The Navier-Stokes equations for compressible fluids incorporate the conservation equations for mass, momentum, and energy, taking into account variations in fluid density due to changes in pressure and temperature. The three-dimensional form of the compressible Navier-Stokes equations can be expressed as follows:

Continuity Equation (Conservation of Mass):

∂ρ/∂t + ∇ · (ρv) = 0

Momentum Equations (Conservation of Momentum):

∂(ρv)/∂t + ∇ · (ρvv) = -∇p + ∇ · τ + ρg

Energy Equation (Conservation of Energy):

∂(ρE)/∂t + ∇ · (ρEv) = -∇ · (p v) + ∇ · (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.

- τ represents the stress tensor, accounting for the viscous forces.

- g represents the acceleration due to gravity.

- E represents the total energy per unit volume.

- 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 conservation of mass, momentum, and energy for a compressible fluid. The continuity equation ensures that mass is conserved, with the time derivative of density and the divergence of the density times velocity being zero.

The momentum equations describe the conservation of momentum, accounting for the forces acting on the fluid. The left-hand side represents the time derivative of the momentum, and the right-hand side includes the pressure gradient, the viscous stress tensor, and the gravitational force.

The energy equation governs the conservation of energy within the fluid. It relates the time derivative of the total energy to the pressure gradient, the diffusion of heat through conduction, and any external heat sources or sinks present in the system.

Solving the compressible Navier-Stokes equations analytically is challenging, and closed-form solutions are generally limited to simplified cases. Therefore, numerical methods, such as finite volume or finite element methods, are commonly employed to discretize and solve the equations on a computational grid or mesh. These numerical solutions enable the study of compressible fluid flows and provide valuable insights into phenomena such as shock waves, supersonic and hypersonic flows, and combustion processes.