Navier-Stokes equation for a dummies
I was recently on a holiday with a close friend of mine. We started talking about his research in Computational Physics. We were waiting to board the plane back to SFO, and what better time than that to talk about Fluids — given that most popular forms of long distance travel are through air and water.
Very recently my understanding about how aeroplanes worked was debunked. My previous understanding was that it was due to the lift caused by the difference in pressure above and below a curved surface. Also known as Bernoulli’s Principle. This is incorrect. For the correct explanation, refer: Nasa Article and Dr. Babinsky’s Journal article.
Curious about what equations govern water and more generally fluids, I asked him to, in a few minutes, give me a better model to think about fluids. Following is an attempt at summarizing what I understood:
Kaushik, what do you know about solids?
- Continuity Equation: Mass cannot be created nor destroyed.
- Newton’s second law: F = ma, otherwise known as Newton’s second law
- Conservation of energy: Energy cannot be created nor destroyed.
What makes fluids complex is interaction between the layers. We end up with a modified version of the momentum equation. This modified version should at least account for:
- Forces between the layers of fluids: Pressure, Viscosity
- Energy and its transfer: Thermal, Potential, Kinetic
The Navier-Stokes equations are an expression of Newton’s Second Law for fluids, stating that mass times the acceleration of fluid particles is proportional to the forces acting on them. If we take the Navier-Stokes equations for incompressible flow as an example, which we can write in the form:
The L.H.S is the product of fluid density times the acceleration that particles in the flow are experiencing. This term is analogous to the term m a, mass times acceleration — per the momentum equation. R.H.S has forces that are responsible for particle acceleration:
- Pressure gradient
- Viscous shear stresses
- Volume forces. If the volume forces are due to gravity alone, then we have f = ρg, where g stands for the gravitational acceleration vector.
It’s interesting to note that this equation is very difficult to solve.
Hopefully you have a slightly better understanding of how fluids are modeled.