transportteorem

The transportteorem, commonly known as the Reynolds transport theorem, is a result in continuum mechanics that relates the time rate of change of an integral over a region whose boundary may move or deform with time to the local time derivative and to the flux across the moving boundary. It provides a link between Lagrangian and Eulerian descriptions of a field.

For a time-dependent scalar field f(x,t) defined on a moving region V(t) with boundary ∂V(t) moving with velocity w(x,t), the theorem states:

d/dt ∫_{V(t)} f dV = ∫_{V(t)} ∂f/∂t dV + ∮_{∂V(t)} f (w · n) dS,

where n is the outward normal to the boundary and dS is the surface element.

A common specialization is when the boundary moves with the fluid velocity u, so w = u. Applying the divergence theorem gives:

d/dt ∫_{V(t)} f dV = ∫_{V(t)} [∂f/∂t + ∇·(f u)] dV.

If V(t) is fixed in space (w = 0), the theorem reduces to d/dt ∫_{V} f dV = ∫_{V} ∂f/∂t dV.

Applications include deriving conservation laws in fluid dynamics and continuum mechanics, such as mass, momentum, and energy balances. The theorem also underpins the passage between Eulerian and Lagrangian descriptions and is named after Osborne Reynolds, who popularized its use in the 19th century. It remains a foundational tool in modeling transport phenomena and deforming media.