Hagen-Poiseuille s law

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... 

Experimentally, the viscosity of dilute polymer solutions is, in most cases, determined with glass capillary viscometers, making application of the Hagen-Poiseuille s law for laminar flow of liquids. The time required for a specific volume of a liquid to flow through a capillary of... 

Let A be its half spatial dimension in the cross-section of the tube. Again a Cartesian coordinate system is used for the description where x points in the direction of the width and y does in the direction of the length of the tube. Equation 7.51 is now replaced by Hagen-Poiseuille s law that describes the flow rate Qy in the tube ... 

The effective pore size of the PNA membranes at any pH and temperature can be calculated using a simple Hagen-Poiseuille s law and the ratio of flux or permeability coefficient of the virgin and graft membrane. 

The fluid flow in the capillary is given by Hagen-Poiseuille s law ... 

Schon and Georgi (2003) developed a capillary-based model for dispersed shaly sand that shows an analogy to the Waxman-Smits equation (Section 8.5.3) for electrical properties. This model (Fig. 2.32) accounts for the reduction in porosity and decrease in the pore cross-sectional area with the content of clay and the associated immobile water. Starting with Hagen-Poiseuille s law, the flow rate for a cross-section is reduced by a film of clay particles. Permeability for dispersed shaly sand can be written as a functimi of the clean sand permeability sd and the dispersed shale content Fsh... 

Equation 9.11 is usually referred to as Poiseuille s law and sometimes as the Hagen-Poiseuille law. It assumes that the fluid in the cylinder moves in layers, or laminae, with each layer gliding over the adjacent one (Fig. 9-14). Such laminar movement occurs only if the flow is slow enough to meet a criterion deduced by Osborne Reynolds in 1883. Specifically, the Reynolds number Re, which equals vd/v (Eq. 7.19), must be less than 2000 (the mean velocity of fluid movement v equals JV, d is the cylinder diameter, and v is the kinematic viscosity). Otherwise, a transition to turbulent flow occurs, and Equation 9.11 is no longer valid. Due to frictional interactions, the fluid in Poiseuille (laminar) flow is stationary at the wall of the cylinder (Fig. 9-14). The speed of solution flow increases in a parabolic fashion to a maximum value in the center of the tube, where it is twice the average speed, Jv. Thus the flows in Equation 9.11 are actually the mean flows averaged over the entire cross section of cylinders of radius r (Fig. 9-14). 

Jean Louis Marie Poiseuille (1799-1869), a French physician and physiologist, investigated the flow of human blood in narrow tubes. In 1828 he earned his doctoral degree with a dissertation entitled Recherches sur la force du coeur aortique. In the 1840s, he formulated Poiseuille s law, today named the Hagen-Poiseuille law in order to honor also Cotthilf... 

More properly called the Hagen-Poiseuille law, it was developed independently by Gotthilf Heinrich Ludwig Hagen (1797-1884) and Jean Louis Marie Poiseuille. Poiseuille s law was experimentally derived in 1838 and formulated and published in 1840 and 1846 by Jean Louis Marie Poiseuille (1797-1869). Hagen also carried out experiments in 1839. While there are a number of derivations, we follow a simple one here from Physical Chemistry by Castellan [5]. 

From the above relationships it can also be shown that the pressure drop AP (Pa) in the laminar flow of a Newtonian fluid ofviscosity // (Pa s) through a straight round tube of diameter d (m) and length L (m) at an average velocity of v (ms ) is given by Equation 2.9, which expresses the Hagen-Poiseuille law ... 

With the aid of the straight capillaric model and according to the Hagen-Poiseuille equation and Darcy s law, a is proportional to the viscosity of the fluid. Thus, Eq. (5.322) becomes... 

In the simplest situation of a flow through a straight cylindrical pore, Darcy s law, based on the Hagen-Poiseuille equation, describes the process with the expression [73]... 

K being a constant, the fluidity ( ). Bingham does not think the name Hagen-Poiseuille law should be used, since Poiseuille published his first paper on the circulation of blood ten years before Hagen s paper appeared, and worked steadily on the subject, publishing a series of papers until 1847. Newton and Stokes and others antedate Poiseuille and have a greater claim than Hagen. ... 

Hagen-Poiseuille law, 72 Hagenbach coefficient, 74 correction, 73, 75 hanging drop, 183, 189 level viscometer, 80 Hare s apparatus, 12 Harkins s equation, 155 heat capacity of electrolyte solution, 225 content of electrolyte and non-electrolyte solutions, 225-6 content of vapour, 348 Heilborn s specific heat formula, 218 Henning s latent heat formula, 307 Herwig s method for density of saturated vapour, 325... 

The Hagen-Poiseuille law is mathematically analogous to the Ohm s Law. In addition, the conservation of mass (or flow for incompressible fluid) of fluid is analogous to the law of conservation of charge and current in electrical systems. This analogy allows for the use of Kirchoffs equations for calculation of the distribution of the volumetric flow of liquid between channels in a microfluidic network once we know the resistances of all the channels in the network and the pressures at the inlet and outlet, we can calculate the speed of flow in any part of the network (Fig. 1). 

Considering the Hagen-Poiseuille formula we find that the departures from the behavior expected from Newton s law can be represented, if we substitute for the Newtonian expression a more general relation between velocity gradient and shear stress. Many workers have endeavored to find an expression which would satisfy experimental facts. Rabino-witsch, Reiner and Weissenberg, particularly, have interested themselves recently in the anomalous behavior of viscous liquids. 

For efificient viscosimetric measurements, only ISO cups (ISO 2431) should be used. The length of the capillary is 20 mm. The shear rates lie—for efflux times f=30-100 s—between y=60 and 1500 s" They depend on the capillary radius Ry which can be varied through interchangeable outlet nozzles. The calculation of the mean shear rate is made using the Hagen-Poiseuille-Law ... 

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