Hagen–Poiseuille equation

With the law of Hagen -Poiseuille [ Po'a zœj ː ] ( for God Help Heinrich Ludwig Hagen, 1797-1884 and Jean Léonard Marie Poiseuille, 1797-1869 ), the volume flow - that is, the flowed volume V per unit of time - in a laminar steady flow of a Newtonian homogeneous fluid through a pipe ( capillary) of radius r and of length l described.

Formulation

The law is

With

This law follows directly from the steady parabolic flow profile through a tube that can be derived from the Navier -Stokes equations - or directly from the definition of viscosity, see below. Noteworthy is the dependence of the volume flow of the fourth power of the radius of the tube. Thus, the flow resistance is strongly dependent on the radius of the tube, it would, for example, a reduction of the pipe diameter increase to half the flow resistance on 16 times.

The law applies only for laminar flows. At a higher flow conduit associated with higher flow rates and larger sizes, there will be turbulent flow with a substantially higher flow resistance than by the Hagen -Poiseuille would be expected. The specific conditions of turbulent flows, among others described by the formulas of Blasius, Prandtl - Nikuradse or Colebrook.

Derivation

Here is the consideration from the Hagen -Poiseuille law and its underlying flow profile follows: Denote the flow velocity at the point of a circular pipe with radius. Consider a hollow cylinder of the length and the wall thickness between the radii. The cylinder should be located in the state of equilibrium, so experience no acceleration, therefore, is the sum of all forces acting on the surfaces equal to zero. Of the friction on the outer or inner surface or the coefficient of static friction and the pressure differential to the hollow cylindrical base results in the power equation:

.

The friction with the adjacent out flow cylinder having the radius. The speed difference is distributed on the layer thickness and acts along the outer surface. This applies analogously to the friction at the inner surface of the cylinder adjacent inward flow.

In the border crossing there is a second order differential equation for:

.

The solution must satisfy the boundary condition and is thus uniquely determined:

.

This is exactly called quadratic flow profile. By integration it follows the law of Hagen- Poiseuille:

.

Non- circular channel cross-sections

For a rectangular channel with dimensions and can specify this law in the following form:

This is

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

Some sample values ​​calculated in the third approximation:

Formulas for other cross-sectional shapes are eg in derived.

Applications

In the scope of the law about the narrowing of a round cross -section by 10% causes a throughput drop! To reach the original flow at a reduced cross-section again, the pressure difference must increase by 52%.

Moreover, the law of Hagen - Poiseuille law provides the basis of a plurality of model equations for the flow of bulk materials.

Limited validity in the blood

The law of Hagen- Poiseuille refers to Newtonian fluids. For Newtonian fluids, the viscosity is a constant material properties ( and only dependent on the temperature). An example of such liquid is water. The blood plasma is a Newtonian fluid, but not the blood: It is an inhomogeneous suspension of the various cells in the plasma. Here, the viscosity ( the flow rate so ) is the magnitude of the shear stress -dependent. Furthermore, the deformability of the red blood cells play a role. These can be, for example, money roll-like ' in thin vessels aggregate.

This particular field of rheology of blood is called hemorheology (English Hemorheology ).