08 / CHAPTER 07
Material derivative
Follow a parcel through an Eulerian velocity field.
ANCHOR EQUATION
Material derivative
In symbols: Dφ/Dt = ∂φ/∂t + u·∇φ
The field-following derivative used throughout continuum physics.
The material derivative follows matter
A fluid description at fixed coordinates is Eulerian; a parcel-following description is Lagrangian. The material derivative translates between them and is the acceleration operator for fluid velocity.
Apply the chain rule to a parcel
A parcel at x(t) moves with dx/dt = u. For any field φ, Dφ/Dt = ∂φ/∂t + u·∇φ. Setting φ = u gives Du/Dt, the parcel acceleration. This is why a steady flow can still accelerate a parcel when speed changes along a streamline.
In a steady nozzle u(x) = 2 + x m/s along x, a parcel has Du/Dt = u du/dx = (2+x)(1) m/s². At x = 1 m, acceleration is 3 m/s² even though ∂u/∂t = 0.
Local and material acceleration can cancel
A parcel can move through a field whose spatial and temporal changes oppose. Dropping the advective term u·∇u is valid only for uniform flow or a justified approximation, not merely because the flow is steady.
Work it: For u(x,t) = 3t + 2x², calculate ∂u/∂t, u∂u/∂x, and Du/Dt at x = 1, t = 2. Identify the local and advective contributions.