What are the Navier-Stokes Equations
This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
The Navier-Stokes equations describe the motion of fluid substances and are fundamental in fluid mechanics. For incompressible flow with a Newtonian model of viscosity, the Navier-Stokes equations are given by:
\rho\frac{\partial{\vec{u}}}{\partial t} + \rho{\vec{u}}\cdot\nabla{\vec{u}} = -\nabla p + \eta\nabla^2{\vec{u}} + \rho {\vec{g}}
In this equation:
- \rho is the fluid density.
- {\vec{u}} is the velocity vector of the fluid.
- t is time.
- p is the pressure.
- \eta is the dynamic viscosity of the fluid.
- {\vec{g}} represents body forces (like gravity) acting on the fluid.
The left side of the equation represents the change in momentum of the fluid at a point in space, whereas the right side accounts for the forces acting on the fluid, including pressure forces, viscous forces, and body forces.
In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:
Textbook sections:
- Appendix F (RELEVANT)
- Chapter 5.6: Euler Equations: Summary of Streamwise, Normal, and Binormal Equations (RELEVANT)
- Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 1 (RELEVANT)
- Chapter 5.8: Models (RELEVANT)
- Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (RELEVANT)
- Appendix D (MARGINALLY RELEVANT)
- Chapter 5.1: Inviscid Incompressible Flow (MARGINALLY RELEVANT)
- Chapter 11.2: Couette Flow (MARGINALLY RELEVANT)
- Chapter 11 Fundamentals: Laminar and Turbulent Flow (MARGINALLY RELEVANT)
- Chapter 11.3: Poiseuille Flow (MARGINALLY RELEVANT)
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