Potential flow theory

At any position within this Newtonian fluid, one calculates the r 9 component of the viscous stress tensor as follows  

Obviously, integration constant A must be zero to satisfy the zero shear condition 3 at the gas-liquid interface. Now condition 2 is satisfied when 2B = —l Vbubbie-The final results for the stream function and the fluid velocity profile are 

Generalized vector analysis is presented in this section for fluid flow adjacent to zero-shear interfaces in the laminar regime. The following adjectives have been used to characterize potential flow inviscid, irrotational, ideal, and isentropic. Ideal fluids experience no viscous stress because their viscosities are exceedingly small (i.e., ii 0). Hence, the V r term in the equation of motion is negligible 

For solid-body rotation at constant angular velocity, the vorticity vector, defined by I (V X v), is equivalent to the angular velocity vector of the solid. For two-dimensional flow in cylindrical coordinates, with Vr(r,0) and V0 r,9), the volume-averaged vorticity vector. 

The second integral of (8-201) vanishes, due to the periodicity of at 0 = 0 and 2ji. Hence, the volume-averaged vorticity 

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Airflow near the hood can be described using the incompressible, irrota-tional flow (i.e., potential flow) model. The potential flow theory is based on... 

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. 

Potential flow theory is used to predict the velocity components (V, V.)... 

As Wallis (1969) points out, the upper limit of region 4 is with very large bubbles when their rise is dominated by inertial forces. Under these conditions, the terminal rise velocity is readily calculated from potential flow theory and is given by... 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

Predictions for Round Wires. Similar computations were carried out for diffusion through the boundary layer around a round wire, assuming that the gas velocity just outside the boundary layer (U) was given by potential flow theory U 2V sin 0, where V is the velocity in the undisturbed gas upstream of the wire, and 9 is the angle relative to the forward stagnation point. 

The middle layer height hm divides the outer layer into a lower part, where vorticity in the background flow is dynamically important, and an upper part where the flow perturbations can be calculated by potential flow theory, is defined by the relationship,... 

When we sub.stitute and solve for dp, the result is dp 10 /im. Particles smaller than this will not deposit by impaction on the tubes according to calculations based on potential flow theory. In reality, there may be a small contribution by impaction due to boundary layer effects and direct interception. Diffusion may also contribute to deposition. 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

A second, equally dramatic difference between the potential-flow theory and experimental observation was that the flow patterns were often completely different. In the case of streaming flow past a circular cylinder, for example, the potential-flow solution is fore aft symmetric with no indication of a wake downstream of the body. To show this we simply solve the potential-flow equation in cylindrical coordinates,... 

The conclusion to be drawn from the preceding discussion is that the potential-flow theory (10-9) [or, equivalently, (10 12) and (10 13)] does not provide a uniformly valid first approximation to the solution of the Navier Stokes and continuity equations (10-1) and (10 2) for Re 1. Furthermore, our experience in Chap. 9 with the thermal boundary-layer structure for large Peclet number would lead us to believe that this is because the velocity field near the body surface is characterized by a length scale 0(aRe n), instead of the body dimension a that was used to nondimensionalize (10-2). As a consequence, the terms V2co and u V >, in (10 6), which are nondimensionalized by use of a, are not 0(1) and independent of Re everywhere in the domain, as was assumed in deriving (10-7), but instead are increasing fimctions of Re in the region very close to the body surface. Thus in... 

We see then that (10-7) - and potential-flow theory - provides a leading-order approximation for Re 1 in an outer region away from the body but is not valid very near the body surface. Physically, to retain the essential effects ofviscosity in the vicinity of the body, there is an internal readjustment in the flow that produces very large gradients in oo (or u)... 

Outer region, where variations of velocity are characterized by the length scale a of the body and potential-flow theory provides a valid first approximation in an asymptotic expansion of the solution for Re -> oo. 

Re < 40, and the predicted pressure distribution from potential-flow theory. The ability of the boundary-layer theory to predict separation is probably its most important characteristic. 

Assuming that (10-112) can be solved subject to (10-96) and (10-97), the question is whether ue = xm corresponds to any physically realizable body shapes. To answer from first principles, we would have to solve an inverse problem in potential-flow theory. We do not propose to do that here. Rather, we simply state the result, which is that... 

An example of the pressure distribution predicted by potential-flow theory and the experimentally measured distribution for a case in which the boundary-layer separates was shown in Fig. 10-3. 

Suppose that a is sufficiently small, i.e., We is sufficiently large, that surface tension plays no role in determining the bubble shape, except possibly locally in the vicinity of the rim where the spherical upper surface and the flat lower surface meet. Further, suppose that the Reynolds number is sufficiently large that the motion of the liquid can be approximated to a first approximation, by means of the potential-flow theory. Denote the radius of curvature at the nose of the bubble as R(dX 6 = 0). Show that a self-consistency condition for existence of a spherical shape with radius R in the vicinity of the stagnation point, 0 = 0, is that the velocity of rise of the bubble is... 

Boundary Layer Concept. The transfer of heat between a solid body and a liquid or gas flow is a problem whose consideration involves the science of fluid motion. On the physical motion of the fluid there is superimposed a flow of heat, and the two fields interact. In order to determine the temperature distribution and then the heat transfer coefficient (Eq. 1.14) it is necessary to combine the equations of motion with the energy conservation equation. However, a complete solution for the flow of a viscous fluid about a body poses considerable mathematical difficulty for all but the most simple flow geometries. A great practical breakthrough was made when Prandtl discovered that for most applications the influence of viscosity is confined to an extremely thin region very close to the body and that the remainder of the flow field could to a good approximation be treated as inviscid, i.e., could be calculated by the method of potential flow theory. 

Wittke and Chao [187] considered heat-transfer-controlled condensation on a moving bubble. They assumed that the bubble was a rigid sphere that moved with a constant velocity. They assumed that potential flow theory was valid. Isenberg et al. [188] corrected this model for no slip at the bubble surface and arrived at ... 

We note that Higbie s penetration theory (HI5), with contact time assumed as that required by the drop to traverse a distance of one diameter (W6), gives an expression identical with Eq. (18). Although potential-flow theory, unlike the penetration theory, takes interfacial acceleration into account, the two are actually physically identical, both being based on diffusion into an element of fluid sliding over the constant-temperature interface. 

This potential-flow theory also describes the flow of a viscous fluid in a porous medium, which has considerable practical significance in petroleum reservoir engineering, hydrology, filters, etc. 

There are no dimensionless numbers in this potential flow equation because convective forces per unit volume and dynamic pressure forces per unit volume both scale as pV /L. Furthermore, potential flow theory provides the formalism to calculate p and the dimensionless scalar velocity potential such that the vorticity vector vanishes and overall fluid mass is conserved for an incompressible fluid. Hence,... 

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