What is reynolds transport theorem?

Reynolds transport theorem is a fundamental concept in fluid mechanics and transport phenomena that is used to derive the conservation equations for mass, momentum, and energy in a fluid flow system. It is named after the British scientist Osborne Reynolds.

The theorem states that the rate of change of any extensive property of a system within a control volume is equal to the sum of the rate of change of that property within the control volume and the net flow rate of that property into or out of the control volume through its boundaries. Mathematically, the theorem can be written as:

d/dt ∫ ρφ dV = ∫ ρ (∂φ/∂t + ∇·(φv)) dV + ∫ (∂φ/∂n) dA,

where ρ is the density of the fluid, φ is the extensive property being studied, v is the velocity field of the fluid, ∂φ/∂t represents the time rate of change of φ, ∂φ/∂n represents the flux of φ through the boundaries of the control volume, and ∇·(φv) represents the divergence of the quantity φ multiplied by the velocity field v.

Reynolds transport theorem is used to derive the continuity equation, the Navier-Stokes equations, and the energy equation for fluid flow problems. It is a powerful tool for analyzing and solving complex fluid flow systems in engineering and physics.