JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III HIGH SPEED AERODYNAMICS Previous Question Paper (August - 2022 - R18)

To derive the momentum equation for compressible flow, consider an infinitesimal fluid element of dimensions dx, dy, and dz in a Cartesian coordinate system. Applying Newton's Second Law, the rate of change of momentum of the fluid element equals the sum of the forces acting on it.

The forces acting on the element include surface forces (pressure and viscous stresses) and body forces (such as gravity). Focusing on the x-direction, the net surface force due to pressure and viscous stress components is given by the divergence of the stress tensor.

In this equation, ρ is the fluid density, u, v, w are velocity components, p is the static pressure, and τ_ij represents the viscous stress components. The term on the left represents the material derivative of the velocity component, accounting for both local and convective acceleration.

For a Newtonian fluid, the viscous stresses are related to the velocity gradients through the dynamic viscosity μ and the second coefficient of viscosity λ (often expressed as λ = -2/3μ via the Stokes hypothesis). Substituting these constitutive relations into the momentum balance yields the Navier-Stokes equations for compressible flow:

Similar expressions are derived for the y and z directions by cyclic permutation of the variables. These equations describe the conservation of momentum for a compressible fluid by accounting for pressure gradients, viscous dissipation, and body forces acting on the control volume.