Viscosity Hagen-Poiseuille law

Notice that Eq. (7.72) reduces to the Hagen-Poiseuille Law for n = 1, where K = fi, the Newtonian viscosity. 

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 Otherwise, a transition to turbulent flow occurs, and Equation 9.11 is no longer valid.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). 

From the above relationships it can also be shown that the pressure drop AP (Pa) in the laminar flow of a Newtonian fluid of viscosity ji (Pas) through a straight, round tube of diameter d (m) and length I (m) at an average velocity ofv (m s 1) is given by Equation 2.9, which expresses the Hagen-Poiseuille law ... 

This is the Hagen-Poiseuille law for laminar flow in tubes. Its conversion to a form applicable to experimental measurements will be given later. For the present we note that the quantities R, aP and I can all be obtained by direct measurement and Eqn 4-8 therefore can be used in the absolute experimental method for the determination of the viscosity of a liquid in any physically rational system of units. When the... 

The viscosity of a liquid or solution can be measured by using a viscometer whose design is based on the Hagen-Poiseuille law. Essentially, this involves the measurement of the flow rate of the liquid through a capillary tube which is part of the viscometer. Consequently, by measuring the flow time of the solution, t, and that of the pure solvent, to, the relative viscosity can be determined ... 

This indicates a departure in the opposite sense from the earlier one, i.e. in the direction that, at infinitely small shearing forces the Newtonian behavior appears as the limiting value, while the solution becomes increasingly more fluid and mobile with increasing r. Here, also, the coefficient of viscosity, calculated formally from the Hagen-Poiseuille law, decreases with increasing shearing stress but in an entirely different way from that in elastic liquids. 

However, in viscosimetry using a capillary viscosimeter normally the viscosity is not calculated from the maximum shear rate and the shear stress but rather from the Hagen-Poiseuille-Law (Eq. 3.1). 

In most cases viscosity is measured by capillary viscometers or rotating viscometers. In a capillary viscometer one measures the pressure drop by means of constant laminar flow in a capillary the constant flow can be achieved by a pump and the pressure drop is obtained by a differential pressure transmitter whose plus and minus sides are connected to the capillary. The pressure drop is then directly proportional to the viscosity according to the Hagen-Poiseuille law [4, 11] [Eq. (30), where p is the viscosity, r is the capillary radius, I is the capillary length, Ap is the pressure drop, and is the mass flow rate]. The capillary viscometer may also be employed in-line for monitoring of molecular weight in polymerizations, as described in Ref. 14. 

The pressure required for this process is determined by two effects. The first one is the pressure drop resulting from the flow of the disperse phase through the pores. It can be estimated by the Hagen-Poiseuille law (Equation 13.1) [8]. The pressure drop, Ap, depends on the viscosity of the disperse phase, p, the diameter and length of the pores, dp and Ip, and the flow velocity, v, in the pores ... 

Doolittle s torsional viscosimeter was essentially a damped, oscillating Couette viscometer, and Doolittle chose as the unit of viscosity the "number of degrees of retardation between the first and second complete arcs." There is no mention In this paper of the Hagen-Poiseuille law, the Navier-Stokes equations, the treatise of Lamb (1879) or the work of Basset (1888), and it would appear that Doolittle was unaware of the studies of fluid mechanics carried out in the previous century and half. Nearly two decades later, Gillet (1909) began his paper entitled "Analysis and Friction Tests of Lubricating Greases" with the comment ... 

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... 

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... 

In order to define the effective viscosity for a power-law fluid, the expression for the viscosity of a Newtonian fluid is extended to a non-Newtonian fluid. For a Newtonian fluid, rearranging the Hagen-Poiseuille equation gives... 

B) The Hagen-Poiseuille (pronounced Pwah-z0-yah) law for the volumetric flow rate, Q, of a fluid with viscosity /x through a pipe of radius r,... 

Two hundred years after the early contributions of Newton and Hooke, various laws of real fluids emerged as well as a quantitative description of flow and the measurement of viscosity, including the work of Euler, Cauchy, Coulomb, Poiseuille, Hagen, Couette, Reynolds, and Bingham. In 1890, the first rotational rheometer was invented by Couette. In 1929, Reiner and Bingham founded the first rheological society. 

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