Poiseuille profile

Fig. 4.3.6 Velocity maps and profiles at differ- mark the NMR foldbacks from the stationary ent heights of the Fano column. The dark ring fluid at the inner surface of the fluid reservoir, surrounding the pipe at z= 1.5 mm (larger In the velocity profiles, the solid curves are the white arrow) is due to a layer of stationary fluid calculated Poiseuille profiles in tube flow, adhering to the pipe exterior following the Velocity images are reprinted from Ref. [20], dipping of the pipe into the reservoir at the with permission from Elsevier, start of the experiment. The small white arrows...

Figure 9. Model of convective diffusion inside a tube channel (radius R) towards the walls of the tube considered as perfect active surfaces. A Poiseuille profile for the velocity (equation (35)) is also schematically shown with arrows on the right-hand side. The maximum velocity v is reached at the centre of the tube (r = 0)...

Next we consider a fluid flowing through a circular tube with material at the wall diffusing into the moving fluid. This situation is met with in the analysis of the mass transfer to the upward-moving gas stream in wetted-wall-tower experiments. Just as in the discussion of absorption in falling films, we consider mass transfer to a fluid moving with a constant velocity profile and also flow with a parabolic (Poiseuille) profile (see Fig. 5). 

Figure 5.14 illustrates the nondimensional velocity profiles (Eq. 5.122) for different values of the cross-stream velocity V. As should be anticipated, for sufficiently low injection velocity V, the parabolic Poiseuille profile is obtained. As the injection velocity increases, the axial velocity profile is skewed toward the upper wall. 

Since the velocity profile approaches the Hagen-Poiseuille profile asymptotically, the factor 0.05 depends on a criterion for deciding how close the profile needs to be to the Hagen-Poiseuille profile (e.g., the maximum velocity is within 1% of the steady state maximum velocity). In any case, the entry length scales linearly with the Reynolds number. 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

The formation of a surface wave in Fig. 1.8a requires the lateral displacement of liquid. Assuming a non-slip boundary condition at the substrate surface (lateral velocity v(z) = 0 at the surface (z = 0)), and the absence of normal stresses at the liquid surface, this implies a parabolic velocity profile (half-Poiseuille profile) in the film... 

Laminar flow is the usual flow regime met in monolith reactors, given that the typical Reynolds number has values below SOO. The radial velocity profile in a single channel develops from the entrance of the monolith onward and up to the position where a complete Poiseuille profile has been established. The length of the entrance zone may be evaluated from the following relation [3] ... 

Obviously, as t - cxd, this solution reverts to the steady-state Poiseuille profile. To obtain other details of this velocity profile, it is necessary to evaluate the infinite series numerically for each value of t and r. A typical numerical example of the results is shown in Fig. 3-10, where uz has been plotted versus r for several values of 7. It can be seen that the initial profile for 7 = 0.05 is flat, with uz approximately independent of r except for r very close to the tube walls. Right at the tube wall, uz = 0, and it can be seen that this manifestation of the no-slip condition gradually propagates across the tube by means of the diffusion process discussed earlier. The region in which the wall is felt increases in width at a rate proportional to Jvt as is typical of diffusion or conduction processes with diffusivity v. 

By substituting (6.4.8) into (6.4.4)-(6.4.7), one can find the basic characteristics of motion of a power-law fluid in a circular tube. The results of the corresponding calculations [452, 508] are presented in Table 6.5 and are shown in Figure 6.2. One can see that the velocity profiles become more and more filled as the rheological parameter n decreases. The limit case n -> 0 is characterized by a quasisolid motion of the fluid with the same velocity in the entire cross-section of the tube (it is only near the wall that the velocity rapidly decreases to zero). The parabolic Poiseuille profile corresponds to the Newtonian fluid (n = 1). The limit dilatant flow (n — oo) has a triangular profile, which is characterized by a linear law of velocity variation along the radius of the tube. 

