Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/99

Substituting from (1) for ${\frac {Du}{Dt}},{\frac {Dv}{Dt}},$ and ${\frac {Dw}{Dt}}$ we have:—

${\frac {du}{dt}}+u{\frac {du}{dx}}+v{\frac {du}{dy}}+w{\frac {du}{dz}}$	$=-{\frac {dp}{\rho dx}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
${\frac {dv}{dt}}+u{\frac {dv}{dx}}+v{\frac {dv}{dy}}+w{\frac {dv}{dz}}$	$=-{\frac {dp}{\rho dy}},$
${\frac {dw}{dt}}+u{\frac {dw}{dx}}+v{\frac {dw}{dy}}+w{\frac {dw}{dz}}$	$=-{\frac {dp}{\rho dz}}.$

In the steady state ${du}/{dt}$ is zero, and the equations become:—

$u\ {\frac {du}{dv}}+v\ {\frac {du}{dy}}+w\ {\frac {du}{dz}}=-{\frac {dp}{\rho dx}},\ {\mbox{etc.}}$

When there is no motion or motion of translation only in the fluid, the last three terms of the left-hand side of the equation are zero, and the equations become:—

${\frac {du}{dt}}=-{\frac {dp}{\rho dx}},\ {\mbox{etc.}}$

The further development and employment of these equations is outside the scope of the present work, but the physical significance can be gathered by comparison with §§ 60 and 88.

The mathematical superstructure founded on the above consists in the main of finding solutions to the equations of motion in a number of well-defined cases, and in the general development of the theory in its application to the motions of bodies of stated geometrical form under known boundary conditions.^[1]

 

§ 60. Velocity Potential ($\phi$ Function).—If a force be applied to a body initially at rest in a fluid, a circulation of the fluid is set up, the flow taking place along paths of curvilinear form by which the displaced fluid is conveyed from one side of the body to the other. We may regard the initial direction of flow, produced in this manner, as denoting a "field of force," the

1. For the full mathematical treatment reference should be made to "Hydrodynamics," H. Lamb, Camb. University Press.

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