COMPLEX NUMBERS AND EULER CONCEPTS OF HYDRODYNAMICS

Author Horia Orasanu

ABSTRACT

Up to now the focus has been fluids at rest. This section deals with fluids that are in motion in a steady fashion such that the fluid velocity at each given point in space is not changing with time. Any flow pattern that is steady in this sense may be seen in terms of a set of streamlines, the trajectories of imaginary particles suspended in the fluid and carried along with it. In steady flow, the fluid is in motion but the streamlines are fixed. Where the streamlines crowd together, the fluid velocity is relatively high; where they open out, the fluid becomes relatively stagnant.

When Euler and Bernoulli were laying the foundations of hydrodynamics, they treated the fluid as an idealized inviscid substance in which, as in a fluid at rest in equilibrium, the shear stresses associated with viscosity are zero and the pressure p is isotropic. They arrived at a simple law relating the variation of p along a streamline to the variation of v (the principle is credited to Bernoulli, but Euler seems to have arrived at it first), which serves to explain many of the phenomena that real fluids in steady motion display. To the inevitable question of when and why it is justifiable to neglect viscosity, there is no single answer. Some answers will be provided later in this article, but other matters will be taken up first.

Consider a small element of fluid of mass m, which—apart from the force on it due to gravity—is acted on only by a pressure p. The latter is isotropic and does not vary with time but may vary from point to point in space. It is a well-known consequence of Newton’s laws of motion that, when a particle of mass m moves under the influence of its weight mg and an additional force F from a point P where its speed is vP and its height is zP to a point Q where its speed is vQ and its height is zQ, the work done by the additional force is equal to the increase in kinetic and potential energy of the particlehttps://adsprod1.pixfuture.com/www/delivery/ck.php?oaparams=2__bannerid=6__zoneid=6__cb=da30a68b6e__oadest=https://www.steepto.com/ghits/2639113/i/7064/0/pp/1/2?h=ViXM2Yz5AVmIwLxBC4HVO_F-kT6EFG8s_njl9TkzvoKVeALruZ-7X1s-QIHgATJi&rid=03a5aafb-2541-11e8-bc82-141877677254&tt=Direct&k=61863fczy7eva_A*fZ!fWIVrn5yfWIVrv7Nf.%3BfT%3AfP%3Df*ff%3BfP%24fl*fY%24faYWKxAfP%3Dfgf%3AfaHR0cHM6Ly9hZHNlcnZlci5hZHRlY2h1cy5jb20vYWRpZnJhbWUvMy4wLzk4MzQuM%248%3DfaHR0cHM6Ly93aW5keW1lZC5jb20vZD9pPWI4MW1wbXVmNmd6dW9pZTU1N3AmYT1kZDY%3DfKysvZD9pPWI4MW1wbXVmNmd6dW9pZTU1N3AmYT1kZDYxNjM3ZGRkZDdiZWE0NDI1OTJhNw%3D%3Df%3BfT%3AfP%3DfS~f%3BfcfT*fUAfQAfWIVrpEhf!fTW96aWxsY%2481LjAgKFdpbmRvd3MgTlQgNi4xOyBydjo1OC4wK%24BHZWNrby8yMDEwMDEwM%24BGaXJlZm94LzU4LjA%3Dff!fV2luMzI%3DfMTIwf%3BfUAfPYfi%3Bfl%3Af*f%2Bf*f*

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