Ideal Fluids in Motion

Ideal Fluids in Motion

The motion of real fluids is very complicated and not yet fully understood. Instead, we shall discuss the motion of an ideal fluid, which is simpler to handle mathematically and yet provides useful results. Here are four assumptions that we make about our ideal fluid; they all are concerned with flow:

1. Steady flow In steady(or laminar) flow,the velocity of the moving fluid at any fixed point does not change with time. The gentle flow of water near the center of a quiet stream is steady; the flow in a chain of rapids is not. Figure 14-12 shows a transition from steady flow to nonsteady(or nonlaminaror turbulent) flowfor a rising stream of smoke. The speed of the smoke particles increases as they rise and, at a certain critical speed, the flow changes from steady to nonsteady.
2. Incompressible flow We assume, as for fluids at rest, that our ideal fluid is incompressible; that is, its density has a constant, uniform value.
3. Nonviscous flow Roughly speaking, the viscosity of a fluid is a measure of how resistive the fluid is to flow. For example, thick honey is more resistive to flow than water, and so honey is said to be more viscous than water. Viscosity is the fluid analog of friction between solids; both are mechanisms by which the kinetic energy of moving objects can be transferred to thermal energy. In the absence of friction, a block could glide at constant speed along a horizontal surface. In the same way, an object moving through a nonviscous fluid would experience no vis-cous drag force— that is, no resistive force due to viscosity; it could move at con-stant speed through the fluid. The British scientist Lord Rayleigh noted that in an ideal fluid a ship’s propeller would not work, but, on the other hand, in an ideal fluid a ship (once set into motion) would not need a propeller!
4. Irrotational flow Although it need not concern us further, we also assume that the flow is irrotational. To test for this property, let a tiny grain of dust move with the fluid. Although this test body may (or may not) move in a circu-lar path, in irrotational flow the test body will not rotate about an axis through its own center of mass. For a loose analogy, the motion of a Ferris wheel is ro-tational; that of its passengers is irrotational.

We can make the flow of a fluid visible by adding a tracer. This might be a dye injected into many points across a liquid stream (Fig. 14-13) or smoke parti-cles added to a gas flow (Fig. 14-12). Each bit of a tracer follows a streamline, which is the path that a tiny element of the fluid would take as the fluid flows.

Recall from Chapter 4 that the velocity of a particle V is always tangent to the path taken by the particle. Here the particle is the fluid element, and its velocity is always tangent to a streamline (Fig. 14-14). For this reason, two streamlines can never intersect; if they did, then an element arriving at their intersection would have two different velocities simultaneously — an impossibility.

Fig. 14-12 At a certain point, the rising flow of smoke and heated gas changes from steady to turbulent. (Will McIntyre/Photo Researchers)

Fig. 14-13 The steady flow of a fluid around a cylinder, as revealed by a dye tracer that was injected into the fluid upstream of the cylinder.(Courtesy D.H. Peregrine, University of Bristol)