Flow nets streamlines

The equation of continuity and the Bernoulli theorem together show, for a stream of incompressible fluid, that (a) where the cross-sectional area is large and the streamlines are widely spaced, the velocity is low and the pressure is high, and (6) where the cross-sectional area is small and the streamlines are crowded together, the velocity is high and the pressure is low Hence a flow net gives a picture not only of the velocity field but also of the pressure field. 

Flow nets consist of two types of lines streamlines and isopotentials. Streamlines are lines that follow the path of representative parcels of water water always flows parallel to streamlines. Isopotentials, drawn perpendicular to streamlines, are lines along which the hydraulic head, h, is constant. Therefore, water always flows perpendicular to isopotentials. Flow nets are often drawn to represent the horizontal movement of groundwater and associated chemicals in an aquifer the plane of the flow net then represents the horizontal aquifer surface, and it is assumed that underneath each point on the surface, flow is essentially the same at all depths in the aquifer. An example of such a flow net is shown in Fig. 3-8. 

The groundwater flow into a well also can be analyzed by using a flow net. Because the well is symmetrical, lines of constant drawdown (isopotentials) are circles centered at the center of the well (Fig. 3-11). Streamlines are arranged radially around the well if there is no regional flow in the aquifer. 

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. 

Rather than discuss the penetration of the flow streamlines into the molecular domain of a polymer in terms of viscosity, we shall do this for the overall friction factor of the molecule instead. The latter is a similar but somewhat simpler situation to examine. For a free-draining polymer molecule, the net friction factor f is related to the segmental friction factor by... 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Figure 6.6 uses a particular problem to illustrate some of the salient differences between the semi-infinite and finite-gap configurations. In both cases the net flow rates are the same, and the streamlines in both plots have the same values. The streamlines for the finite-gap problem show pure axial flow at the inlet, whereas the semi-infinite case shows radial spreading everywhere. The radial-velocity profiles are quite different, with the finite-gap profile showing no slip at both boundaries. 

The characteristic feature of flow FFF is the superimposition of a second stream of liquid perpendicular to the axis of separation. This cross-flow drives the injected sample plug toward a semipermeable membrane that acts as the accumulation wall. The cross-flow liquid permeates across the membrane and exits the channel, whereas the sample is retained inside the channel in the vicinity of the membrane surface. Sample displacement by the cross-flow is countered by diffusion away from the membrane wall. At equilibrium, the net flux is zero and sample clouds of various thicknesses are formed for different sample species. As with other FFF techniques, a larger diffusion coefficient D leads to a thicker equilibrium sample cloud that, on average, occupies a faster streamline of the parabolic flow profile and subsequently elutes at a shorter retention time t,. For well-retained samples analyzed by flow FFF, t, can be related to D and the hydrodynamic diameter d by... 

The sphere in pure extensional flow (X = 1). In this case, the streamlines are open (hyperbola at infinity), and the net hydrodynamic torque on the sphere is zero. Thus, if no external torque is apphed to the sphere, it will not rotate. 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

A stream tube, or stream filament, is a tube of small or large cross section and of any convenient cross-sectional shape that is entirely bounded by streamlines. A stream tube can be visualized as an imaginary pipe in the mass of flowing fluid through the walls of which no net flow is occurring. 

The selection of the type of particle collector is somewhat arbitrary, usually limited by practical considerations. The stream lines of fluid flow around the collectors should be similar to those in a porous medium and yet simple to calculate. As a particle carried by the fluid comes into the proximity of the collector and experiences the net interaction force between itself and the collector, the trajectory of the particle deviates from that of the fluid streamlines. The actual path of the particle is... 

The SI unit for the coefficient of viscosity ps kg m s or N s m the commonly used unit, the poise, is one-tenth of the SI unit. The type of flow to which (11.95) applies is called laminar, streamline, or Newtonian flow. In flow of this kind there is a net component of velocity in the direction of flow superimposed on the random molecular velocities. Streamline flow is observed if the velocity of flow is not too large with very rapid flow the motion becomes turbulent and (11.95) no longer applies. 

At low degrees of surface heterogeneity (streamline pattern shown in Fig. 6a was obtained. As can be seen a net counter-clockwise flow perpendicular to the applied electric field is present at the first transition plane (i. e. at the initial discontinuity in the heterogeneous surface pattern) and a clockwise flow at the second transition plane. This flow circulation is a pressure induced effect that arises as a result of the transition from the higher local fluid velocity (particularly in the double layer) over the homogeneous surface on the right hand side at the entrance, to the left hand side after the first transition plane (and vice versa at the second transition plane). To satisfy continuity then there must be a net flow from... 

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