Eulerian and Lagrangian descriptions, the material derivative

There are two ways to describe a moving fluid. In the Lagrangian description you label each fluid particle by where it started, \(\mathbf{a}\), and follow its position \(\mathbf{X}(\mathbf{a},t)\). In the Eulerian description you stand at a fixed point \(\mathbf{x}\) and record the velocity \(\mathbf{u}(\mathbf{x},t)\) of whatever particle happens to be there. The two are linked by the statement that particles move with the flow: \[ \frac{\partial \mathbf{X}}{\partial t}(\mathbf{a},t) = \mathbf{u}(\mathbf{X}(\mathbf{a},t),t), \qquad \mathbf{X}(\mathbf{a},0) = \mathbf{a}. \] The map \(\mathbf{a}\mapsto\mathbf{X}(\mathbf{a},t)\) is called the flow map. Both descriptions go back to Euler; the names are a later convention.

Physical laws are stated for particles (a parcel of fluid is accelerated by the forces on it), but measurements and equations are easiest at fixed points. The bridge is the chain rule. If \(f(\mathbf{x},t)\) is any field, its rate of change along a particle path is \[ \frac{d}{dt} f(\mathbf{X}(\mathbf{a},t),t) = \frac{\partial f}{\partial t} + \frac{\partial X_j}{\partial t}\frac{\partial f}{\partial x_j} = \frac{\partial f}{\partial t} + (\mathbf{u}\cdot\nabla) f. \] This operator is the material derivative, \(\dfrac{D}{Dt} = \dfrac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\). The first term is the change seen by a fixed observer; the second is the change caused by being carried to a place where the field is different.

Applied to the velocity itself it gives the acceleration of a fluid particle, \[ \frac{D\mathbf{u}}{Dt} = \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}. \] The second term is quadratic in \(\mathbf{u}\): it is the nonlinearity of fluid mechanics. It also means a steady flow can accelerate its particles. In the rotation \(\mathbf{u} = (-y, x)\) nothing depends on time, yet \((\mathbf{u}\cdot\nabla)\mathbf{u} = (-x,-y)\), the centripetal acceleration of a particle going round a circle.

A second useful fact is how volumes change. Let \(J = \det(\partial X_i/\partial a_j)\) be the Jacobian of the flow map, the factor by which a small blob of particles has changed volume. Differentiating the determinant gives Euler's expansion formula \[ \frac{DJ}{Dt} = (\nabla\cdot\mathbf{u})\,J. \] So the divergence of the velocity is the relative rate of volume growth of a fluid parcel, which is the key to incompressibility.

Picture it: a thermometer bolted to a bridge pier reads \(\partial T/\partial t\); a thermometer on a drifting buoy reads \(DT/Dt\). Think it: \(D/Dt\) is a directional derivative in space-time, along the vector \((\mathbf{u},1)\). For \(T = x^2+y^2\) in the rotation \(\mathbf{u} = (-y,x,0)\), \(DT/Dt = 0\): the temperature a particle carries does not change, because particles move along the circles where \(T\) is constant.