### Poiseuille's Law

In 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 a fluid flowing through a long cylindrical pipe.
It can be successfully applied to air flow in lung alveoli, for the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Gotthilf Heinrich Ludwig Hagen in 1839 and Jean Léonard Marie Poiseuille in 1838, and published by Poiseuille in 1840 and 1846.

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.

## Contents

## Equation

### Standard fluid dynamics notation

In standard fluid dynamics notation:^{[1]}^{[2]}

- $\backslash Delta\; P\; =\; \backslash frac\{8\; \backslash mu\; L\; Q\}\{\; \backslash pi\; r^4\}$

or

- $\backslash Delta\; P\; =\; \backslash frac\{128\; \backslash mu\; L\; Q\}\{\; \backslash pi\; d^4\}$

where:

- $\backslash Delta\; P$ is the pressure loss
- $L$ is the length of pipe
- $\backslash mu$ is the dynamic viscosity
- $Q$ is the volumetric flow rate
- $r$ is the radius
- $d$ is the diameter
- $\backslash pi$ is the mathematical constant Pi

### Physics notation

- $\backslash Phi\; =\; \backslash frac\{dV\}\{dt\}\; =\; v\; \backslash pi\; R^\{2\}\; =\; \backslash frac\{\backslash pi\; R^\{4\}\}\{8\; \backslash eta\}\; \backslash left(\; \backslash frac\{-\; \backslash Delta\; P\}\{\backslash Delta\; x\}\backslash right)\; =\; \backslash frac\{\backslash pi\; R^\{4\}\}\{8\; \backslash eta\}\; \backslash frac\{\; |\backslash Delta\; P|\}\{L\}$

where in compatible units (e.g., S.I.):

- $\backslash Phi$ is the volumetric flow rate (denoted as $Q$ above)
- $V(t)$ is the volume of the liquid transferred as a function of time, $t$
- $v$ is mean fluid velocity along the length of the tube
- $x$ is distance in direction of flow
- $R$ is the internal radius of the tube
- $\backslash Delta\; P$ is the pressure difference between the two ends
- $\backslash eta$ is the dynamic fluid viscosity (S.I. unit: pascal-second (Pa·s)),
- $L$ is the length of the tube

The equation does not hold close to the pipe entrance.^{[3]}^{: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 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

$\backslash Phi\_\{max\}\; =\; \backslash pi\; R^2\; \backslash sqrt\{2\; \backslash Delta\; P\; /\; \backslash rho\}$.

## 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 phenomenological 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:

- $\backslash Lambda\; =\; \{64\backslash over\; \{\backslash it\; \backslash mathrm\{Re\}\}\}\; \backslash ;\; ,\; \backslash quad\backslash quad\; \backslash mathrm\{Re\}\; =\; \{2\backslash rho\; v\; r\backslash over\; \backslash eta\}\; \backslash ;\; ,$

where *Re* is the Reynolds number and *ρ* fluid density. Note that *v* is the mean flow velocity, which is half the maximum 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 maximum 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 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. Although more lengthy than directly using the Navier–Stokes equations, an alternative method of deriving the Hagen–Poiseuille equation is as follows.

### Liquid flow through a pipe

Assume the liquid exhibits Laminar 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 the no-slip condition).

To figure out the motion of the liquid, all forces acting on each lamina must be known:

- The pressure force pushing the liquid through the tube is the change in pressure multiplied by the area: $F\; =\; -\backslash Delta\; P\; A$. This force is in the direction of the motion of the liquid. The negative sign comes from the conventional way we define $\backslash Delta\; P\; =\; P\_\{end\}-P\_\{top\}\; <\; 0$.
- Viscosity effects will pull from the faster lamina immediately closer to the center of the tube.
- Viscosity effects will drag from the slower lamina immediately closer to the walls of the tube.

### 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 $\{\backslash Delta\; v\_x\}/\{\backslash Delta\; y\}$, and a proportionality constant *η* (viscosity) and is given by

- $F\_\{\backslash text\{viscosity,\; top\}\}\; =\; -\; \backslash eta\; A\; \backslash frac\{\backslash Delta\; v\_x\}\{\backslash Delta\; y\}.$

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.

### 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 $r$, thickness $dr$, and length Δx. The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder: $A\; =\; 2\; \backslash pi\; r\; \backslash Delta\; x$ . 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 $r$. 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

- $F\_\{\backslash text\{viscosity,\; fast\}\}\; =\; -\; \backslash eta\; 2\; \backslash pi\; r\; \backslash Delta\; x\; \backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_r$

where the vertical bar and subscript *r* following the derivative indicates that it should be taken at a radius of $r$.

### 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 *r*+*dr* instead of *r*. 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).

- $F\_\{\backslash text\{viscosity,\; slow\}\}\; =\; \backslash eta\; 2\; \backslash pi\; (r+dr)\; \backslash Delta\; x\; \backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_\{r+dr\}$

### 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

- $0\; =\; F\_\{\backslash text\{pressure\}\}\; +\; F\_\{\backslash text\{viscosity,\; fast\}\}\; +\; F\_\{\backslash text\{viscosity,\; slow\}\}$

or

- $0\; =\; -\; \backslash Delta\; P2\; \backslash pi\; rdr\; -\; \backslash eta\; 2\; \backslash pi\; r\; \backslash Delta\; x\; \backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_r\; +\; \backslash eta\; 2\; \backslash pi\; (r+dr)\; \backslash Delta\; x\; \backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_\{r+dr\}.$

First, to get everything happening at the same point, use the first two terms of a Taylor series expansion of the velocity gradient:

- $\backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_\{r+dr\}\; =\; \backslash left\; .\; \backslash frac\{dv\}\{dr\}\; \backslash right\; \backslash vert\_r\; +\; \backslash left\; .\; \backslash frac\{d^2\; v\}\{dr^2\}\; \backslash right\; \backslash vert\_r\; dr\; .$

