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Governing Navier-Stokes Equations

The fundamental set of partial differential equations describing fluid motion. For incompressible flow, the momentum equation is ρ(u⃗⋅∇)u⃗=−∇p+μ∇2u⃗+f⃗\rho (\vec{u} \cdot \nabla) \vec{u} = -\nabla p + \mu \nabla^2 \vec{u} + \vec{f}, where ρ\rho is density and μ\mu is dynamic viscosity.
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The statement of the theorem

For an incompressible, Newtonian fluid (∇⋅u=0\nabla \cdot \mathbf{u} = 0), the governing equations are:\n\n1. **Continuity Equation:**\n∇⋅u=0 \nabla \cdot \mathbf{u} = 0 \n2. **Momentum Equation:**\nρ(u⋅∇)u=−∇p+μ∇2u+f \rho (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f} \nwhere ρ\rho is the density, μ\mu is the dynamic viscosity, u\mathbf{u} is the velocity vector, pp is the pressure, and f\mathbf{f} represents external body forces.
Source: Wikipedia