Solutions of Eulers equations

Euler s equations of motion are not linear differential equations so that their general solution is not generally straight-foward and the principle of superposition cannot be used to obtain additional solutions. However, first integrals can readily be obtained for a number of situations and some of these will now be examined. 

The simplest condition is that where the velocity u is zero everywhere in the liquid. Then the equations of motion become the equations of equilibrium. In vector notation  

For a liquid of constant density, that is, an incompressible liquid  

Many instruments for measuring pressures in liquids measure only the increase in pressure over the pressure at the surface. Therefore, p in equation [5.28] is sometimes referred to as the absolute pressure and p — po as the gauge pressure. 

It is of interest to examine the forces exerted on a solid body that is immersed in a liquid that is at rest. Consider first the pressure on a plane area placed at any angle in a liquid of density p. The force on an element of this surface of area bS is (po + pgz) 65, where po is the pressure on the liquid surface and z is the vertical distance from the element to the surface this force is perpendicular to the area 65. It follows that the total force over the whole surface is  

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Meshcheryakov, M. V. "Solutions of Euler equations on singular orbits of simple Lie groups. In Geometry and Topology in Global Nonlinear Problems. Voronezh, Voronezh Univ. Press, (1984), 158-162. 

Kroll N, Rossow C-C. Foundations of numerical methods for the solution of Euler equations. Prepared for the Lecture F6.03 of the CCG. Braunschweig DLR 1989. 

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