Hagen–Poiseuille equation
In fluid dynamics, the Hagen–Poiseuille equation is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe. The assumptions of the equation are that the flow is laminar, viscous and incompressible and the flow is through a constant circular cross-section that is substantially longer than its diameter. It is also assumed that there is no acceleration of liquid in the pipe. The equation is also known as the Hagen–Poiseuille law, Poiseuille law and Poiseuille equation.
The fluid flow will be turbulent for velocities and pipe diameters above a threshold, leading to larger pressure drops than would be expected according to the Hagen–Poiseuille equation.
Contents
Equation
Standard fluid dynamics notation
In standard fluid dynamics notation:[1][2] \[ \Delta P = \frac{8 \mu L Q}{ \pi r^4} \] or \[ \Delta P = \frac{128 \mu L Q}{ \pi d^4}\]
where: \[\Delta P \] is the pressure drop \[L\] is the length of pipe \[ \mu \] is the dynamic viscosity \[Q\] is the volumetric flow rate \[r\] is the radius \[d\] is the diameter
Physics notation
\[ \Phi = \frac{dV}{dt} = v \pi R^{2} = \frac{\pi R^{4}}{8 \eta} \left( \frac{- \Delta P}{\Delta x}\right) = \frac{\pi R^{4}}{8 \eta} \frac{ |\Delta P|}{L} \]
where: \[ \Phi \] is the volumetric flow rate (denoted as \(Q\) above) \[V\] is a volume of the liquid poured (cubic meters) \[t\] is the time (seconds) \[v\] is mean fluid velocity along the length of the tube (meters/second) \[x\] is a distance in direction of flow (meters) \[R\] is the internal radius of the tube (meters) \[\Delta P \] is the pressure difference between the two ends (pascals) \[\eta \] is the dynamic fluid viscosity (pascal-second (Pa·s)), \[L\] is the total length of the tube in the x direction (meters).
The equation does not hold close to the pipe entrance.[3]:3
Relation to Darcy–Weisbach
This result is also a solution to the phenomenological Darcy–Weisbach equation in the field of hydraulics, given a relationship for the friction factor in terms of the Reynolds number:
\[ \Lambda = {64\over {\it \mathrm{Re}}} \; , \quad\quad \mathrm{Re} = {2\rho v r\over \eta} \; , \]
where Re is the Reynolds number and ρ fluid density. 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 specially in hemorheology and hemodynamics, both fields of physiology.[4]
The Poiseuilles' law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work.
Derivation
The Hagen–Poiseuille equation can be derived from the Navier–Stokes equations.
Viscosity
The derivation of Poiseuille's law is surprisingly simple, but it requires an understanding of viscosity. When two layers of liquid in contact with each other move at different speeds, there will be a shear force between them. This force is proportional to the area of contact A, the velocity gradient in the direction of flow \({\Delta v_x}/{\Delta y}\), and a proportionality constant η (viscosity) and is given by
\[ F_{\text{viscosity, top}} = - \eta A \frac{\Delta v_x}{\Delta y}.\]
The negative sign is in there because we are concerned with the faster moving liquid (top in figure), which is being slowed by the slower liquid (bottom in figure). By Newton's third law of motion, the force on the slower liquid is equal and opposite (no negative sign) to the force on the faster liquid. This equation assumes that the area of contact is so large that we can ignore any effects from the edges and that the fluids behave as Newtonian fluids.
Liquid flow through a pipe
In a tube we make a basic assumption: the liquid in the center is moving fastest while the liquid touching the walls of the tube is stationary (due to friction).
To figure out the motion of the liquid, we need to know all forces acting on each lamina:
- The force pushing the liquid through the tube is the change in pressure multiplied by the area\[F = -\Delta P A\]. This force is in the direction of the motion of the liquid - the negative sign comes from the conventional way we define \(\Delta P = P_{end}-P_{top} < 0 \).
- The pull from the faster lamina immediately closer to the center of the tube
- The drag from the slower lamina immediately closer to the walls of the tube.
