Navier–Stokes equations

ConvectiveAcceleration vectorized

NSConvection vectorial

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Navier–Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances, such as liquids and gases. These equations are fundamental in the field of fluid dynamics, providing a mathematical model for the behavior of fluids in various conditions and applications, from weather forecasting to aircraft design. The equations are named after Claude-Louis Navier and George Gabriel Stokes, who made significant contributions to fluid mechanics in the 19th century.

Overview

The Navier–Stokes equations express the principles of conservation of momentum and conservation of mass for fluids. They can be used to model a wide range of fluid dynamics problems such as ocean currents, blood flow in the human body, and the flow of air over an aircraft wing. The equations are complex and, in most cases, cannot be solved exactly. Solutions are often obtained through numerical methods and simulations.

Mathematical Formulation

The Navier–Stokes equations can be written in several forms, but the most common is the vector form, which for an incompressible fluid is:

\[ \nabla \cdot \mathbf{v} = 0 \]

\[ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{v} + \mathbf{f} \]

where:

• \(\mathbf{v}\) is the velocity field of the fluid,
• \(t\) is time,
• \(\rho\) is the density of the fluid,
• \(p\) is the pressure field,
• \(\nu\) is the kinematic viscosity of the fluid, and
• \(\mathbf{f}\) represents external forces acting on the fluid, such as gravity.

Challenges and Unsolved Problems

One of the most famous unsolved problems in mathematics is related to the Navier–Stokes equations. The Millennium Prize Problems, established by the Clay Mathematics Institute, include a problem concerning the existence and smoothness of solutions to the Navier–Stokes equations in three dimensions. A proof or disproof of this problem remains one of the most significant open questions in the field of mathematical physics.

Applications

The Navier–Stokes equations are used in many scientific and engineering disciplines. In meteorology, they are fundamental in predicting weather patterns and understanding the dynamics of the atmosphere. In aerospace engineering, they are used to design more efficient aircraft by analyzing air flow around wings. In civil engineering, understanding fluid flow through pipes and channels is essential for the design of water supply systems.

See Also

• Fluid mechanics
• Turbulence
• Computational fluid dynamics
• Bernoulli's principle