Poiseuille-flow

Here we consider Poiseuille flow due to an applied pressure gradient down a cylindrical capillary tube of radius R as shown in Fig. 5.9. It is natural to choose the usual cylindrical coordinates as introduced above in the Couette flow examples in Section 5.7 with the 2 -axis coincident with the axis of the capillary. In Section 5.8.1 

We shall next consider the rigorous theory of Poiseuille flow, i.e., the steady laminar flow, caused by a pressure gradient, of an incompressible fluid through a tube of circular cross-section. We shall suppose that the tube is of infinite length so that end effects can be ignored. Let us choose a cylindrical polar coordinate system r(pz with z along the axis of the tube. In the steady state the only component of the velocity gradient is = dv/dr. It is natural to expect that the director is everywhere in the rz plane 

If a magnetic field with components 0, H ) is applied, the external body force 

In ordinary nematics p and p are both negative and the flow alignment angle 0 is usually small. This equilibrium orientation of the director is 

By scaling the radius and the time as r = hr and t = kt respectively, with k = h3, it is easily shown that in the absence of a magnetic field Q/R is a unique function of aR. Consequently rj plotted versus Q/R should be a universal curve for all tube radii and flow rates. This has been confirmed 

Equations (3.6.11) and (3.6.12) with H = 0 have been solved numerically by Tseng, Silver and Finlayson assuming the boundary conditions 

If there is a difference in total pressure across a particle, then there will be a direct contribution to the adsorption flux from forced laminar flow through the macropores. This effect is generally negligible in a packed bed since the pressure drop over an individual particle is very small. The effect may be of greater significance in the direct laboratory measurement of uptake rates in a vacuum system. From Poiseuille s equation it may be shown that the equivalent diffusivity is given by 

the dependence of shear stress on radius (denoted r(r)) is 

For a tube with inner radius R (dropping the rx subscript on shear stress), 

FIGURE 14.10 Force balance on an element in a cylindrical tube. 

Therefore, the shear stress varies linearly with radius from zero at the tube center to a maximum of = -iR 2) (dP/dx) at the tube wall. Note that this result does not depend in any way on the fluid properties. 

Since the axial fluid velocity m is a function of the radial position only (denoted u r)), we may write 

Integrating with the boundary conditions u R) = 0 (i.e., the fluid sticks to the wall) and u at radius r = w(r) gives 

---

The situation in electroosmosis may be reversed when the solution is caused to flow down the tube, and an induced potential, the streaming potential, is measured. The derivation, again due to Smoluchowski [69], begins with the assumption of Poiseuille flow such that the velocity at a radius x from the center of the tube is... 

Example 4 Plnne Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to he found. This example is found in most fluid mechanics textbooks the solution presented here closely follows Denn. 

Poiseuille flow Parabolic laminar flow in a straight tube. 

In creeping flow with the inertia term neglected, the velocity distribution rapidly reaches a steady value after a distance of r0 inside a capillary tube. At this stage the velocity distribution showed the typical parabolic shape characteristic of a Poiseuille flow. In the case of inviscid flow where inertia is the predominant term, it takes typically (depending on the Reynolds number) a distance of 20 to 50 diameters for the flow to be fully developed (Fig. 34). With the short capillary section ( 4r0) in the present design, the velocity front remains essentially unperturbed and the velocity along the symmetry axis, i.e. vx (y = 0), is identical to v0. 

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows ... 

Thus, the measurements of integral flow characteristics, as well as mean velocity and rms of velocity fluctuations testify to the fact that the critical Reynolds number is the same as Rccr in the macroscopic Poiseuille flow. Some decrease in the critical Reynolds number down to Re 1,500— 1,700, reported by the second group above, may be due to energy dissipation. The energy dissipation leads to an increase in fluid temperature. As a result, the viscosity would increase in gas and decrease in liquid. Accordingly, in both cases the Reynolds number based on the inlet flow viscosity differs from that based on local viscosity at a given point in the micro-channel. 

The comparison of experimental results accounting for effects discussed above to those obtained by conventional theory is correct when the experimental conditions were consistent with theoretical ones, and agrees well with Poiseuille flow predictions. 

Leite RJ (1959) An experimental investigation of the stability of Poiseuille flow. J Fluid Mech 5 81-96... 

Qcon = flow rate of the continuum Poiseuille flow D / 6 ... 

Velocity profiles of Plane Poiseuille Flow Continuum-------First-order slip Second-order slip - -... 

Fig. 3—Comparison of velocity profiles of plane Poiseuille flow between different models.

Equation (46) takes into consideration only the viscous drag due to Poiseuille flow inside the tube. 

The consequences of the wetting ridge in the capillary penetration of a liquid into a small-diameter tube have been evaluated. Viscoelastic braking reduces the liquid flow rate when viscoelastic dissipation outweighs the viscous drag resulting from Poiseuille flow. 

Saner, SG Locke, BR Arce, P, Effects of Axial and Orthogonal Applied Electric Eields on Solnte Transport in Poiseuille Flows. An Area Averaging Approach, Indnstrial and Engineering Chemistry Research 34, 886,1995. 

To ensure a better separation, molecular sieving will act much better This size exclusion effect will require an ultramicroporous (i.e pore size D < 0.7 nm) membrane Such materials should be of course not only defect-free, but also present a very narrow pore size distribution. Indeed if it is not the case, the large (less separative and even non separative, if Poiseuille flow occurs) pores will play a major role in the transmembrane flux (Poiseuille and Knudsen fluxes vary as and D respectively). The presence of large pores will therefore cancel any sieving effect... 

Figure 7 shows that N2 permeability strongly depends on the pore size. For the macroporous support (curve 1) Poiseuille flow occurs, leading to an increase of the permeance... 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

Travis, K. P., Gubbins, K. E., Poiseuille flow of Lennard-Jones fluids in narrow slit pores, J. Chem. Phys. 112, 4 (2000) 1984-1994. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

Quantitative Visualization of Taylor-Couette-Poiseuille Flows with MRI+... 

Quantitative Visualization ofTaylor-Couette—Poiseuille Flows with MRI 419... 

Taylor-Couette-Poiseuille Flow 4.4.2.1 Fundamental Hydrodynamics... 

Fig. 4.4.2 The discrete data points represent Taylor-Couette-Poiseuille flow regimes observed with MRI for r = 0.5 [41]. The curved boundaries were obtained for r = 0.77 with optical techniques [38]. The two inserts show MRI spin-tagging FLASH images of the SHV and PTV hydrodynamic modes.

Quantitative Visualization ofTaylor-Couette-Poiseuille Flows with MRI 421... 

Fig. 4.4.3 Experimental set-up for the MRI investigation of Taylor-Couette-Poiseuille flow. The electrical motor driving the shaft, as well as the pump, are placed 7.3 m away from the scanner to avoid interference with the magnetic fringe field.

Fig. 4.4.5 Gradual blurring (staring on locations marked by arrow) of MRI spin-tagging spin-echo images of Taylor—Couette—Poiseuille flow as the axial flow is increased (from left to right). The images correspond to longitudinal sections of the flow and the axial flow is upwards. The dashed line marks the location of one of the stationary helical vortices which characterize the SHV mode. This flow regime corresponds to the transition from the SHV (steady) to partial PTV (unsteady) regimes as Re increases, as shown in Figure 4.4.2.

Particle Dispersion in Oscillatory Taylor-Couette-Poiseuille Flow... 

---