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Law describing the pressure drop in an incompressible and Newtonian fluid
The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen-Poiseuille equation.
Poiseuille's Equation describes the pressure drop due to the viscosity of the fluid; Other types of pressure drops may still occur in a fluid (see a demonstration here). For example, the pressure needed to drive a viscous fluid up against gravity would contain both that as needed in Poiseuille's Law plus that as needed in Bernoulli's equation, such that any point in the flow would have a pressure greater than zero (otherwise no flow would happen).
Another example is when blood flows into a narrower constriction, its speed will be greater than in a larger diameter (due to continuity of volumetric flow rate), and its pressure will be lower than in a larger diameter (due to Bernoulli's equation). However, the viscosity of blood will cause additional pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's Law). Both effects contribute to the actual pressure drop.
The equation does not hold close to the pipe entrance.:3
The equation fails in the limit of low viscosity, wide and/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the Darcy-Weisbach equation. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen-Poiseuille law to be valid. If the pipe is too short, the Hagen-Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle, under less restrictive conditions, by
because it is impossible to have less-than-zero (absolute) pressure (not to be confused with gauge pressure) in an incompressible flow.
Relation to the Darcy-Weisbach equation
Normally, Hagen-Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic. However, the result for the pressure drop can be extended to turbulent flow by inferring an effective turbulent viscosity in the case of turbulent flow, even though the flow profile in turbulent flow is strictly speaking not actually parabolic. In both cases, laminar or turbulent, the pressure drop is related to the stress at the wall, which determines the so-called friction factor. The wall stress can be determined phenomenologically by the Darcy-Weisbach equation in the field of hydraulics, given a relationship for the friction factor in terms of the Reynolds number. In the case of laminar flow, for a circular cross section:
where Re is the Reynolds number, ? is the fluid density, and v is the mean flow velocity, which is half the maximal flow velocity in the case of laminar flow. It proves more useful to define the Reynolds number in terms of the mean flow velocity because this quantity remains well defined even in the case of turbulent flow, whereas the maximal flow velocity may not be, or in any case, it may be difficult to infer. In this form the law approximates the Darcy friction factor, the energy (head) loss factor, friction loss factor or Darcy (friction) factor? in the laminar flow at very low velocities in cylindrical tube. The theoretical derivation of a slightly different form of the law was made independently by Wiedman in 1856 and Neumann and E. Hagenbach in 1858 (1859, 1860). Hagenbach was the first who called this law Poiseuille's law.
The radial and azimuthal components of the fluid velocity are zero ( ).
The flow is axisymmetric ( ).
The flow is fully developed ( ). Here However, this can be proved via mass conservation, and the above assumptions.
Then the angular equation in the momentum equations and the continuity equation are identically satisfied. The radial momentum equation reduces to , i.e., the pressure is a function of the axial coordinate only. For brevity, use instead of . The axial momentum equation reduces to
where is the dynamic viscosity of the fluid. In the above equation, the left-hand side is only a function of and the right-hand side term is only a function of , implying that both terms must be the same constant. Evaluating this constant is straightforward. If we take the length of the pipe to be and denote the pressure difference between the two ends of the pipe by (high pressure minus low pressure), then the constant is simply defined such that is positive. The solution is
Since needs to be finite at , . The no slip boundary condition at the pipe wall requires that at (radius of the pipe), which yields Thus we have finally the following parabolicvelocity profile:
The maximum velocity occurs at the pipe centerline (), . The average velocity can be obtained by integrating over the pipe cross section,
The easily measurable quantity in experiments is the volumetric flow rate . Rearrangement of this gives the Hagen-Poiseuille equation
Startup of Poiseuille flow in a pipe
When a constant pressure gradient is applied between two ends of a long pipe, the flow will not immediately obtain Poiseuille profile, rather it develops through time and reaches the Poiseuille profile at steady state. The Navier-Stokes equations reduce to
Plane Poiseuille flow is flow created between two infinitely long parallel plates, separated by a distance with a constant pressure gradient is applied in the direction of flow. The flow is essentially unidirectional because of infinite length. The Navier-Stokes equations reduce to
Therefore, the velocity distribution and the volume flow rate per unit length are
Poiseuille flow through some non-circular cross-sections
Joseph Boussinesq derived the velocity profile and volume flow rate in 1868 for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section.Joseph Proudman derived the same for isosceles triangles in 1914. Let be the constant pressure gradient acting in direction parallel to the motion.
The velocity and the volume flow rate in a rectangular channel of height and width are
The velocity and the volume flow rate of tube with equilateral triangular cross-section of side length are
The velocity and the volume flow rate in the right-angled isosceles triangle are
The velocity distribution for tubes of elliptical cross-section with semi-axis and is
Here, when , Poiseuille flow for circular pipe is recovered and when , plane Poiseuille flow is recovered. More explicit solutions with cross-sections such as snail-shaped sections, sections having the shape of a notch circle following a semicircle, annular sections between homofocal ellipses, annular sections between non-concentric circles are also available, as reviewed by Ratip Berker [tr; de].
