Saturday, March 3, 2012

Euler on Fluid Motion

A paper by Euler, General Principles on the Movement of Fluids (1757), describes the motion of a fluid within a "vessel" which provides the bounds for the flow. He starts by deriving the equations for the flow of an incompressible fluid. In modern vector form the equations are as follows,

There are two equations. The first is a vector equation involving the forces on a element of fluid including the force of acceleration. The second is the scalar continuity equation which gives the condition for the conservation of fluid mass. If the fluid is incompressible, i.e., its density is constant, the the equations of motion can be integrated to give an equation for the direction of flow for the element.

The paper concludes as follows,

"Everything is reduced to finding suitable values ​​for the three velocities u, v, & w, which satisfy our two equations, which contain all that regards our knowledge of the movement of fluids. Since with those three speeds, we can determine the path, that each element of the fluid travels through by its movement. Consider the particle which is at present in Z, and to find the path, it already traveled, and it will travel again, since its three speeds, u, v, & w, are assumed known, we will have to its place in the next moment, dx = u dt, dy = v dt & dz = w dt. We eliminated from these three equalities the time t, and we have two more equations between the three coordinates x, y and z, which seek to determine the path of the fluid element, which is currently in Z, and in general we know the route, that each particle described, and will describe again.

"The determination of these routes is of extreme importance, and must be used to apply the Theory to each case prospect. For if the figure of the vessel, in which the fluid moves, is given, the particles of the fluid, which touch the surface of the vessel, must necessarily follow the direction: & therefore the velocities u, v, & w, must be such that the routes will be deduced to, fall within the same surface. Now we see from what suffices, how we are still far from a complete understanding of the movement of fluids, & that what I have just explained, contains only a weak beginning. But all that which the Theory of fluids includes, is contained in the two equations reported above (§ XXXIV.), so it is not the principles of Mechanics that we are lacking in the pursuit of this research, but only the Analysis, which is not yet adequately cultivated, for this purpose: & therefore it is clear, there are discoveries we have yet to make in this Science, before we can arrive at a more perfect Theory of the movement of fluids."

This paper is sometimes cited as the first reference to tubes of flow. The statement of the problem is not quite the same as that of the imaginary tubes defined by the lines of flow found in Maxwell's works on Electricity and Magnetism and those of hydrodynamics but appears to be more like boundary conditions for the walls of a vessel confining the fluid flow. The boundary conditions usually used today specify that the velocity is zero at the walls of a tube or pipe through which the fluid flows.

This paper by Euler was published only a few years before Lambert's Photometria (1760). Cavendish, who was a Fellow of the Royal Society of London, could be considered a comtemporary of Euler and Lambert although his mathematics does not appear to be as sophisticated.

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