, r, t), the pressnre distribution in the mold, P z, r, t), and the plnnger force, Fn z, r, t) can be obtained as follows ... 

Molecular theories of flow behavior are applied on the assumption that the macroscopic velocity field can be considered to apply without modification right down to the molecular scale. In continuum theories the components of relative velocity in an arbitrarily small neighborhood of any material point are taken to be linear functions of the spatial coordinates measured from that point, i.e., the flow is assumed to be locally homogeneous. The local velocity field is calculated from the macroscopic velocity field. This property of local homogeneity of flow is an obvious prerequisite for any meaningful macroscopic analysis, and perhaps the fact that analyses are at all successful and that flow properties can be determined which are independent of apparatus geometry constitutes a fair test of the assumption. 

Suppose for example that there is no unperturbed velocity field, V°(R) =0. Then the frictional force on the, /th bead is determined by not only its own velocity ft , but also by the velocities ft, of all the other beads. The only exception can be when the friction tensor , is strictly diagonal and this may be expected to be true only in the complete absence of hydrodynamic interaction. 

We focus our attention on a packet of fluid, or a fluid particle, whose size is small compared to the length scales over which the macroscopic velocity varies in a particular flow situation, yet large compared to molecular scales. Consider air at room temperature and atmospheric pressure. Using the ideal-gas equation of state, it is easily determined that there are approximately 2.5 x 107 molecules in a cube that measures one micrometer on each side. For an ordinary fluid mechanics problem, velocity fields rarely need to be resolved to dimensions as small as a micrometer. Yet, there are an enormous number of molecules within such a small volume. This means that representing the fluid velocity as continuum field using an average of the molecular velocities is an excellent approximation. 

Net Forces on a Differential Control Volume Based on a differential control volume (i.e vanishingly small dimensions in each of three spatial coordinates), we write the forces on each of the six faces of the control volume. The forces are presumed to be smooth, continuous, differentiable, functions of the spatial coordinates. Therefore the spatial variations across the control volume in each coordinate direction may be represented as a first-order Taylor-series expansion. When the net force is determined on the differential control volume, each term will be the product a factor that is a function of the velocity field and a factor that is the volume of the differential control volume 8 V. 

We have determined the components of the rotation-rate vector dQ/dt for a general velocity field. However, it is conventional in fluid mechanics to represent rotation in the form of a derived variable called vorticity, which is denoted as the vector u. By definition,... 

By substitution of the velocity field into the constant-viscosity, incompressible, Navier Stokes equations, determine an expression for the pressure field around the sphere (i.e p(z, r)). 

Steady parallel flow can be realized in ducts of essentially arbitrary cross section. A linear elliptic partial differential equation must be solved to determine the velocity field and the shear stresses on the walls. For an incompressible, constant-viscosity fluid, the axial momentum equation states that... 

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