Hagen–Poiseuille equation
Hagen–Poiseuille equation
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838[1] and Gotthilf Heinrich Ludwig Hagen,[2] and published by Poiseuille in 1840–41 and 1846.[1] The theoretical justification of the Poiseuille law was given by George Stokes in 1845.[3]
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.
Equation
where:
- Δpis the pressure difference between the two ends,Lis the length of pipe,μis thedynamic viscosity,Qis thevolumetric flow rate,Ris the piperadius.
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 Darcy–Weisbach equation. 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
Relation to Darcy–Weisbach
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, ρ 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 Λ 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 the Poiseuille's law.
The law is also very important in hemorheology and hemodynamics, both fields of physiology.[8]
Poiseuille's law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work.
Derivation
The flow is steady ( ).
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.
- Thepressureforce pushing the liquid through the tube is the change in pressure multiplied by the area:F = −A Δp. This force is in the direction of the motion of the liquid. The negative sign comes from the conventional way we defineΔp = pend− ptop< 0.
- Viscosityeffects will pull from the faster lamina immediately closer to the center of the tube.
- Viscosityeffects will drag from the slower lamina immediately closer to the walls of the tube.
| Elaborate derivation starting directly from first principles |
|---|
Although more lengthy than directly using theNavier–Stokes equations, an alternative method of deriving the Hagen–Poiseuille equation is as follows.Liquid flow through a pipeAssume the liquid exhibitslaminar flow. Laminar flow in a round pipe prescribes that there are a bunch of circular layers (lamina) of liquid, each having a velocity determined only by their radial distance from the center of the tube. Also assume the center is moving fastest while the liquid touching the walls of the tube is stationary (due to theno-slip condition). To figure out the motion of the liquid, all forces acting on each lamina must be known:Viscosity Two fluids moving past each other in thexdirection. The liquid on top is moving faster and will be pulled in the negative direction by the bottom liquid while the bottom liquid will be pulled in the positive direction by the top liquid. Faster laminaAssume that we are figuring out the force on the lamina withradiusr. From the equation above, we need to know theareaof contact and the velocitygradient. Think of the lamina as a ring of radiusr, thicknessdr, and lengthΔx. The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder:A = 2πr Δx. We don't know the exact form for the velocity of the liquid within the tube yet, but we do know (from our assumption above) that it is dependent on the radius. Therefore, the velocity gradient is thechange of the velocity with respect to the change in the radiusat the intersection of these two laminae. That intersection is at a radius ofr. So, considering that this force will be positive with respect to the movement of the liquid (but the derivative of the velocity is negative), the final form of the equation becomesSlower laminaNext let's find the force of drag from the slower lamina. We need to calculate the same values that we did for the force from the faster lamina. In this case, the area of contact is atr + drinstead ofr. Also, we need to remember that this force opposes the direction of movement of the liquid and will therefore be negative (and that the derivative of the velocity is negative).Putting it all togetherTo find the solution for the flow of a laminar layer through a tube, we need to make one last assumption. There is noaccelerationof liquid in the pipe, and byNewton's first law, there is no net force. If there is no net force then we can add all of the forces together to get zero |
Startup of Poiseuille flow in a pipe[9]
with initial and boundary conditions,
The velocity distribution is given by
Poiseuille flow in annular section[10]
Poiseuille flow in annular section
![{\displaystyle {\begin{aligned}u(r)&={\frac {G}{4\mu }}(R_{1}^{2}-r^{2})+{\frac {G}{4\mu }}(R_{2}^{2}-R_{1}^{2}){\frac {\ln(r/R_{1})}{\ln(R_{2}/R_{1})}},\\Q&={\frac {G\pi }{8\mu }}\left[R_{2}^{4}-R_{1}^{4}-{\frac {(R_{2}^{2}-R_{1}^{2})^{2}}{\ln R_{2}/R_{1}}}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df04125cec4c29798d3e1a970da7ce8ae7821a04)
Poiseuille flow in a pipe with oscillating pressure gradient
![{\displaystyle u(r,t)={\frac {G}{4\mu }}(R^{2}-r^{2})+[\alpha F_{2}+\beta (F_{1}-1)]{\frac {\cos \omega t}{\rho \omega }}+[\beta F_{2}+\alpha (F_{1}-1)]{\frac {\sin \omega t}{\rho \omega }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1a90f753b2628525978d5656a0255cbf07feeb)
where
Plane Poiseuille flow
Plane Poiseuille flow
with no-slip condition on both walls
Therefore, the velocity distribution and the volume flow rate per unit length are
Poiseuille flow through some non-circular cross-sections[15]
![{\displaystyle {\begin{aligned}u(y,z)&={\frac {G}{2\mu }}(y+z)(\pi -y)-{\frac {G}{\pi \mu }}\sum _{n=1}^{\infty }{\frac {1}{\beta _{n}^{3}\sinh(2\pi \beta _{n})}}\{\sinh[\beta _{n}(2\pi -y+z)]\sin[\beta _{n}(y+z)]-\sinh[\beta _{n}(y+z)]\sin[\beta _{n}(y-z)]\},\quad \beta _{n}=n-{\frac {1}{2}},\\Q&={\frac {G\pi ^{4}}{12\mu }}-{\frac {G}{2\pi \mu }}\sum _{n=1}^{\infty }{\frac {1}{\beta _{n}^{5}}}[\coth(2\pi \beta _{n})+\csc(2\pi \beta _{n})].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a01fa73e4464a440e1e6022b2ca2b50568cc4fd)
Poiseuille flow through arbitrary cross-section
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 compressible fluids
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.
Electrical circuits analogy
- .
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 = 1.6×10−19 C.) 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 current I. Therefore, I = nQq*. Consequently, Q = I/nq*, and
This is exactly Ohm's law, where the resistance R = 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 = πr2 of the resistor, which is wrong according to the electrical analogy. The correct relation is
where ρ is the specific resistance; i.e. the resistance R is inversely proportional to the cross section area S of the resistor.[21] The reason why Poiseuille's law leads to a wrong 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 flow rate of intravenous 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 to the radius. Peripheral IV cannulas are typically available as (from large to small) 14G, 16G, 18G, 20G, 22G. As an example, the flow of a 14G cannula is typically 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, the change in pressure can be used to speed up flow rate by pressurizing the bag of fluid, squeezing the bag, or hanging the bag higher from the level of the cannula. It is also useful to understand that viscous fluids will flow slower (e.g. in blood transfusion).
See also
Couette flow
Darcy's law
Pulse
Wave
Hydraulic circuit