Poiseuille

The surface viscosity can be measured in a manner entirely analogous to the Poiseuille method for liquids, by determining the rate of flow of a film through a narrow canal under a two-dimensional pressure difference Ay. The apparatus is illustrated schematically in Fig. IV-7, and the corresponding equation for calculating rj is analogous to the Poiseuille equation [99,100]... 

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... 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

This is Che required boundary condition for the mass mean velocity, Co be applied at the tube surface r = a. With a non-vanishing value for v (a), Che Poiseuille solution (4.5) must now be replaced by the simple modification. 

In pneumatic nebulizers, the relative velocity of gas and liquid first induces a reduction in pressure above the surface of the liquid (see the calculation in Figure 19.4). The reduction in pressure is sufficient to cause liquids to flow out of capillary tubes, in accord with Poiseuille s formula (Figure 19.5). As the relative velocity of a liquid and a gas increases — particularly if the mass of liquid is small — this partial vacuum and rapid flow cause the surface of the liquid to be broken into droplets. An aerosol is formed. 

Using Poiseuille s formula, the calculation shows that for concentric-tube nebulizers, with dimension.s similar to those in use for ICP/MS, the reduced pressure arising from the relative linear velocity of gas and liquid causes the sample solution to be pulled from the end of the inner capillary tube. It can be estimated that the rate at which a sample passes through the inner capillary will be about 0.7 ml/min. For cross-flow nebulizers, the flows are similar once the gas and liquid stream intersection has been optimized. 

In Chap. 9 we shall examine the flow of a solution through a capillary tube. The rate of volume delivery in that case is given by Poiseuille s law [Eq. (9.29)], which states that the time required for a constant volume of liquid to drain out of the capillary is proportional to r jp. Accordingly, the viscosity is proportional to the product pt, and when the delivery times for two liquids are compared in the same capillary. 

Some exceptionally gifted researchers have made contributions to the topics we discuss here, and their impact on these and other areas of inquiry have earned them special recognition. Specifically, we note that Einstein,. Svedberg, Staudinger, and Flory have all been awarded the Nobel Prize, and Stokes and Poiseuille have been honored by having units named after them. These are but a few of the superstars whose work we encounter in this chapter. 

This result is called the Poiseuille equation, after Poiseuille, who discovered this fourth-power dependence of flow rate on radius in 1844. The poise unit of viscosity is also named after this researcher. The following example illustrates the use of the Poiseuille equation in the area where it was first applied. 

Poiseuille was a physician-physiologist interested in the flow of blood through blood vessels in the body. Estimate the viscosity of blood from the fact that blood passes through the aorta of a healthy adult at rest at a rate of about 84 cm sec , with a pressure drop of about 0.98 mmHg m". Use 9 mm as the radius of the aorta for a typical human. 

The Poiseuille equation provides a method for measuring 77 by observing the time required for a liquid to flow through a capillary. The apparatus shown in Fig. 9.6 is an example of one of many different instruments designed to use this relationship. In such an experiment the time required for the meniscus to drop... 

Experiments based on the Poiseuille equation make intrinsic viscosity an easily measured parameter to characterize a polymer. In the next section we consider how this property can be related to the molecular weight of a polymer. 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

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. 

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation... 

Simplified theoiy for both batch and continuous filtration is based on the time-honored Hagen-Poiseuille equation ... 

For example, consider an open tubular column with the dimensions previously defined, operated at constant mass flow rate of helium (which is normal for temperature programming purposes), then, from Poiseuille s equation, after time (t),... 

The radius of the column cannot be reduced indefinitely due to the pressure difference that will increase as (R) is reduced. From Poiseuille s equation the pressure drop across the tube is given by... 

The length of the column is also defined by the Poiseuille equation that describes the flow of a fluid through an open tube in terms of the tube radius, the pressure applied across the tube (column), the viscosity of the fluid and the linear velocity of the fluid. Thus, for a compressible fluid. 

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