The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances, taking into account conservation of mass, momentum, and energy. They are named after Claude-Louis Navier and George Gabriel Stokes, who made significant contributions to fluid mechanics. The three-dimensional form of the Navier-Stokes equations for an incompressible fluid can be expressed as follows:
Continuity Equation (Conservation of Mass):
∇ · (ρv) = 0
Momentum Equations (Conservation of Momentum):
ρ[(∂v/∂t) + (v · ∇)v] = -∇p + μ∇²v + ρg
Energy Equation (Conservation of Energy):
ρ[(∂T/∂t) + (v · ∇)T] = ∇ · (k∇T) + Q
In these equations:
- ρ represents the density of the fluid.
- v = (u, v, w) represents the velocity vector of the fluid in the x, y, and z directions, respectively.
- t represents time.
- p represents the pressure.
- μ represents the dynamic viscosity of the fluid.
- g represents the acceleration due to gravity.
- k represents the thermal conductivity of the fluid.
- T represents the temperature of the fluid.
- Q represents any heat source or sink.
The equations describe the interactions between the fluid's velocity, pressure, density, and temperature fields. The continuity equation ensures that mass is conserved, stating that the divergence of the velocity field must be zero, meaning that the rate of change of fluid mass within a given region is balanced.
The momentum equations describe the conservation of momentum, accounting for the forces acting on the fluid. The left-hand side represents the change in momentum over time, the convective term accounts for the advection of momentum, the pressure gradient contributes to the overall force, the viscous term represents the diffusion of momentum due to internal friction, and the gravitational term represents the effect of gravity.
The energy equation governs the conservation of energy within the fluid. It relates the rate of change of temperature to the advection of heat, the diffusion of heat through conduction, and any external heat sources or sinks present in the system.
Solving the Navier-Stokes equations analytically is often infeasible for most practical applications due to their complex nature. Therefore, numerical methods, such as finite difference, finite volume, or finite element methods, are used to discretize and solve the equations on a computational grid or mesh. These numerical solutions provide valuable insights into fluid behavior, allowing engineers and scientists to study and analyze various flow phenomena.