By Sam Howison

Drawing from an exhaustive number of mathematical topics, together with genuine and intricate research, fluid mechanics and asymptotics, this ebook demonstrates how arithmetic may be intelligently utilized in the particular context to quite a lot of commercial makes use of. the amount is directed to undergraduate and graduate scholars.

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**Example text**

Example: the drag on a cylinder Suppose a cylinder of length L and radius a is held in a viscous fluid moving with far-field velocity U normal to the axis of the cylinder. How does the drag force depend on the parameters of the problem? What happens as L → ∞? There are six independent physical quantities in this problem: r L and a, which are properties of the cylinder and which both have dimension [L]; 11 From which the name of the theorem comes. 3 Buckingham’s Pi-theorem 41 r µ and ρ, which are properties of the fluid and have dimensions [M][L]−1 [T]−1 , [M][L]−3 respectively; r the force F on the cylinder ([M][L][T]−2 ) and the free-stream velocity U ([L][T]−1 ); F is an output (or dependent) variable, and we aim to express it in terms of the inputs.

It is often easier to analyse advection because we can usually decouple the question of finding the fluid flow from the heat-flow problem. In convection, the buoyancy force that drives the flow is strongly temperature dependent – indeed, without it there would be no flow – and the problem is correspondingly more difficult. For our first example, we’ll consider the two-dimensional flow of an incompressible liquid with a given (that is, we can calculate it separately) velocity u past a circular cylinder of radius a, with a free-stream velocity (U , 0) at large distances.

The electrostatic potential φ satisfies ∇ 2 φ = 0 for r > a, where r is the distance from the centre of the sphere. If the sphere is raised to a voltage V relative to a potential of zero at infinity, we have φ = V on r = a and φ → 0 as r → ∞. We (you) can write down φ immediately. Now use Gauss’ flux theorem, also known as the divergence theorem, on a sphere r = a+ to show that the total charge on the sphere is −ε0 r =a ∂φ dS ∂r and deduce that the capacitance of the sphere is 4π ε0 a. A capacitor with capacitance C is charged up to voltage V and discharged to earth (voltage 0) through a resistor of resistance R.