Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/79

Rh there is for a given value of $$R$$ a corresponding value of $$V$$ such that $$V\ l$$ is constant; this is merely an expression of Equation (7) $$V\ l = c\ \nu .$$ But this holds good equally when the curve is read as an $$R\ l$$ curve, the value of the constant product of $$V\ l$$ being unaffected; consequently in reading the curve either as $$R\ V$$ or $$R\ l$$ the units are interchangeable. It may be noted that $$l$$ may be the length, i.e., the axial dimension of the body, or the transverse or some other dimension, without affecting the result, provided it is in all cases the same, and thus truly represents the linear size of the body.

§ 43. Other Relations.—In considering the relations of the curve of resistance we have hitherto taken the kinematic viscosity $$\nu$$ as constant; we will now study the consequences of taking this as a variable. So far the treatment has covered the case of variations of the velocity and linear dimensions of bodies in a fluid of constant physical properties; in supposing the viscosity to vary-we are introducing the condition of a change of fluid, or at any rate such a change in the physical state of the fluid as is equivalent thereto.

Now, (7) $$V = c\ \frac{\nu}{l}$$ is the equation to similar systems, so that the similar system when $$\nu$$ varies is found when $$V$$ or $$l$$ or their product $$V\ l$$ varies in like ratio, that is the scale value of the axis of $$x$$ varies with the kinematic viscosity. But by (5) $$R = c\ \nu^q\ (l\ V)^r ,$$ or for similar systems where $$l\ V \propto \nu ,$$ we have $$R \propto \nu^q\ \nu^r ,$$ where $$q + r = 2 ,$$ that is, $$R \propto \nu^2 ,$$ or the scale value of the axis of $$y$$ varies as the square of the kinematic viscosity.

The conclusion may therefore be stated that:—The resistance of a body of any definite geometrical form, in a stated aspect, may he represented as a function of its linear dimension (that is its size) and its velocity, by means of a single curve which may be termed its characteristic curve of resistance, the form of which is constant whether the abscissae represent linear dimension or velocity, and whatever the value of the kinematic viscosity may be. And further,