Hagen-Poiseuille's law

With the law of Hagen-Poiseuille [ poaː'zœj ] (after Gotthilf Heinrich Ludwig Hagen , 1797–1884 and Jean Léonard Marie Poiseuille , 1797–1869) the volume flow - d. H. the flowed volume V per unit of time - with a laminar steady flow of a homogeneous Newtonian fluid through a tube ( capillary ) with the radius and length described. ${\displaystyle {\dot {V}}}$  ${\displaystyle r}$ ${\displaystyle l}$

formulation

The law reads

${\displaystyle {\dot {V}}={\frac {dV}{dt}}={\frac {\pi \cdot r^{4}}{8\cdot \eta }}{\frac {\Delta p}{l}}=-{\frac {\pi \cdot r^{4}}{8\cdot \eta }}{\frac {\partial p}{\partial z}}}$

With

variable	meaning	SI unit
${\displaystyle {\dot {V}}}$	Volume flow through the pipe	${\displaystyle {\frac {\rm {m^{3}}}{\rm {s}}}}$
${\displaystyle r}$	Inner radius of the pipe	m
${\displaystyle l}$	Length of pipe	m
${\displaystyle \eta }$	dynamic viscosity of the flowing liquid	Pa · s
${\displaystyle \Delta p}$	Pressure difference between the beginning and the end of the pipe	Pa
z	Flow direction

Laminar flow profile

This law follows directly from the steady parabolic flow profile through a pipe, which can be derived from the Navier-Stokes equations - or directly from the definition of viscosity , see below. The dependence of the volume flow on the fourth power of the radius of the pipe is remarkable. As a result, the flow resistance depends very much on the radius of the pipe, for example reducing the pipe diameter by half would increase the flow resistance 16 times.

The law only applies to laminar flows. With a larger flow through a pipeline, combined with higher flow speeds or larger dimensions, turbulent flows with significantly higher flow resistance than would be expected according to Hagen-Poiseuille occur. The concrete conditions of turbulent flows are u. a. with the formulas of Blasius , Nikuradse and Prandtl - Colebrook .

The law of Hagen-Poiseuille basically only applies to a fully developed hydrodynamic flow profile (parabolic velocity profile). For example, if liquid flows out of a tank through a pipe, the pipe must be long enough for the Hagen-Poiseuille law to apply. In the run-up to the flow has to the parabolic flow profile with additional pressure drop ( "energy consumption") that is formed first. The pressure difference in the formula above for the volume flow therefore relates to the pressure difference of a fully developed flow.

In very thin tubes, in which the boundary layer has a decisive influence on the flow profile and is not very small compared to the radius, this greatly simplified mathematical model of the flow cannot be used either.

A modified law applies to compressible fluids (such as gases).

Derivation

Here is the consideration from which the Hagen-Poiseuille law and the underlying flow profile follow: Designate the flow velocity at the point of a circular tube with a radius . Let us consider a hollow cylinder of length and wall thickness between the radii and . The cylinder should be in a state of equilibrium, i.e. not experience any acceleration, so the sum of all forces acting on the surfaces is zero. The force equation results from the friction on the outer or inner surface or with the dynamic viscosity and the pressure difference on the hollow cylinder base area : ${\displaystyle v(s)}$ ${\displaystyle s}$ ${\displaystyle R}$ ${\displaystyle l}$ ${\displaystyle ds}$ ${\displaystyle s}$ ${\displaystyle s^{+}=s+ds}$ ${\displaystyle A^{+}}$ ${\displaystyle A}$ ${\displaystyle \eta }$ ${\displaystyle \Delta p}$ ${\displaystyle dA=2\pi \,s\,ds}$

${\displaystyle \eta (A^{+}dv^{+}-A\,dv)/ds+dA\Delta p=0}$.

There is friction with the outwardly adjacent flow cylinder, which has the radius . The speed difference is distributed over the layer thickness and acts along the outer surface . This applies analogously to the friction on the inner surface with the flow cylinder adjacent to the inside. ${\displaystyle \eta A^{+}dv^{+}/ds}$ ${\displaystyle s+ds}$ ${\displaystyle {\frac {dv^{+}}{ds}}={\frac {dv}{ds}}\vert _{s+ds}}$ ${\displaystyle ds}$ ${\displaystyle A^{+}=2\pi s^{+}l}$

At the limit , there is a second order differential equation for : ${\displaystyle ds\to 0}$ ${\displaystyle v(s)}$

${\displaystyle {\begin{aligned}2\pi \,l\,\left(s+ds\right){\frac {dv}{ds}}\vert _{s+ds}-2\pi \,l\,s\,{\frac {dv}{ds}}\vert _{s}+{\frac {2\pi \,s\,ds\,\Delta p}{\eta }}&=0\\\left(s+ds\right){\frac {dv}{ds}}\vert _{s+ds}-s\,{\frac {dv}{ds}}+{\frac {s\,ds\,\Delta p}{\eta \,l}}&=0\\s\,{\frac {dv}{ds}}+s\,{\frac {d^{2}v}{ds^{2}}}\,ds+{\frac {ds\,dv}{ds}}+{\mathcal {O}}\left(ds^{2}\right)-s\,{\frac {dv}{ds}}+{\frac {s\,ds\,\Delta p}{\eta \,l}}&=0\\s\,{\frac {d^{2}v}{ds^{2}}}\,ds+{\frac {ds\,dv}{ds}}+{\frac {s\,ds\,\Delta p}{\eta \,l}}&=0\\{\frac {d^{2}v}{ds^{2}}}+{\frac {dv}{s\,ds}}+{\frac {\Delta p}{\eta \,l}}&=0\\\end{aligned}}}$

