In this series:
- The Material Derivative in Cartesian Coordinates
- The Material Derivative in Cylindrical Coordinates
- The Material Derivative in Spherical Coordinates
- The Material Derivative in Vector Form
- The Reynolds Transport Theorem
- How Euler Derived the Continuity Equation
- The Continuity Equation in Cartesian Coordinates
- The Continuity Equation in Cylindrical Coordinates
- The Momentum Equation in Cartesian Coordinates
- The Momentum Equation in Cylindrical Coordinates
Cite as:
Saad, T. "Derivation of the Navier Stokes and Fluid Flow Equations." Weblog entry from
Please Make A Note.
http://pleasemakeanote.blogspot.com/2008/08/derivation-of-navier-stokes-equations_16.html
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3 comments:
You are going to have enough material for a book when you are done with these posts.
I found the derivation of the substantive derivative in cylindrical coordinates extremely helpful. (Why are those derivations omitted from the texts? ) Too bad the post for the momentum equations in cylindrical c.s. has not been created yet.
Here is what I am struggling with. Consider a case of Newtonian, incompressible fluid. The momentum equations for the radial and angular directions have two additional viscosity terms. Where do they come from and what is their physical significance?
Hi Anton,
Thank you for your feedback. I am very glad to hear that these posts have been of help to you.
The question you raised is a very interesting one and I was going to discuss it in the posts for the momentum equations. Unfortunately, I haven't had a chance to do so as I am extremely busy at the current time.
However, to quickly address your question, please consider the following argument.
As you may already know, the viscous terms stem from the surface forces on a fluid element which are described by the divergence of the stress tensor. By expanding that in curvilinear coordinates, you will end up with derivatives of unit vectors (just like in the case of the material derivative). At the outset, you will recover some additional terms. I have worked out the math for that, and I will post it as soon as possible.
This is the mathematical reason of why these terms show up in curvilinear coordinates. Nonetheless, as far as I know, I don't think that they hold any specific physical significance. They are only a byproduct of the coordinate system that you have chosen so that the net viscous force on a fluid element will be the same regardless.
I hope that answers your question. If I find something else on the physicality of these terms, I will let you know.
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