Kinematics: streamlines, pathlines and streaklines

A velocity field can be drawn in three ways, and they answer three different questions. A streamline is a curve tangent to \(\mathbf{u}\) at one frozen instant: along it \(d\mathbf{x}\) is parallel to \(\mathbf{u}\), so in two dimensions \[ \frac{dy}{dx} = \frac{v(x,y,t)}{u(x,y,t)}, \qquad t \text{ fixed}. \] A pathline is the actual trajectory of one particle, the solution of \(d\mathbf{X}/dt = \mathbf{u}(\mathbf{X},t)\). A streakline is what you see when dye is released continuously from a fixed point \(\mathbf{x}_0\): at time \(t\) it is the set of positions of all particles that passed through \(\mathbf{x}_0\) at earlier times \(s \le t\).

For a steady flow the three coincide: the field never changes, so a particle simply runs along a streamline and every later particle from the same point follows it. For an unsteady flow they differ, and the difference is easy to compute. Take \(\mathbf{u} = (1, t)\). At time \(t\) the streamlines are straight lines of slope \(t\). The particle that starts at the origin has \(X = t\), \(Y = t^2/2\), so its pathline is the parabola \(y = x^2/2\). A particle released from the origin at time \(s\) is at \(X = t-s\), \(Y = (t^2-s^2)/2\) at time \(t\); eliminating \(s\) gives the streakline \(y = tx - x^2/2\) for \(0 \le x \le t\). Three different curves from one field.

Streamlines of the steady rotation \(\mathbf{u} = (y,-x)\) satisfy \(dy/dx = -x/y\), a separable equation whose solutions are the circles \(x^2+y^2 = C\). Streamlines of the strain \(\mathbf{u} = (x,-y)\) satisfy \(dy/dx = -y/x\), giving the hyperbolas \(xy = C\). A bundle of streamlines through a small closed curve forms a stream tube; no fluid crosses its wall, because the velocity is tangent to it.

Points where \(\mathbf{u} = 0\) are stagnation points, and near one the flow looks like its linearisation \(\mathbf{u} \approx (\nabla\mathbf{u})\,\mathbf{x}\). The eigenvalues of the velocity gradient classify the picture: real of opposite signs gives a saddle (the strain \((x,-y)\) has eigenvalues \(1\) and \(-1\)), purely imaginary gives a centre (the rotation).

Picture it: iron filings on a sheet show streamlines; a long-exposure photograph of one lit particle shows a pathline; smoke from a chimney on a gusty day shows a streakline. Think it: streamlines are integral curves of a time-dependent vector field at a frozen time, pathlines are integral curves in space-time, and streaklines are images of a curve of release times under the flow map. Only the pathline is a solution of an ODE in time.