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The French physicist Jean Leonard Marie Poiseuille theorised Poiseuille’s law, which describes the pressure drop due to the viscosity of the fluid. Hagen-Poiseuille law or Poiseuille law defines the relationship between pressure gradient, flow rate, and resistance of the fluid. Poiseuille’s Law says that the flow of a liquid or gas which flows from a capillary is linearly proportional to the variance of the pressure of that liquid or gas (fluid) and conversely proportional to the opposition of flow (viscosity).
This law is used to obtain the pressure drop in an incompressible and Newtonian fluid in laminar flow in a long cylindrical pipe with a constant cross-sectional area.
Poiseuille’s Law Equation
The Poiseuille equation is an important equation in fluid mechanics that helps us understand the pressure variation in a flowing fluid. Poiseuille’s formula for flow in a tube is expressed below.
Q=PR48L
Here, Q is the flow rate, p is the pressure difference, R is the radius of the tube, L is the length of the tube and is the fluid viscosity or coefficient of viscosity.
The volume flow rate by this law can also be expressed in the following manner.
Q=PRflow
In which Rflow is the resistance of the flow such that;
Rflow=8LR4
The expression of Poiseuille’s formula for the pressure difference is given below.
P=8LQR4
P=8LQA2
Here, A is the cross-sectional area of the tube.
Derivation of Poiseuille Equation
The two main forces which are involved in the flow of viscous incompressible fluid are:
- Driving Force: This force is responsible for the flow of fluid as it pushes the fluid.
The following expression can give the driving force on the surface of a cylindrical pipe.
Fpressure=P (r2)
Here, r is the inner radius of the tube, and P is the pressure difference.
- Viscous Drag: The viscous drag force works opposite to the driving force and tries to slow down the flow.
The total area of the surface of the cylindrical pipe, which has a measure of length equal to L is 2rL. The viscous drag force caused due to the viscosity of the fluid is given as follows:
Fviscosity=-(2rL)dvdr
In the case of a balanced position of steady velocity, where no net force is acting on it, the sum of driving force and viscous drag force becomes zero.
Fpressure+Fviscosity=0
P (r2)-(2rL)dvdr=0
P (r2)=(2rL)dvdr
Rewrite the equation of velocity gradient.
dv=(P2L).r dr
At the edge points, the limit of velocity varies from v to 0, and radii vary from r to R. Integrate the equation to get the value of velocity.
v0dv=rR(P2L).r dr
v(r)=(P4L).[R2–r2]
The continuity equation of volume flux for the variable speed is:
dvdt=v.dA
Plug in the value of velocity in the above formula.
dvdt=0R(P4L).[R2–r2].[2r.dr]
dvdt=P2L[R22–R44]
dvdt=PR48L
Q=PR48L
This is Poiseuille’s equation.
Limitation of Poiseuille’s Law
There are some specific limitations of Poiseuille’s law.
- This law is applicable for only long tubes.
- The tube should be cylindrical.
- The radius of the tube should be uniform throughout the pipe.
Example problems
- A fluid flows through a cylindrical tube. How is the flow rate of the fluid affected when the pressure difference between the ends of the tube is doubled?
Ans: Poiseuille’s formula for flow in a tube is expressed with the following formula.
Q=PR48L
Here,
Q= Flow rate,
p= Pressure difference,
R=Radius of the tube,
L= Length of the tube
= Fluid viscosity.
The pressure difference between the ends of the tube is doubled. Thus, the above equation can be rewritten with the known information.
Q=(2P)R48L
Q=2PR48L
Q=2Q
The flow rate is doubled.
- A fluid flows through a cylindrical tube. What happens to the flow rate of the fluid when the radius of the tube is doubled, and the length of the tube is halved?
Ans: Poiseuille’s formula for flow in a tube is expressed with the following formula.
Q=PR48L
Here,
Q= Flow rate,
p= Pressure difference,
R=Radius of the tube,
L= Length of the tube
= Fluid viscosity.
The radius of the tube is doubled, and the length of the tube is halved. Here, the new values of radius is 2R and length is L2 .
Thus, the above equation can be rewritten with the known information.
Q=P(2R)28L2
Q=2P(4R2)8L
Q=8Q
The flow rate has increased eight times compared to the initial flow rate.
Conclusion
Poiseuille’s law deals with the steady laminar flow rate through a capillary tube. Poiseuille’s law is used to calculate the volume flow rate in a cylindrical tube in the case of laminar flow. There are many applications of this law, such as calculating blood flow in capillaries, finding the pressure drop in a cylindrical pipe, etc.
