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Linking Viscosity and Poiseuille's Law

Key Takeaways

  • As viscosity increases, more force is required to move the fluid. It implies that the flow rate of the fluid is limited due to the property of viscosity.

  • The two forces that initiate flow in viscous incompressible fluids are viscous drag force due to viscosity and pressure difference. 

  • Poiseuille’s law is the fluid mechanical equation that states the relationship between the velocity of the fluid flow, its pressure difference, geometrical dimensions, and viscosity.

Testing viscosity

My 4-year-old niece recently asked me why honey does not flow like water. How do you explain that it is because of viscosity to a 4-year-old? Viscosity is a fluid property that is produced due to the friction within fluid particles and between the fluid and its contact surface. Since the viscosity of honey is greater than that of water, it flows less than water does.

The flow rate of a fluid depends on the resistance to flow. The resistance to flow is directly proportional to viscosity, and Poiseuille’s law states the relationship between the flow, pressure, and resistance or viscosity. Let’s see how viscosity influences the flow rate by examining Poiseuille’s law.

Determining Viscosity  

According to the two plates model, the viscosity of a fluid () can be given as:

Viscosity equation

F is the force to keep the top plate moving in the two plates model, v is the constant velocity at which the top plate is moving, L is the distance between the plates, and A is the area between the plates.

As viscosity increases, more force is required to move the fluid. It implies that the flow rate of the fluid is limited due to the property of viscosity.

Next, we will take a look at the two main forces influencing fluid flow. 

Forces Influencing Fluid Flow: Viscosity and Pressure Difference  

Let’s concentrate on the laminar flow of viscous incompressible Newtonian fluids. The two forces that initiate flow in viscous incompressible fluids are:

  1. Viscous drag force due to viscosity
  2. Pressure difference 

When fluid is stagnant, the pressure difference and viscous drag are at an equilibrium condition. Both the forces are in balance and the fluid remains still. When a fluid flows through a tube or pipe, there is a pressure difference between the upstream and downstream. The fluid flows from high pressure to low pressure. The difference in pressure from high to low can induce a push for the fluid, forcing it to flow while the viscous drag force opposes it. When the pressure difference becomes stronger to overcome the viscous drag force, the fluid flows.

Each layer of the fluid experiences a viscous drag force. The viscous drag force is exerted on the sandwiched layer of the fluid between the slow-moving outer layer and the fast-moving inner layer. The velocity difference in the layers causes the viscous drag force; the lower the viscous drag force, the higher the flow rate.

How the flow rate is mathematically related to viscosity and pressure difference can be derived from Poiseuille’s law.

Linking Viscosity and Poiseuille’s Law

Poiseuille’s law is the fluid mechanical equation that states the relationship between the velocity of the fluid flow, its pressure difference, geometrical dimensions, and viscosity. The law is helpful in understanding the variation in flow characteristics with pressure drop or rise, viscosity, and dimensional changes. Poiseuille’s law is the statement of physics that specifically deals with viscous, incompressible, Newtonian fluids with non-accelerated laminar flow. Poiseuille’s law does have a limitation though; it can only be applied to long cylindrical tubes with uniform radius.

Here is the mathematical expression of Poiseuille’s law linking the viscosity and flow rate. According to this law, the flow rate Q is given by equation:

Poiseuille’s law expression

P is the pressure difference between higher pressure P2 and lower pressure P1 and R is the resistance to flow. The resistance to flow R is dependent on the viscosity of the fluid; the higher the viscosity, the higher the resistance to flow. As the viscosity decreases, the fluid friction reduces and resistance to flow decreases. At zero viscosity, this resistance is also zero.

The resistance to flow also relies on other factors such as:

  1. Length of the pipe -  The longer the tube or pipe length, the greater the resistance to flow.
  2. Turbulence - Resistance to flow increases with turbulence.
  3. Diameter of the pipe - Increase the diameter of the tube to decrease the resistance to flow.

In a horizontal tube of uniform radius (r) and length (l) with an incompressible Newtonian fluid of viscosity( )  flowing through it, the resistance of flow R can be given by:

Equation describing the resistance of flow

Substituting equation 3 for equation 1, the flow rate can be described using Poiseuille’s law as:

 Flow rate and viscosity relationship

It is evident that the resistance to flow is directly proportional to viscosity and the mathematical expression of Poiseuille’s law gives an inverse relationship between viscosity and flow rate. With CFD simulations, it is easy to determine the flow rate and its dependency on pipe dimensions and viscosity.

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