The Navier-Stokes equations describe the motion of viscous fluid substances. They are a set of nonlinear partial differential equations that predict how the velocity field of a fluid evolves over time: \[ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \nabla \cdot \mathbf{T} + \mathbf{f} \] where \( \mathbf{u} \) is the velocity field, \( p \) is the pressure, \( \mathbf{T} \) is the stress tensor, and \( \mathbf{f} \) represents body forces.
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