Poiseuille equation

Hagen-Poiseuille equation (Poiseuille equation) n. The equation of steady, laminar, Newtonian flow through circular tubes ... 

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]... 

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... 

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. 

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 ... 

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. 

Thus he modified Poiseuille s equation (Eq. 23) to describe the penetration of such a liquid into a closed pore of length L ... 

In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is ... 

Equation (4.44) is the Hagen-Poiseuille law, which shows that the pressure loss during laminar flow is linearly proportional to the flow velocity. 

For laminar flow in a circular tube of radius R, the pressure gradient is given by a differential form of the Poiseuille equation ... 

This equation has the same functional dependence on p (namely none) and m as the Poiseuille equation that governs laminar flow in an empty tube. Thus, laminar flow packed beds scale in series exactly like laminar flow in empty tubes. See the previous sections on series scaleup of liquids and gases in laminar flow. 

Solution of Equation (8.63) for the case of constant viscosity gives the parabolic velocity profile. Equation (8.1), and Poiseuille s equation for pressure drop. Equation (3.14). In the more general case of /r = /r(r), the velocity profile and pressure drop are determined numerically. 

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

In the capillary method, the time required for a liquid to flow through a capillary tube is determined. The melt under investigation flows with a constant rate through a tube with a small, definite cross-sectional area, such as a cylindrical capillary. The viscosity can be measured in an absolute way from the pressure drop. This method can yield the most reliable absolute data, the viscosity being given by a modified Hagen-Poiseuille equation ... 

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 ... 

Streaming Potential When the solution is forced through the porous solid under the effect of an external pressure P, the character of liquid motion in the cylindrical pores will be different from that in electroosmotic transport. Since the external pressure acts uniformly on the full pore cross section, the velocity of the liquid will be highest in the center of the pore, and it will gradually decrease with decreasing distance from the pore walls (Fig. 31.5). The velocity distribution across the pore is quantitatively described by the Poiseuille equation. 

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... 

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 ... 

Cf., Poiseuille s equation, according to which the fluid velocity through a capillary is linearly dependent on the pressure gradient along it. 

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