How do I derive viscosity equation

[jubaaa]

Hi.everybody
How do I derive viscosity equation F=6π*r*μ?

 

[JaredJames]

What are your thoughts on this? How would you start? You must show some effort to gain help.

 

[Studiot]

A somewhat elderly thread methinks.

 

[JaredJames]

Only by two months, figured it deserved at least a response plus bump for the OP.

Haven't quite gotten into the realms of necroposting yet.

 

[PJC01]

Hello! The viscosity equation, also known as the Hagen-Poiseuille equation, can be derived through the application of fluid mechanics principles. It describes the relationship between the flow rate of a fluid, the pressure difference across a pipe or channel, and the viscosity of the fluid. The equation is typically used to calculate the flow rate of a Newtonian fluid, which is a fluid that has a constant viscosity regardless of the applied shear stress.

To derive the viscosity equation, we start with the Navier-Stokes equations, which are a set of equations that describe the motion of a fluid. We then apply the assumptions of steady, laminar flow and incompressible fluid to simplify the equations. This results in the following equation:

ΔP = 8μQ/πr^4

Where ΔP is the pressure difference, μ is the viscosity of the fluid, Q is the volumetric flow rate, and r is the radius of the pipe or channel.

Next, we rearrange the equation to solve for Q, which gives us:

Q = (πr^4/8μ)ΔP

Finally, we can simplify the equation by substituting in the definition of the area of a circle (A=πr^2) and the definition of shear stress (τ=ΔP/A) to get:

Q = (πr^3/8μ)τ

This is the final form of the viscosity equation, where F (force) is equal to πr^3τ/8, or 6πrμ, as stated in the original question.

I hope this helps explain the derivation of the viscosity equation. Keep in mind that this is a simplified version and there are more complex derivations that take into account factors like fluid density and pipe length. Thank you for your question!