08 / CHAPTER 02
Partial derivatives and total change
Distinguish holding a coordinate fixed from following a moving system.
ANCHOR EQUATION
Total derivative
In symbols: dφ/dt = ∂φ/∂t + v·∇φ
A moving observer sees local change plus change from moving through the field.
Partial and total derivatives answer different questions
A fixed probe measures local time change at one point; a moving fluid parcel experiences local change plus the change caused by moving into a different part of the field. Confusing them changes the physical question.
Apply the chain rule along a path
For φ(x(t),y(t),z(t),t), dφ/dt = ∂φ/∂t + (dx/dt)∂φ/∂x + (dy/dt)∂φ/∂y + (dz/dt)∂φ/∂z. With velocity u, this is dφ/dt = ∂φ/∂t + u·∇φ. The first term is local evolution; the second is advection.
Let T = 20 + 0.5t + 2x and a probe move at u = 1 m/s. A fixed probe sees ∂T/∂t = 0.5°C/s; the moving probe sees dT/dt = 0.5 + 2(1) = 2.5°C/s.
Uniform fields erase advection
If ∇φ = 0, motion through the field changes nothing. If the field is steady at fixed points, ∂φ/∂t = 0, but a moving parcel can still experience change.
Work it: For φ = x² + t and a particle x = 3t, calculate partial and total time derivatives at t = 2. Explain which derivative a thermometer attached to the particle records.