Meccanica dei fluidi

The governing equations of motion for a viscous fluid are the Navier-Stokes equations. These are a set of nonlinear partial differential equations that arise from applying Newton’s second law to fluid motion, combined with the assumption that fluid stress is the sum of a diffusing viscous term and a pressure term. For a compressible Newtonian fluid, the vector equation is: [latex]\rho(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \nabla \cdot \mathbf{T} + \mathbf{f}[/latex], where [latex]\rho[/latex] is density, [latex]\mathbf{v}[/latex] is velocity, [latex]p[/latex] is pressure, [latex]\mathbf{T}[/latex] is the stress tensor, and [latex]\mathbf{f}[/latex] represents body forces. Solving these equations is a central challenge in the field.

Fluid behavior is often characterized by dimensionless numbers. The most famous is the Reynolds number (Re), which describes the ratio of inertial forces to viscous forces and is used to predict the transition from smooth, orderly laminar flow to chaotic turbulent flow. Other important numbers include the Mach number for compressible flows and the Froude number for flows with a free surface. Due to the complexity of the governing equations, especially for turbulent flows, computational fluid dynamics (CFD) has become an essential tool, using numerical methods to solve and analyze problems involving fluid flows.