By Derek F. Lawden
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Extra resources for A Course in Applied Mathematics, Vols 1 & 2
Qn ). ). Then there are n − r dimensionless11 combinations Πi (Qj ) and a function 11 And eponymous. CHAPTER 4. DIMENSIONAL ANALYSIS 52 g such that Π1 = g(Π2 , . . , Πn−r ). Example: the drag on a cylinder Suppose a cylinder of length L and radius a is held in 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 6 independent physical quantities in this problem: • L and a, which are properties of the cylinder and both have dimensions [L]; • µ and ρ, which are properties of the ﬂuid and have dimensions [M][L]−1 [T]−1 , [M][L]−3 respectively; • The force F on the cylinder ([M][L][T]−2 ) and the free stream velocity U ([L][T]−1 ).
So here’s a selection of three relatively simple physical situations where we can see the technique in action. 1 Example: advection-diﬀusion We’ll start with a combination of two very familiar models, heat conduction and ﬂuid ﬂow. When you stand in front of a fan to cool down, two mechanisms come into play: heat is conducted (diﬀuses) into the air, and is then carried away by it. The process of heat transfer via a moving ﬂuid is called 39 CHAPTER 4. 1 Both advection and convection are major mechanisms for heat transfer in systems such as the earth’s atmosphere, oceans and molten core; almost any industrial process (think of cooling towers as a visible example); car engines; computers; you name it.
We exploit this by arbitrarily (in fact it is a cunning choice from the practical point of view) setting µ0 = 4π × 10−7 H m−1 and then deﬁning the ampere as the current that makes F exactly equal to 2 × 10−7 N m−1 . We think of a current as generating a magnetic ﬁeld, denoted by B. Remember iron The Lorenz force law states that the force on a charge q moving with ﬁlings experiments to show the velocity v in an electric ﬁeld E and magnetic ﬁeld B is F = q(E + v ∧ B). Deduce the fundamental units of B (SI unit the tesla, T).
A Course in Applied Mathematics, Vols 1 & 2 by Derek F. Lawden