Poiseuille Flow

The law of flow through cylindrical pipes was experimentally derived by Jean Louis Marie Poiseuille, a French physicist and physiologist. Poiseuille was interested in the functioning of human blood flow, which eventually made him observe the flow through cylindrical pipes.

What is Poiseuille Flow?

There is only a single component of velocity that is non-zero, and that component has the same direction everywhere and is independent of distance in the flow direction. This means that the pipe flow is a unidirectional concept.

Using cylindrical polar coordinates, the geometry of Poiseuille flow is analyzed and the length of the long cylindrical pipe is ‘l’ and radius ‘a’ so the suitable coordinate system is cylindrical polar (r, θ, z). The pressure gradient, dP/dz, is constant everywhere in the pipe since P1 and P0 are the pressures at each end of the pipe. Here, ur will be equal to 0 and uθ will be equal to 0 because of the unidirectional nature of the problem. Therefore, the continuity equation is summarized to ∂u_z/∂z = 0.

This means that at any value of z, the velocity must have both an identical velocity profile and a constant value, because of its incompressibility limitation.

Also, any change in the flow will occur everywhere in the pipe suddenly. You can understand this with a daily life example that when you take a shower which does not have a pressure regulator and at that point of time when somebody else opens a tap or flushes a toilet or something, then to refill the toilet tank the cold water is diverted, and the pressure gradient drops in the pipes for cold water which eventually decreases the flow of cold water.

Equation of Continuity:

Derivation of Poiseuille Flow

Navier-Stokes equation that has an inertial term even in the more general case:

ρ( ∂u_z/∂t + ∂u_z/∂z), you can see that for steady flow ∂u/z∂t = 0, the incompressibility of the fluid and the geometry of the problem agree that the inertial term is exactly zero. So Poiseuille flow occurs at higher Re and is not limited to the Stokes regime, we can understand that this is important.

In cylindrical coordinates, this 1-dimensional type of the momentum equation is:

The, we can try a solution of the form:

considering the boundary conditions of finite force over the fluid length and no-slip side walls

Solving for the constants we get:

This means that, with a maximum in the center (r = 0) and zero at the pipe walls the velocity profile of the flow has a parabolic shape. Also it should be noted that the flow is independent of the fluid density. As it is mentioned that the velocity at the center is maximum, which we can calculate

When P1 > P0, u_max is a positive quantity because pressure gradients are generally defined to be negative, such that the flow of water is from high pressure to low pressure. It is also very useful to determine the total flow rate through the pipe, therefore, over the a cross-section of the pipe we integrate the velocity:

For a given pressure gradient and viscosity, the volumetric flow rate shows that the flow through the pipe is directly proportional to the fourth power of the radius of the pipe. This is what Poiseuille experimentally illustrated. Normalized by the cross-sectional area of the pipe, the mean velocity is simply the total flow .

The mean velocity is totally the result of the net force exerted on the fluid by the gradient of pressure acting on the fluid to overcome the viscous drag from the pipe walls. The force (per unit length) from pressure is:

This shows that the pressure force by u = F_P /(8πη) is related to the mean flow u, therefore it is linearly inversely proportional to η. Similarly, viscous drag is proportional to the shear (tangential) stress, σzr, for a Newtonian fluid, which can be evaluated near the wall of the pipe, r = a,

Similarly we can determine a friction factor, f, which describes the effect of drag to the non-dimensional drag coefficient of the Stokes sphere, c_D. Evaluated at the wall as a characteristic stress we use the shear stress and normalize that value by a characteristic pressure (1/2 ρfu²) in which we use the mean velocity

SUbstituting for one value of u, we get f:

If characteristic length scale is chosen as the diameter of the pipe, D = 2a, then we get:

At Re ∼ 2000 − 3000, this relationship holds good until the transition into the turbulent flow regime.

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