Divergence and flux

Divergence measures local volume imbalance

Divergence tells whether a vector field carries more flow out of a tiny volume than into it. Positive divergence acts like a local source; negative divergence acts like a sink.

Compare flux through a small box

For u = (u_x,u_y,u_z), net outward flux from a box ΔxΔyΔz is approximately [∂u_x/∂x + ∂u_y/∂y + ∂u_z/∂z]ΔV. Divide by volume and shrink the box: ∇·u = ∂u_x/∂x + ∂u_y/∂y + ∂u_z/∂z.

For u = (2x,−y,3z), divergence = 2 − 1 + 3 = 4 s⁻¹. A small material volume expands locally at a rate proportional to 4 per second if this velocity field is present.

Incompressible does not mean velocity is constant

Incompressibility requires ∇·u = 0, allowing speed and direction to vary as long as local volume is preserved. Density variation adds the mass-balance term ∂ρ/∂t + ∇·(ρu) = 0.

Work it: Compute divergence for u = (x², xy, −2z) at (1,2,1). Decide whether a constant-density fluid could follow this field incompressibly.