Navier-Stokes and why DP/Dz is constant

In summary, the professor explains that in solving for Poiseuille's Law, the differential equation becomes DP/dz = a function of R only. They further state that if DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, then the only way they can be equal is if both are constants. This is because if f(r) = g(z), then according to the given information, f(r) and g(z) must both be constants.
  • #1
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So in trying to solve for Poiseuille's Law using the continuty eq and Navier-Stokes, the differential EQ becomes DP/dz = some function of R only. The professor says that because DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, the only way they can be equal is if both are constants.

What's the intuition or reasoning behind this?
 
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  • #2
If the flow is fully developed, then the fluid velocity does not depend on z. Also, the flow velocity is purely axial. Are you comfortable with this?

Chet
 
  • #3
Oh sorry, I should have labeled. the P means pressure of flow and z is the direction.
I have found a similar explanation on this video which may give you a better understanding of what I'm asking. This is at time 1:50.

Sorry, the explanation still isn't working for me!
 
  • #4
What I was trying to do is to explain why your professor is correct. For fully developed flow in a horizontal pipe, the velocity vector is given by:
[tex]\vec{V}=v(r)\vec{i}_z[/tex]
where ##\vec{i}_z## is the unit vector in the axial direction. This relationship satisfies the continuity equation exactly and, if you substitute it into the z component of the momentum equation in cylindrical coordinates, you end up with your professor's equation. The right hand side of this equation is independent of z. That means that dp/dz is independent of z. Right now, I don't see how to prove that dp/dz is not a function of r, but I'll think about it. I know that it is.

Chet
 
  • #5
After thinking about it some more, I realized that dp/dz not being a function of r comes from the components of the NS equation in the other two directions. For these other two directions, the pressure variations are hydrostatic.

Chet
 
  • #6
I just reread my question and it's more vague than I thought it was, sorry about that!
I do agree with all the equations in the video and what you've posted, but here's my real question:

I'm curious as to why, at time 1:50 in the video, a function of Z only being equal to a function of R only means that both of them have to be constants for this to be true. It doesn't haveWhat is the logic behind this?
 
  • #7
yosimba2000 said:
I just reread my question and it's more vague than I thought it was, sorry about that!
I do agree with all the equations in the video and what you've posted, but here's my real question:

I'm curious as to why, at time 1:50 in the video, a function of Z only being equal to a function of R only means that both of them have to be constants for this to be true. It doesn't haveWhat is the logic behind this?
If f(r) = g(z), give me a specific example of a case in which f and g are not constants.

Chet
 
  • #8
yosimba2000 said:
So in trying to solve for Poiseuille's Law using the continuty eq and Navier-Stokes, the differential EQ becomes DP/dz = some function of R only. The professor says that because DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, the only way they can be equal is if both are constants.

What's the intuition or reasoning behind this?

Start with: [tex]
\frac{\partial p}{\partial z} = f(r).
[/tex] Using the given information, [tex]
\frac{\partial f}{\partial r} = \frac{\partial}{\partial r}\left(\frac{\partial p}{\partial z}\right) = 0[/tex] and [tex]
\frac{\partial}{\partial z}\left(\frac{\partial p}{\partial z}\right) =\frac{\partial f}{\partial z} = 0.[/tex] Hence [itex]f(r) = \frac{\partial p}{\partial z}[/itex] is constant.
 

1. What is Navier-Stokes equation?

The Navier-Stokes equation is a mathematical equation that describes the motion of fluids, such as liquids and gases. It takes into account factors such as fluid velocity, pressure, and density, and is widely used in fluid dynamics to model and predict the behavior of fluids in various scenarios.

2. How is Navier-Stokes equation derived?

The Navier-Stokes equation is derived from the fundamental laws of physics, namely the conservation of mass, momentum, and energy. It is a set of nonlinear partial differential equations that can be solved using various numerical methods.

3. What is the significance of Navier-Stokes equation?

The Navier-Stokes equation is of great significance in many fields of science and engineering, including fluid mechanics, aerodynamics, weather forecasting, and oceanography. It allows us to understand and predict the behavior of fluids in different situations, which is crucial for many practical applications.

4. Why is DP/Dz constant in Navier-Stokes equation?

In the Navier-Stokes equation, the term DP/Dz represents the gradient of pressure with respect to the vertical coordinate z. This value is typically assumed to be constant, as the pressure in a fluid is affected by gravity, and the vertical change in pressure is small compared to the horizontal change. This assumption simplifies the equation and makes it easier to solve.

5. What are the limitations of Navier-Stokes equation?

The Navier-Stokes equation is a simplified model of fluid flow and has some limitations. It assumes that fluids are continuous and have constant properties, which may not always be the case in real-world scenarios. It also neglects factors such as turbulence and viscosity, which can greatly affect the behavior of fluids. Additionally, the equation is only valid for Newtonian fluids and cannot accurately describe non-Newtonian fluids.

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