The Flow Field

A RIGID SPHERE IN AXISYMMETRIC, EXTENSIONAL FLOW I. The Flow Field 

A more direct motivation for studying linear flows is that we are frequently interested in applications of creeping-flow results for particles that are very small compared with the length scale L that is characteristic of changes in the undisturbed velocity gradient for a general flow. In this case, we may approximate the undisturbed velocity field in the vicinity of the particle by means of a Taylor series approximation, namely, 

Thus the undisturbed flow that is seen in a frame of reference that translates with the particle is just 

however, this can be expressed as the sum of a linear straining flow and a purely rotational flow, that is, 

A plot showing streamlines for this flow with a spherical body at the origin (calculated later in this section) is shown in Fig. 7-13. For A 0, there is flow outward away from the sphere along the axis of symmetry and flow inward in the plane orthogonal to this axis. This flow is called uniaxial extensional flow. For E 0, the direction of fluid motion is reversed and the undisturbed flow is known as biaxial extensional flow. In either case, with an axially 

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Two-dimensional models can be used to provide effective approximations in the modelling of polymer processes if the flow field variations in the remaining (third) direction are small. In particular, in axisymraetric domains it may be possible to ignore the circumferential variations of the field unlaiowns and analytically integrate the flow equations in that direction to reduce the numerical model to a two-dimensional form. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

Atomization. A gas or Hquid may be dispersed into another Hquid by the action of shearing or turbulent impact forces that are present in the flow field. The steady-state drop si2e represents a balance between the fluid forces tending to dismpt the drop and the forces of interfacial tension tending to oppose distortion and breakup. When the flow field is laminar the abiHty to disperse is strongly affected by the ratio of viscosities of the two phases. Dispersion, in the sense of droplet formation, does not occur when the viscosity of the dispersed phase significantly exceeds that of the dispersing medium (13). 

One-equation models relax the assumption that production and dissipation of turbulence are equal at all points of the flow field. Some effects of the upstream turbulence are incorporated by introducing a transport equation for the turbulence kinetic energy k (20) given by... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

Experimental techniques to visualize flows have been extensively used to define fluid flow in pipes and air flow over lift and control surface of airplanes. More recently this technology has been appHed to the coating process and it is now possible to visualize the flow patterns (16,17). The dimensions of the flow field are small, and the flow patterns both along the flow and inside the flow are important. Specialized techniques such as utilizing small hydrogen bubbles, dye injection, and optional sectioning, are required to visualize these flows. 

Turbulent flame speed, unlike laminar flame speed, is dependent on the flow field and on both the mean and turbulence characteristics of the flow, which can in turn depend on the experimental configuration. Nonstationary spherical turbulent flames, generated through a grid, have flame speeds of the order of or less than the laminar flame speed. This turbulent flame speed tends to increase proportionally to the intensity of the turbulence. 

Benisek, E., 1998. Experimental and Analytical Investigation for the Flow Field of a Turbocharger Turbine, IMechE, Paper No. 0554/027/98. 

From the eontours of the stream funetion and a eloseup of the flow field near the shroud and top eover plate, we note that the shroud eauses signifieant reeireulation at the top end of the eatalyst bed. Also, the downward veloeity field in the annular region between eatalyst bed and reaetor shell is found not to be uniform. The eomparative... 

The parts of local ventilation systems, situated inside rooms, that influence the flow field are described here. This presumes that the inlet and outlet openings are properly connected to duct systems either directly or through flexible connections (tubes). These ducts and tubes and other parts of importance for the function of these systems are described in other chapters. 

The effects of obstacles in the flow field were also studied. A bar with dimensions of 0.04 m high x 0.003 m wide x 1 m long was used. Capture efficiency was... 

Method A Calculating Contaminant and Exhaust Velocities at All Points in the Flow Field Local exhaust hoods are used to remove contaminants at the point of generation before they escape into the workplace air. The efficiency of any local exhaust system is greatly affected by the flow field generated by the exhaust opening. Therefore, accurate modeling of this flow field is essential for reliable predictions. However, solving the airflow field is a formidable task and often must be done numerically. 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Semi-Theoretical and Empirical Velocity Fields Since the use of formulas to calculate the velocities outside an arbitrary opening could be very tedious, only some examples of these formulae are given. These calculations are best done on computers and there are some dedicated programs to calculate and visualize the flow fields outside exhaust openings. There could sometimes be problems when calculating the velocity field outside an opening close to... 

IJnflanged Circular, Rectangular, and Slot Openings For unflanged openings no explicit equations exist for the flow fields. However, it is possible to calculate the velocity field outside a specific BEO using computers. These... 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

Partial enclosures are a compromise between containment and access. Most people misunderstand the function of partial enclosures. It is not possible to completely separate the interior from the surroundings with partial enclosures. Complete separation is only possible with total enclosures. The function of a partial enclosure is as dependent on the flow rate, the flow field, the working procedures, the contaminant generation process, etc. as is the function of exterior hoods. The advantage with a partial enclosure is that the physical walls diminish the possibilities for the contaminants to escape from the hood to the surroundings. Thus these hoods could be used when relatively high demands are put on the contaminant concentration outside the hood. Some of the most commonly used enclosures, such as fume cupboards and booths, are described. Many variations of these exist, e.g., enclosure of the complete process, and some of these are described here. 

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