Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/263

§ 169. Influence of Viscosity.—The influence of viscosity in the resistance of bodies is to cause a departure from the V-square law. It has been shown (Chap. II.) that the resistance in a viscous fluid can be expressed as a power of the velocity whose index must be less than 2, this form of expression not representing a definite law that holds good over any wide range, but rather defining the rate of change in the quantities round about the values for which the index value may have been determined (§ 40).

Adopting this approximate form of expression, we shall have in prop. i. $x\propto V^{n},$ and assuming (as we are probably justified in assuming) that viscosity has only a negligible influence on the aerodynamic resistance, we have:—

$x\propto V^{n}\quad y\propto {\frac {1}{V^{2}}},$  ∴ $\ x\propto \left({\frac {1}{y}}\right)^{\frac {n}{2}}$

∴ differentiating, we have—

${\frac {dx}{dy}}={\boldsymbol {-}}{\frac {nx}{2y}}.$

Now $x+y$ is minimum when $dx={\boldsymbol {-}}dy,$ that is—

${\boldsymbol {-}}{\frac {nx}{2y}}={\boldsymbol {-}}1,$  or $\ nx=2y,$  or $\ x={\frac {2}{n}}y.$

This is the solution to the equivalent of prop, i., on the modified hypothesis.

Thus if $n=1.75,$ that is to say, if $x=V^{1.75},$ we have the minimum total resistance when $x={\frac {8}{7}}y.$

The necessary modifications to generalise the further propositions in respect of the index of $x$ may be easily effected; the matter, however, has not been pursued further in the present work, the approximate assumption of $n=2$ being deemed sufficient for the needs of the practical application of the theory.

§ 170. The Weight as a Function of the Sail Area.—It has been pointed out, in § 166, that we cannot, strictly speaking, regard the