The expression is valid for all laminae. Grouping like terms and dropping the vertical bar since all derivatives are assumed to be at radius *r*,

- $0\; =\; -\; \backslash Delta\; P2\; \backslash pi\; rdr\; +\; \backslash eta\; 2\; \backslash pi\; dr\; \backslash Delta\; x\; \backslash frac\{dv\}\{dr\}\; +\; \backslash eta\; 2\; \backslash pi\; r\; dr\; \backslash Delta\; x\; \backslash frac\{d^2\; v\}\{dr^2\}\; +\; \backslash eta\; 2\; \backslash pi\; (dr)^2\; \backslash Delta\; x\; \backslash frac\{d^2\; v\}\{dr^2\}.$

Finally, put this expression in the form of a differential equation, dropping the term quadratic in *dr*.

- $\backslash frac\{1\}\{\backslash eta\}\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; =\; \backslash frac\{d^2\; v\}\{dr^2\}\; +\; \backslash frac\{1\}\{r\}\; \backslash frac\{dv\}\{dr\}$

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 at the center of the tube, where *r* = 0). Using the chain rule, the equation may be re-arranged to:

- $\backslash frac\{1\}\{\backslash eta\}\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; =\; \backslash frac\{1\}\{r\}\; \backslash frac\{d\}\{dr\}\; r\; \backslash frac\{dv\}\; \{dr\}.$

This differential equation is subject to the following boundary conditions:

- $v(r)\; =\; 0$ at $r\; =\; R$ -- "no-slip" boundary condition at the wall

- $\backslash frac\{dv\}\; \{dr\}\; =\; 0$ 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 $\backslash frac\{dv\}\{dr\}$ is zero at r = 0.

The differential equation can be integrated to:

- $v(r)\; =\; \backslash frac\{1\}\{4\; \backslash eta\}r^2\backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; +\; A\backslash ln(r)\; +\; B.$

To find A and B, we use the boundary conditions.

First, the symmetry boundary condition indicates:

- $\backslash frac\{dv\}\{dr\}\; =\; \backslash frac\{1\}\{2\; \backslash eta\}\; r\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; +\; A\; \backslash frac\{1\}\{r\}\; =\; 0$ at r = 0.

A solution possible only if A = 0. Next the no-slip boundary condition is applied to the remaining equation:

- $v(R)\; =\; \backslash frac\{1\}\{4\; \backslash eta\}\; R^2\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; +\; B\; =\; 0$

so therefore

- $B\; =\; -\; \backslash frac\{1\}\{4\; \backslash eta\}\; R^2\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}.$

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

- $v\; =\; -\; \backslash frac\{1\}\{4\; \backslash eta\}\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}\; (R^2\; -\; r^2)$

or, at the center of the tube where the liquid is moving fastest (*r* = 0) with *R* being the radius of the tube,

- $v\_\{max\}\; =\; -\; \backslash frac\{1\}\{4\; \backslash eta\}\; \backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; x\}R^2.$

### 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.

- $\backslash Phi\; (r)dr\; =\; \backslash frac\{1\}\{4\; \backslash eta\}\; \backslash frac\{|\backslash Delta\; P|\}\{\backslash Delta\; x\}\; (R^2\; -\; r^2)\; 2\; \backslash pi\; rdr\; =\; \backslash frac\{\backslash pi\}\{2\; \backslash eta\}\; \backslash frac\{|\backslash Delta\; P|\}\{\backslash Delta\; x\}\; (rR^2\; -\; r^3)dr$

Finally, we integrate over all lamina via the radius variable *r*.

- $\backslash Phi\; =\; \backslash frac\{\backslash pi\}\{2\; \backslash eta\}\; \backslash frac\{|\backslash Delta\; P|\}\{\backslash Delta\; x\}\; \backslash int\_\{0\}^\{R\}\; (rR^2\; -\; r^3)\backslash ,\; dr\; =\; \backslash frac\{|\backslash Delta\; P|\; \backslash pi\; R^4\}\{8\; \backslash eta\; \backslash Delta\; x\}$

## 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

- $\backslash Phi\; =\; \backslash frac\{dV\}\{dt\}\; =\; v\; \backslash pi\; R^\{2\}\; =\; \backslash frac\{\backslash pi\; R^\{4\}\; \backslash left(\; P\_\{i\}-P\_\{o\}\; \backslash right)\}\{8\; \backslash eta\; L\}\; \backslash times\; \backslash frac\{\; P\_\{i\}+P\_\{o\}\}\{2\; P\_\{o\}\}\; =\; \backslash frac\{\backslash pi\; R^\{4\}\}\{16\; \backslash eta\; L\}\; \backslash left(\; \backslash frac\{\; P\_\{i\}^\{2\}-P\_\{o\}^\{2\}\}\{P\_\{o\}\}\; \backslash right)$

where:

- $P\_\{i\}$ inlet pressure
- $P\_\{o\}$ outlet pressure
- $L$ is the length of tube
- $\backslash eta$ 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 $\backslash frac\{P\_\{i\}+P\_\{o\}\}\{2\}\; \backslash times\; \backslash frac\{1\}\{P\_\{o\}\}$ 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 $\backslash Delta\; P$ is analogous to the voltage $V$ and volumetric flow rate $\backslash Phi$ is analogous to the current $I$. Then the resistance

- $R\; =\; \backslash frac\{\; 8\; \backslash eta\; \backslash Delta\; x\}\{\backslash pi\; r^4\}.$

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.

## See also

## Notes

## References

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## External links

- Poiseuille's law for power-law non-Newtonian fluid
- Poiseuille's law in a slightly tapered tube