A general theorem on the motion of a fluid with friction and a few results derived from it

Abstract

The equations of motion for an incompressible fluid with friction can be written as follows

$$ \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0 $$

((1))

$$ \left. \begin{gathered} \rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}}} \right) = X + \left( {\frac{{\partial {X_x}}}{{\partial x}} + \frac{{\partial {X_y}}}{{\partial y}} + \frac{{\partial {X_z}}}{{\partial z}}} \right) \hfill \\ \rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}}} \right) = Y + \left( {\frac{{\partial {X_x}}}{{\partial x}} + \frac{{\partial {X_y}}}{{\partial y}} + \frac{{\partial {X_z}}}{{\partial z}}} \right) \hfill \\ \rho \left( {\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}}} \right) = Z + \left( {\frac{{\partial {X_x}}}{{\partial x}} + \frac{{\partial {X_y}}}{{\partial y}} + \frac{{\partial {X_z}}}{{\partial z}}} \right) \hfill \\ \end{gathered} \right\} $$

((2))

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English translation of H.A. Lorentz, Eene algemeene Stelling omtrent de beweging eener vloeistof met wrijving en eenige daaruit afgeleide gevolgen. Zittingsverslag van de Koninklijke Akademie van Wetenschappen te Amsterdam 5 (1896) 168–175.