In physics, continuity equation often reads as ∂ρ/∂t+∇⋅(ρu)=0. Obviously, if velocity field u is solenoidal, the equation degenerates to dρ/dt=∂ρ/∂t+u⋅∇ρ=0. That is, the total derivative of density is zero. Then the question is arise. Two equations above both state that the density is independent of time, why can't that the total derivative vanishes imply continuity equation? Is it possible that there exists some real total derivative equivalent to that in the continuity equation?
Expand divergence of (rho*u) in the continuity equation. You get (u.grad rho) + rho* divergence u . The first of these added to partial rho by partial t combines to become total derivative of rho. Thus for incompressible flows you will get total derivative of rho is zero thus continuity becomes rho* divergence velocity = 0.
d(rho)/dt=0 only means the flow is "incompressible" while the continuity equation deals with compressible and incompressible flows.
Continuity equation is a general form while incompressible flow is a special case.
- Badges
- Science topic
- Similar topics
- Engineering
- Thermal Engineering
- Thermofluid
- Fluid Mechanics