Chapter 2, Problem 24P

Streaklines are traced out by neutrally buoyant marker fluid injected into a flow field from a fixed point in space. A particle of the marker fluid that is at point (x, y) at time t must have passed through the injection point (x₀, y₀) at some earlier instant t = τ. The time history of a marker particle may be found by solving the pathline equations for the initial conditions that x = x₀, y = y₀ when t = τ. The present locations of particles on the streakline are obtained by setting τ equal to values in the range 0 ≤ τ ≤ t. Consider the flow field $\stackrel{\to }{V}=ax(1+bt)\hat{i}+cy\hat{j}$, where a = c = 1 s⁻¹ and b = 0.2 s⁻¹. Coordinates are measured in meters. Plot the streakline that passes through the initial point (x₀, y₀) = (1, 1), during the interval from t = 0 to t = 3 s. Compare with the streamline plotted through the same point at the instants t = 0, 1, and 2 s.