The first of these forces comes from the definition of pressure. The other two forces require us to modify the equations above that we have for viscosity. In fact, we are not modifying the equations, instead merely plugging in values specific to our problem. Let's focus on the pull from the faster lamina (#2) first.
Faster lamina
Assume that we are figuring out the force on the lamina with radius \(s\). From the equation above, we need to know the area of contact and the velocity gradient. Think of the lamina as a ring of radius \(s\) and thickness \(ds\). The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder\[A = 2 \pi s \Delta x\] . 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 the change of the velocity with respect to the change in the radius at the intersection of these two laminae. That intersection is at a radius of \(s\). 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 becomes
\[ F_{\text{viscosity, fast}} = - \eta 2 \pi s \Delta x \left . \frac{dv}{dr} \right \vert_s \]
where the vertical bar and subscript s following the derivative indicates that it should be taken at a radius of \(s\).
Slower lamina
Next 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 at s+ds instead of s. 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).
\[ F_{\text{viscosity, slow}} = \eta 2 \pi (s+ds) \Delta x \left . \frac{dv}{dr} \right \vert_{s+ds} \]
Putting it all together
To find the solution for the flow of liquid through a tube, we need to make one last assumption. There is no acceleration of liquid in the pipe, and by Newton'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
\[ 0 = F_{\text{pressure}} + F_{\text{viscosity, fast}} + F_{\text{viscosity, slow}} \]
or
\[ 0 = - \Delta P2 \pi sds - \eta 2 \pi s \Delta x \left . \frac{dv}{dr} \right \vert_s + \eta 2 \pi (s+ds) \Delta x \left . \frac{dv}{dr} \right \vert_{s+ds}. \]
First, to get everything happening at the same point, use the first two terms of a Taylor series expansion of the velocity gradient:
\[ \left . \frac{dv}{dr} \right \vert_{r+dr} = \left . \frac{dv}{dr} \right \vert_r + \left . \frac{d^2 v}{dr^2} \right \vert_r dr .\]
Also, use r instead of s since the lamina was arbitrary and the expression must be valid for all laminae. Grouping like terms and dropping the vertical bar since all derivatives are assumed to be at radius r,
\[ 0 = - \Delta P2 \pi rdr + \eta 2 \pi dr \Delta x \frac{dv}{dr} + \eta 2 \pi r dr \Delta x \frac{d^2 v}{dr^2} + \eta 2 \pi (dr)^2 \Delta x \frac{d^2 v}{dr^2}. \]
Finally, put this expression in the form of a differential equation, dropping the term quadratic in dr.
\[ \frac{1}{\eta} \frac{\Delta P}{\Delta x} = \frac{d^2 v}{dr^2} + \frac{1}{r} \frac{dv}{dr} \]
It can be seen that both sides of the equations are negative: there is a drop of pressure along the tube (left side) and both first and second derivatives of the velocity are negative (velocity has a maximum value of the center of the tube). Using the chain rule, the equation may be re-arranged to:
\[ \frac{1}{\eta} \frac{\Delta P}{\Delta x} = \frac{1}{r} \frac{d}{dr} r \frac{dv} {dr}. \]
This differential equation is subject to the following boundary conditions:
\[ v(r) = 0 \] at \( r = R \) -- "no-slip" boundary condition at the wall
\[ \frac{dv} {dr} = 0 \] at \( r = 0 \) -- axial symmetry.
Axial symmetry means that the velocity v(r) is maximum at the center of the tube, therefore the first derivative \( \frac{dv}{dr}\) is zero at r = 0.
The differential equation can be integrated to:
\[ v(r) = \frac{1}{4 \eta}r^2\frac{\Delta P}{\Delta x} + A\ln(r) + B. \]
To find A and B, we use the boundary conditions.
First, the symmetry boundary condition indicates:
\[ \frac{dv}{dr} = \frac{1}{2 \eta} r \frac{\Delta P}{\Delta x} + A \frac{1}{r} = 0 \] at r = 0.
