Equations · Physics
Navier–Stokes equation
Newton's second law for a flowing fluid: how velocity changes under pressure, viscosity and outside forces.
ρ(∂u/∂t + u · ∇u) = −∇p + μ∇²u + f
Aethel says
rho times the quantity the partial derivative of u with respect to t plus u dot the gradient of u equals negative the gradient of p plus mu times the Laplacian of u plus f
What it says
The left side is the mass per volume times the acceleration of a fluid parcel. The right side lists what drives it: differences in pressure, internal friction from viscosity, and forces such as gravity.
For an incompressible fluid it is paired with the condition that the divergence of u is zero. Whether smooth solutions always exist in three dimensions is one of the Clay Millennium Prize Problems.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ρ | rho | the density of the fluid |
| u | u | the velocity field |
| p | p | the pressure |
| μ | mu | the dynamic viscosity |
| f | f | body forces per unit volume, such as gravity |
More onρRho∇Nabla (del)μMu
Where you meet it
Fluid mechanics, aerodynamics, weather prediction and blood-flow models.
Questions
How do you say Navier–Stokes equation out loud?
ρ(∂u/∂t + u · ∇u) = −∇p + μ∇²u + f is read: “rho times the quantity the partial derivative of u with respect to t plus u dot the gradient of u equals negative the gradient of p plus mu times the Laplacian of u plus f.” That is exactly what Aethel says when it meets the formula in a book.
What does ρ mean in Navier–Stokes equation?
ρ (said “rho”) is the density of the fluid.
What does u mean in Navier–Stokes equation?
u (said “u”) is the velocity field.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.