Poiseuille flow through arbitrary cross-section
The flow through arbitrary cross-section satisfies the condition that on the walls. The governing equation reduces to
If we introduce a new dependent variable as
then it is easy to see that the problem reduces to that integrating a Laplace equation
satisfying the condition
on the wall.
Poiseuille's equation for an ideal isothermal gas
For a compressible fluid in a tube the volumetric flow rate (but not the mass flow rate) and the axial velocity are not constant along the tube. The flow is usually expressed at outlet pressure. As fluid is compressed or expands, work is done and the fluid is heated or cooled. This means that the flow rate depends on the heat transfer to and from the fluid. For an ideal gas in the isothermal case, where the temperature of the fluid is permitted to equilibrate with its surroundings, an approximate relation for the pressure drop can be derived. Using ideal gas equation of state for constant temperature process, the relation can be obtained. Over a short section of the pipe, the gas flowing through the pipe can be assumed to be incompressible so that Poiseuille law can be used locally,
Here we assumed the local pressure gradient is not too great to have any compressibility effects. Though locally we ignored the effects of pressure variation due to density variation, over long distances these effects are taken into account. Since is independent of pressure, the above equation can be integrated over the length to give
Hence the volumetric flow rate at the pipe outlet is given by
This equation can be seen as Poiseuille's law with an extra correction factor p1 + p2/2p2 expressing the average pressure relative to the outlet pressure.
Electricity was originally understood to be a kind of fluid. This hydraulic analogy is still conceptually useful for understanding circuits. This analogy is also used to study the frequency response of fluid-mechanical networks using circuit tools, in which case the fluid network is termed a hydraulic circuit. Poiseuille's law corresponds to Ohm's law for electrical circuits, V = IR. Since the net force acting on the fluid is equal to , where S = ?r2, i.e. ?F = ?r2 ?P, then from Poiseuille's law, it follows that
For electrical circuits, let n be the concentration of free charged particles (in m-3) and let q* be the charge of each particle (in coulombs). (For electrons, q* = e = .) Then nQ is the number of particles in the volume Q, and nQq* is their total charge. This is the charge that flows through the cross section per unit time, i.e. the currentI. Therefore, I = nQq*. Consequently, Q = I/nq*, and
But ?F = Eq, where q is the total charge in the volume of the tube. The volume of the tube is equal to ?r2L, so the number of charged particles in this volume is equal to n?r2L, and their total charge is Since the voltageV = EL, it follows then
This is exactly Ohm's law, where the resistanceR = V/I is described by the formula
It follows that the resistance R is proportional to the length L of the resistor, which is true. However, it also follows that the resistance R is inversely proportional to the fourth power of the radius r, i.e. the resistance R is inversely proportional to the second power of the cross section area S = ?r2 of the resistor, which is different from the electrical formula. The electrical relation for the resistance is
where ? is the resistivity; i.e. the resistance R is inversely proportional to the cross section area S of the resistor. The reason why Poiseuille's law leads to a different formula for the resistance R is the difference between the fluid flow and the electric current. Electron gas is inviscid, so its velocity does not depend on the distance to the walls of the conductor. The resistance is due to the interaction between the flowing electrons and the atoms of the conductor. Therefore, Poiseuille's law and the hydraulic analogy are useful only within certain limits when applied to electricity. Both Ohm's law and Poiseuille's law illustrate transport phenomena.
Medical applications - intravenous access and fluid delivery
The Hagen-Poiseuille equation is useful in determining the vascular resistance and hence flow rate of intravenous (IV) fluids that may be achieved using various sizes of peripheral and central cannulas. The equation states that flow rate is proportional to the radius to the fourth power, meaning that a small increase in the internal diameter of the cannula yields a significant increase in flow rate of IV fluids. The radius of IV cannulas is typically measured in "gauge", which is inversely proportional[dubious – discuss] to the radius. Peripheral IV cannulas are typically available as (from large to small) 14G, 16G, 18G, 20G, 22G, 26G. As an example, the flow of a 14G cannula is typically around twice that of a 16G, and ten times that of a 20G. It also states that flow is inversely proportional to length, meaning that longer lines have lower flow rates. This is important to remember as in an emergency, many clinicians favor shorter, larger catheters compared to longer, narrower catheters. While of less clinical importance, an increased change in pressure (?p) -- such as by pressurizing the bag of fluid, squeezing the bag, or hanging the bag higher (relative to the level of the cannula) -- can be used to speed up flow rate. It is also useful to understand that viscous fluids will flow slower (e.g. in blood transfusion).
^István Szabó, ;;Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen, Basel: Birkhäuser Verlag, 1979.
^Stokes, G. G. (1845). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Transactions of the Cambridge Philosophical Society, 8, 287-341
^Uchida, S. (1956). "The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe". Zeitschrift für angewandte Mathematik und Physik. 7 (5): 403-422. doi:10.1007/BF01606327.
^Boussinesq, Joseph (1868). "Mémoire sur l'influence des Frottements dans les Mouvements Réguliers des Fluids". J. Math. Pures Appl. 13 (2): 377-424.