A solution possible only if A = 0. Next the no-slip boundary condition is applied to the remaining equation:
\[ v(R) = \frac{1}{4 \eta} R^2 \frac{\Delta P}{\Delta x} + B = 0 \]
so therefore
\[ B = - \frac{1}{4 \eta} R^2 \frac{\Delta P}{\Delta x}. \]
Now we have a formula for the velocity of liquid moving through the tube as a function of the distance from the center of the tube
\[ v = - \frac{1}{4 \eta} \frac{\Delta P}{\Delta x} (R^2 - r^2) \]
or, at the center of the tube where the liquid is moving fastest (r = 0) with R being the radius of the tube,
\[ v_{max} = - \frac{1}{4 \eta} \frac{\Delta P}{\Delta x}R^2. \]
Poiseuille's Law
To get the total volume that flows through the tube, we need to add up the contributions from each lamina. To calculate the flow through each lamina, we multiply the velocity (from above) and the area of the lamina.
\[ \Phi (r)dr = \frac{1}{4 \eta} \frac{|\Delta P|}{\Delta x} (R^2 - r^2) 2 \pi rdr = \frac{\pi}{2 \eta} \frac{|\Delta P|}{\Delta x} (rR^2 - r^3)dr \]
Finally, we integrate over all lamina via the radius variable r.
\[ \Phi = \frac{\pi}{2 \eta} \frac{|\Delta P|}{\Delta x} \int_{0}^{R} (rR^2 - r^3)\, dr = \frac{|\Delta P| \pi R^4}{8 \eta \Delta x} \]
Poiseuille's equation for compressible fluids
For a compressible fluid in a tube the volumetric flow rate and the linear velocity is 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 and 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, and when the pressure difference between ends of the pipe is small, the volumetric flow rate at the pipe outlet is given by
\[ \Phi = \frac{dV}{dt} = v \pi R^{2} = \frac{\pi R^{4} \left( P_{i}-P_{o} \right)}{8 \eta L} \times \frac{ P_{i}+P_{o}}{2 P_{o}} = \frac{\pi R^{4}}{16 \eta L} \left( \frac{ P_{i}^{2}-P_{o}^{2}}{P_{o}} \right) \]
where: \[ P_{i} \] inlet pressure \[ P_{o} \] outlet pressure \[ L \] is the length of tube \[ \eta \] is the viscosity \[ R \] is the radius \[ V \] is the volume of the fluid at outlet pressure \[ v \] is the velocity of the fluid at outlet pressure
This is usually a good approximation when the flow velocity is less than mach 0.3
This equation can be seen as Poiseuille's law with an extra correction factor \( \frac{P_{i}+P_{o}}{2} \times \frac{1}{P_{o}} \) expressing the average pressure relative to the outlet pressure.
Electrical circuits analogy
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 (\(V=IR\)), where the pressure drop \(\Delta P\) is analogous to the voltage \(V\) and volumetric flow rate \(\Phi\) is analogous to the current \(I\). Then the resistance \[R = \frac{ 8 \eta \Delta x}{\pi r^4}.\] This concept is useful because the effective resistance in a tube is inversely proportional to the fourth power of the radius. This means that halving the radius of the tube increases the resistance to fluid movement by a factor of 16.
Both Ohm's law and Poiseuille's law illustrate transport phenomena.
History
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 did his experiments in 1839.
See also
Notes
- ↑ Script error
- ↑ Script error
- ↑ Script error
- ↑ Determinants of blood vessel resistance
References
- Sutera, S. P.; Skalak, R. (1993), "The history of Poiseuille's law", Annual Review of Fluid Mechanics 25: 1–19, Bibcode 1993AnRFM..25....1S, doi:10.1146/annurev.fl.25.010193.000245.
- Pfitzner, J (1976), "Poiseuille and his law.", Anaesthesia 31 (2): 273–5, 1976 Mar, doi:10.1111/j.1365-2044.1976.tb11804.x, PMID 779509, PMID:779509.
- Bennett, C. O.; Myers, J. E. (1962), Momentum, Heat, and Mass Transfer, McGraw-Hill.
External links
- Poiseuille's law for power-law non-Newtonian fluid
- Poiseuille's law in a slightly tapered tube
- Web-based calculator of the Hagen–Poiseuille equation
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