JNTU B.Tech II AERODYNAMICS - I Previous Question Paper (March - 2021 - R18)

The differential form of the momentum equation is derived by applying Newton's Second Law of Motion to an infinitesimal fluid element of volume dx dy dz. The approach involves analyzing the forces acting on this control volume and equating the net force to the rate of change of momentum.

Assumptions:

• The fluid is treated as a continuum.
• The fluid is Newtonian, meaning viscous stresses are linearly proportional to the rate of strain.
• The flow is governed by the conservation of mass and momentum principles.

Derivation Steps:

1. Define a stationary infinitesimal control volume in a Cartesian coordinate system.
2. Identify surface forces (pressure and viscous stresses) and body forces (such as gravity) acting on the element.
3. Apply Newton's Second Law: the sum of forces in a given direction equals the rate of change of momentum for the fluid particle passing through the control volume.
4. Express the net pressure forces as gradients, such as -∂p/∂x dx dy dz.
5. Include viscous stress terms using the stress tensor components, accounting for both normal and shear stresses.
6. Combine these into the Navier-Stokes equations, which represent the momentum balance in differential form.

Salient Points:

The resulting equations, known as the Navier-Stokes equations, are non-linear partial differential equations. They account for convection, pressure gradients, viscous diffusion, and external body forces. For an incompressible flow with constant viscosity, the momentum equation in the x-direction simplifies to:

These equations are fundamental to aerodynamics as they describe the motion of viscous fluids and are essential for calculating drag, lift, and boundary layer development over aerodynamic surfaces.