The Navier–Stokes equations are fundamental equations in fluid dynamics that describe the motion of fluid substances. They can be derived from the Cauchy momentum equation, which is a more general form applicable to various fluid flow situations. Understanding this derivation requires a grasp of both the Cauchy momentum equation and the assumptions that lead to the specific form of the Navier–Stokes equations.
Cauchy Momentum Equation
The Cauchy momentum equation is expressed in its general convective form as:
∂(ρu)/∂t + ∇·(ρu ⊗ u) = -∇p + μ∇²u + f
In this equation:
ρ is the fluid density.
u is the velocity vector of the fluid.
p is the pressure within the fluid.
μ is the dynamic viscosity of the fluid.
f represents body forces (e.g., gravitational forces).
The left-hand side of the equation accounts for the change in momentum due to both local and convective acceleration, while the right-hand side describes the forces acting on the fluid element, including pressure gradients, viscous forces, and external forces.
Deriving the Navier–Stokes Equations
To derive the Navier–Stokes equations from the Cauchy momentum equation, we typically make several assumptions:
The fluid is Newtonian, meaning that the viscous stress is linearly proportional to the strain rate.
The fluid is incompressible, which implies that the density (ρ) is constant.
With these assumptions, the viscous term can be simplified as follows:
μ∇²u = μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²)
Substituting these simplifications into the Cauchy momentum equation, we obtain the incompressible Navier–Stokes equations:
∂u/∂t + (u · ∇)u = -∇p/ρ + ν∇²u + f
Where:
ν = μ/ρ is the kinematic viscosity of the fluid.
This form of the Navier–Stokes equations describes the balance of inertial forces, pressure forces, and viscous forces in an incompressible fluid.
Example Application
The Navier–Stokes equations are used in numerous applications, including:
Weather modeling, where they help predict atmospheric flows.
Aerodynamics, to analyze the flow of air around vehicles and aircraft.
Hydraulics, for understanding flow in pipes and channels.
For instance, in aerodynamics, the behavior of airflow over a wing can be modeled using the Navier–Stokes equations, allowing engineers to optimize the shape for better lift and drag characteristics.
References
Batchelor, G. K. (2000). Introduction to Fluid Dynamics. Cambridge University Press.
Fletcher, C. A. J. (2012). Computational Techniques for Fluid Dynamics. Springer.
White, F. M. (2016). Fluid Mechanics. McGraw-Hill Education.
In summary, the Navier–Stokes equations are derived from the Cauchy momentum equation by applying specific assumptions related to the nature of the fluid and the forces acting upon it. They provide a powerful tool for analyzing fluid motion in a wide range of applications.