4  Problem class 1

Abstraction

It may seem strange to study examples from elementary physics in a course that is supposed to be about Planet Earth. But simple examples are the best ways to learn these important techniques. The full climate equations are often very involved. These toy models still nevertheless capture the spirit of what you must do when attacking any scientific problem.

In this problem class, we will practice some concepts about non-dimensionalising in preparation for the Problem Set 1 in Chapter 26. We will cover these two strategies in choosing scalings.

Scaling principle 1

Select the characteristic scales so that as many of the possible non-dimensional numbers, \(\Pi_i\), \(i = 1, 2, 3, \ldots\) are normalised.

Scaling principle 2

Select characteristic scales so that no terms in the model diverge in the physical limit of interest.

Here are the problems we shall do in the problem class.

4.1 Projectile motion

A projectile of mass \(M\) (in kg) is launched vertically with initial velocity \(V_0\) (in m/s) from a position \(Y_0\) (in m) above the surface. Thus the mass’s position, \(Y(t)\) is governed by Newton’s second law (applied to the mass and the mass of the Earth) and the set of equations \[ \begin{gathered} M Y_{tt} = - \frac{g R_E^2 M}{(R_E + Y)^2}, \\ Y(0) = Y_0, \end{gathered} \] where \(g = 9.81 m/s^2\) and \(R_E = 6.4 \times 10^6 m\) is the radius of the Earth.

  1. Non-dimensionalise the equation using arbitary length and time scales.

  2. Identify the non-dimensional constants, \(\Pi_i\).

  3. Choose a length scale of \(L = Y_0\) and time scale of \(T = (L/g)^{1/2}\). Discuss the resultant equation and the interpretation of choosing these scales.

  4. Does your above choice allow you to easily study the limit of \(R_E \to \infty\)? If the limit can be taken, reduce the governing system to a simpler equation.

  5. Does your choice in 3. allow you to easily study the limit of \(Y_0 \to 0\)? If not, choose an alternative choice of length and time scales and in that case, reduce the set of equations.

4.2 Terminal velocity

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A ball of radius \(R\) (in m) and uniform density \(\rho\) (in kg/m3) falls in a viscous fluid. The fluid has density \(\rho_f\) (in kg/m3) and viscosity (a measure of friction or resistance) \(\mu\) (in kg/(m s)). The equation that governs the velocity is \[ \begin{gathered} \frac{4}{3} \pi R^3 \rho \frac{\mathrm{d}V}{\mathrm{d}t} = \frac{4}{3} \pi R^3 (\rho - \rho_f) g - 6\pi \mu R V, \\ V(0) = V_0. \end{gathered} \]

  1. Choose appropriate velocity and time scales to non-dimensionalise the equation so as to leave only a single non-dimensional number on the drag term (the last term on the right hand-side).

  2. Define the non-dimensional parameter expressing a ratio between drag force and gravity force by the Stokes number (St) and confirm that it is \[ St = \frac{9\mu V_0}{2(\rho - \rho_f) g R^3}. \]

  3. Comment on the two limits of \(St \to 0\) and \(St \to \infty\). Can the problem be reduced in these two limits? If so, reduce and solve.