Poiseuille’s law

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. 

A simple starting point for such a discussion is Poiseuille s law [42] for the flow of a Newtonian liquid of viscosity in a tube of radius r under the influence of a pressure P ... 

Because pore sizes in the cake and filter medium are small, and the liquid velocity through the pores is low, the filtrate flow may be considered laminar hence, Poiseuille s law is applicable. Filtration rate is directly proportional to the difference in pressure and inversely proportional to the fluid viscosity and to the... 

If a solution tends to be independent of shear, then the measurement of viscosity (q) is based on Poiseuille s law can be made easy by grouping all those terms related to a specific viscometer as a calibration constant A. 

The flow rate inside a capillary may be obtained from Poiseuille s law, which relates the liquid flow rate, dVIdt (V = volume, t = time), to the difference of pressure AP pushing the liquid inside the tube and to the dimensions of the tube, in particular, its radius, r. 

Filtration of Liquids Depending on the specific electrochemical reactor type, the filtration rate of a liqnid electrolyte throngfi tfie separator should be either high (to secure a convective snpply of snbstances) or very low (to prevent mixing of the anolyte and catholyte). The filtration rate that is attained under the effect of an external force Ap depends on porosity. For a separator model with cylindrical pores, the volnme filtration rate can be calcnlated by Poiseuille s law ... 

The on-line viscosimeters currently available are adaptations of the classical dilute solution capillary viscosimeters. They work on the principle of measuring the pressure drop across a capillary with a differential pressure transducer. The pressure drop can be related to the reduced or inherent viscosity of the sample via Poiseuille s law.84 Intrinsic viscosity is determined using the equation ... 

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

The velocity profile during slip flow in a cylindrical tube is shown in Figure 21. As in conventional fluid flow, the flow velocity in the z direction, u(r), is parabolic, but rather than reach zero at the tube wall, slip occurs, and the velocity at the wall is greater than zero. The velocity does not reach zero until distance h from the wall surface. The derivation of the mass flux equation proceeds along the same lines as the derivation of Poiseuille s law in conventional hydrodynamics, but in slip flow, u(r) = 0 at r = a + h instead of reaching zero at r = a. 

Ohm s law may be rewritten to include the three factors that affect vascular resistance blood viscosity (q), vessel length (L), and vessel radius (r). The following equation is known as Poiseuille s law ... 

A reduction in diameter of arterioles is of major importance in control of blood pressure, since resistance is directly proportional to the fourth power of the radius (Poiseuille s law), i.e. a change in radius, therefore, produces a massive change in blood flow. 

A low flow rate of mobile phase minimises the dispersion of compounds, which occurs inevitably throughout the instrument. Detection limits are also improved. The laminar flow within the column follows Poiseuille s law, the velocity of the mobile phase being at its maximum in the centre of the tube and zero at the wall. 

The application of the hydrostatic pressure to the solvent of gel causes the permeation flow of the solvent. In this process, the gel behaves as a molecular sieve and imposes a frictional resistance on the flowing water. The permeation flow of water through a gel is a process analogous to the capillary flow of a viscous fluid. The frictional resistance of a single capillary is well described by the Hagen-Poiseuille s law by which the relationship between the dimension of the capillary, the applied pressure, and the flow rate is given as follows... 

The Ostwald U-tube instrument is the most common type of viscometer based upon this principle (Fig. 6.37a and Table 6.7). The viscometer is filled with liquid until the liquid level is such that the bottom of the meniscus in the right-hand limb coincides with the mark C. The fluid is drawn up the left-hand limb to a level about 5 mm above A and then released. The time t taken for the bottom of the meniscus to fall from A to B is recorded. The dynamic viscosity p is determined from Poiseuille s law (Volume 1, equation 3.30 and Section 9.4.3) which under the... 

Some instruments are available for process on-line application (Table 6.7). Capillary types give good results with Newtonian fluids and are based upon Poiseuille s law (Volume 1, equation 3.30). The pressure drop across the capillary... 

With open tubular columns, permeability is given by Poiseuille s law, B = dll32, where dc is the column internal diameter. 

Although reverse osmosis, ultrafiltration and microfiltration are conceptually similar processes, the difference in pore diameter (or apparent pore diameter) produces dramatic differences in the way the membranes are used. A simple model of liquid flow through these membranes is to describe the membranes as a series of cylindrical capillary pores of diameter d. The liquid flow through a pore (q) is given by Poiseuille s law as ... 

If the pores of a microporous membrane are 0.1 xm or larger, gas permeation will take place by normal convective flow described by Poiseuille s law. As... 

This is for the shell at r. The total volumetric flow is v times the cross-sectional area of the shell integrated from 0 to R (Poiseuille s law) ... 

The laminar stationary flow of an incompressible viscous liquid through cylindrical tubes can be described by Poiseuille s law this description was later extended to turbulent flow. Flowing patterns of two immiscible phases are more complex in microcapillaries various patterns of liquid-liquid flow are described in more detail in Chapter 4.3, while liquid-gas flow and related applications are discussed in Chapter 4.4. 

The volumetric flow rate of laminar Newtonian flow of a liquid through a capillary tube is expressed by Poiseuille s law as ... 

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