The coefficient multiplying the Reynolds number for a straight channel is 0.16 (Schlichting 1979). Therefore, for a Reynolds number based on a channel width 2h) of 1000, the Poiseuille profile would develop in about 40 channel widths. Again, since v> D for diffusion in dilute solution, we may expect that the development length is very much longer for the concentration profile than for the velocity profile. 

Following Levich (1962), let us calculate approximately the rate at which the capillary will rise to the height given by Eq. (10.2.4) and the length of time it takes to attain that height. The flow in the capillary is unsteady. To simplify the calculation, we assume that the velocity profile at any instant of time is given by the Poiseuille profile (Eq. (4.2.14), with = 2U and h = alVl)... 

Here, u y) is given by the Poiseuille profile (Eq. 4.2.14). The equation governing the potential distribution follows from current continuity (Eq. 3.4.16), which, with z. = — is... 

For infinite diluted solutions, Sc 1000, therefore Sd O.ldu. Consequently, Sd Su, and the distribution of velocities in the diffusion boundary layer may be determined independently of the appropriate hydrodynamic problem. As an example, consider stationary laminar flow of viscous incompressible liquid in a flat channel. It is known that at some distance from the channel entrance, the velocity profile changes to a parabolic Poiseuille profile (Fig. 6.2) [5]. 

At entry point, the fluid flow profile can be significantly different from the parabolic profile of Poiseuille flows. There is a characteristic distance before the steady-state Poiseuille profile can be established. This distance, between the entry point and the full establishment of Poiseuille flow, is called the entrance length. At the end of the entrance length, the pressure gradient matches that of the fully developed flow. The entrance length, 4. can be expressed in terms of the dimensionless entrance length number El ... 

Figure 9 Molecular dynamics simulation of a Lennard-Jones, bead-spring model, (a) Slip length, 8, as a function of the strength, [ an, of attraction between a hard, corrugated substrate and liquid for temperature, kgT/ [ =. 2. The solid line with circles is obtained from the Couette and Poiseuille profiles (NEMO) according toeqn [37], whereas the dashed line with squares, from the Green-Kubo (GK) relation, eqn [39]. The curve marks the behavior 1/ [ jii in accord with eqn [40]. The inset illustrates the velocity profiles of the Couette and Poiseuille flows, from which the slip length has been estimated for [mii = 0.6, measured in units of the Lennard-Jones parameter, [. Adapted from Servantie, J. Muller, M. Phys. Rev. Lett. 2008, 101,...

In millimetric pipe flow, the increase of the flow rate from the Newtonian to the plateau regime was characterized by a transition from a nearly parabolic Poiseuille profile to an almost flat velocity profile with high shear bands near the tube walls, clearly distinguishable from slip [135]. The thickness of this high shear rate band... 

In 1917, Lucas determined the position of the front of liquid being wicked by a porous material as a function of time. He found that it propagates as a behavior which was previously reported by Bell and Cameron. A few years later, in 1921, Washburn explained Lucas observations assuming (a) that the porous material can be seen as a system of parallel capillaries (b) that the flow in each capillary tube is stationary and axisymmetric and (c) that the flow is well described by a Poiseuille profile with the ptessure difference across the interface given by the Laplace equation. Under these conditions, the position of the meniscus inside a capillary tube varies with time as... 

Capillary rise time is the time required to reach equilibrium capillary rise height, T/q. The performance of many microsystems is dependent on the transient behavior of the capillary flow. Therefore, we intend to calculate the rate at which the capillary will rise to the equilibrium height. Let us assume the velocity profile at any instant of time to be fully developed Poiseuille profile. This assumption is justified from the fact that the developing length is expected to be very small compared to the length of the capillary due to the small diameter of the capillary tube. The instantaneous average velocity of the interface can be expressed from Poiseuille flow relation and the movement of interface as... 

Immobile surfactants do not permit slip, which leads to a parabolic Poiseuille profile. Velocity gradients are thus substantially larger in the presence of immobile surfactants, and there is considerable dissipation along the length of the film. In foams with mobile surfactants, by contrast, dissipation is concentrated in locations where geometry requires the continuous phase to change directions, such as edges and vertices. 

---