The solution must meet the boundary condition and is thus clearly determined: ${\displaystyle v(R)=0}$

${\displaystyle v(s)={\frac {\Delta p}{4\eta \,l}}\,(R^{2}-s^{2})}$.

This is precisely the said square flow profile. Integration then follows the Hagen-Poiseuille law:

${\displaystyle {\dot {V}}=\int _{0}^{2\pi }\int _{0}^{R}v(s)s\,ds\,d\varphi ={\frac {\pi R^{4}}{8\eta }}{\frac {\Delta p}{l}}}$.

Non-circular duct cross-sections

Rectangle channel

For a rectangular duct with the dimensions and , this law can be given in the following form: ${\displaystyle b}$ ${\displaystyle h}$

${\displaystyle {\dot {V}}={\frac {K\cdot \min(b,h)^{3}\cdot \max(b,h)}{12\eta l}}\cdot \Delta p}$

Here is

${\displaystyle K=1-\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{5}}}\cdot {\frac {192}{\pi ^{5}}}\cdot {\frac {\min(b,h)}{\max(b,h)}}\tanh \left((2n-1){\frac {\pi }{2}}{\frac {\max(b,h)}{\min(b,h)}}\right)}$

The deviation from the exact value when calculating K in the first approximation (n = 1) is a maximum of 0.67%, in the second approximation 0.06%, in the third approximation 0.01%.

Some example values, calculated as a third approximation:

${\displaystyle {\frac {\min(b,h)}{\max(b,h)}}}$	0	1/10	1/5	1/4	1/3	1/2	2/3	3/4	1
${\displaystyle K}$	1	0.9370	0.8740	0.8425	0.7900	0.6861	0.5873	0.5414	0.4218

Formulas for other cross-sectional shapes are derived from many textbooks.

Elliptical cross section

For elliptical cross-sections this results

${\displaystyle {\dot {V}}={\frac {\pi }{4\eta }}\cdot {\frac {a^{3}\cdot b^{3}}{a^{2}+b^{2}}}\cdot {\frac {\Delta p}{l}}\,,}$

where and represent the two semi-axes of the ellipse . ${\displaystyle a}$ ${\displaystyle b}$

Note the special case , ${\displaystyle a=b=r}$

${\displaystyle {\dot {V}}={\frac {\pi \cdot r^{4}}{8\cdot \eta }}{\frac {\Delta p}{l}}\,,}$

in which the equation is reduced to the equation for cylindrical tubes.

Applications

In the area of ​​validity of the law, for example, narrowing a round line radius by 10% causes a throughput decrease by . In order to achieve the original flow rate again with a reduced radius, the pressure difference must therefore increase by over 52%. ${\displaystyle 1-0{,}9^{4}=34\,\%}$

In addition, the Hagen-Poiseuille law forms the basis of a large number of model equations for the flow of bulk solids .

Limited validity in the blood

Hagen-Poiseuille's law applies to Newtonian fluids . In the case of Newtonian fluids, the viscosity is not a function of the shear rate. An example of such a liquid is water. Blood plasma is also a Newtonian fluid, but not blood : it is an inhomogeneous suspension of different cells in plasma . Here the viscosity depends on the level of the shear rate (i.e. the flow velocity). The deformability of the erythrocytes also plays a role. These can, for example, aggregate “like a roll of money” in thin vessels. Incidentally, it is not a question of laminar, but rather turbulent flow conditions.

This special field of rheology of the blood is known as hemorheology ( English hemorheology ).

literature

• Wolfgang Beitz; Karl-Heinrich Grote (Ed.): Dubbel. Paperback for mechanical engineering. 20th edition. Springer-Verlag, Berlin / Heidelberg / New York 2001, ISBN 3-540-67777-1
• James P. Hartnett; Milivoje Kostic: Heat Transfer to Newtonian and Non-Newtonian Fluids in Rectangular Ducts. In: Advances in Heat Transfer , Volume 19, 1989
• Rainer Klinke (Ed.): Physiology. Numerous tables. 5th edition. Georg Thieme Verlag, Stuttgart / New York 2005, ISBN 3-13-796005-3

Individual evidence

1. ^ Poiseuille pronunciation: How to pronounce Poiseuille in French
2. ↑ tec-science: energetic consideration of the Hagen-Poiseuille law. In: tec-science. April 2, 2020, accessed on May 7, 2020 (German). 
3. ↑ Henrik Bruus: Theoretical Microfluidics . Oxford University Press, 2008