Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/22

§ 1 The analogy here suggested is not complete. It frequently happens in the case of fluids that the effects of viscosity have to be taken into account as part of the general dynamic system; consequently it is sometimes necessary to devote some attention to these effects even in the preliminary discussion.

The question of compressibility is one on which also it is desirable to have some convention. It is popularly supposed that whereas gases are compressible, liquids are "virtually incompressible; no broad distinction of this kind is justified. The criterion of compressibility in fluid dynamics involves the relative density of the fluid, and on this basis air is only about eighteen times as compressible as water, the ratio of the velocity of sound in water and air being approximately in the proportion of $$\sqrt{18} : 1 .$$ It is shown later that the influence of compressibility only becomes manifest as the velocity of motion approaches the velocity of sound in the fluid in question, or if the pressures developed involve a serious change in density.

The velocities and pressures ordinarily involved in aerial flight are such as will justify the initial assumption that the air is incompressible, that is to say, that the influence of its compressibility is negligible. The possibility of error resulting from this hypothesis will be considered subsequently. (Appendix I.)

§ 2. Two Methods.—There are two ways in which problems in fluid dynamics may be approached: (1) By the method of the Newtonian medium; this, though of great service in certain special cases, is not strictly applicable to real fluids. (2) By the methods of Euler and Lagrange, by which complete equations of motion are obtained, defining the flow of the fluid in the three co-ordinate dimensions of space. This is the method employed in works on analytical hydrodynamics, and discussed in Chap. III, of the present work.

The basis of the Newtonian method is found in the principle of the conservation of momentum, which maybe taken as corollary to the third law of motion as written: When force acts on a body