Question: ρ(\frac{\partial u}{\partial t} + u \cdot \nabla u) = -\nabla p + \mu \nabla^{2} u + f
1. **State the problem:**
We are given the Navier-Stokes equation for fluid dynamics:
$$\rho\left(\frac{\partial u}{\partial t} + u \cdot \nabla u\right) = -\nabla p + \mu \nabla^{2} u + f$$
2. **Explain the terms:**
- $\rho$ is the fluid density.
- $u$ is the velocity vector field.
- $\frac{\partial u}{\partial t}$ is the partial derivative of velocity with respect to time.
- $u \cdot \nabla u$ represents the convective acceleration.
- $-\nabla p$ is the pressure gradient force.
- $\mu \nabla^{2} u$ is the viscous diffusion term, where $\mu$ is dynamic viscosity.
- $f$ is the external body force per unit volume.
3. **Interpretation:**
This equation balances the rate of change of momentum with forces acting on the fluid.
4. **No further simplification requested.**
**Final answer:** The Navier-Stokes equation is as given above, describing fluid motion under forces.