When real fluids flow through pipes, two distinct forces act on them. One is the frictional forces exerted on the fluid by the walls of the pipe and the other is the viscous forces within the fluid. The fluid layers next to the walls of the pipe "stick" slightly to the pipe. As you move further from the walls towards the center of the fluid, this boundary layer ends and the fluid moves faster and more coherently. Viscous forces within the fluid produce a shearing action that results in tiny layers of fluid of ever-increasing speed which eventually reach the speed of the free stream in the center of the pipe. Energy is lost within the fluid to both of these forces.
An ideal fluid is one that meets the following specifications: steady flow, irrotational flow, nonviscous flow, and incompressible flow. Steady flow is laminar flow which means that the particles flow along streamlines - that is, every particle moves along the same path as previous particles followed. Every particle at the same place in a fluid will have the same velocity. Steady flow only occurs at low velocities. When streamlines are forced closer together, the velocity in the fluid is greater. Irrotational flow means that no fluid elements (small volume packets) have angular velocity - that is, there is no turbulence in the form of whirlpools or eddy currents. Nonviscous flow means that viscosity can be neglected - that is, there are no shearing forces within the fluid which subsequently result in the production of heat as the fluid flows. Incompressible flow means that the density of the fluid remains constant.
If a fluid system has no sources providing additional fluid or sinks draining off fluid, the volume of fluid entering the first cross-sectional area must equal the volume of fluid flowing out the later cross-sectional area.
Since density, rho = m/V (or rhoV = m) is a constant in an incompressible fluid, we say that mass is conserved in a closed fluid system.
The Continuity Equation
states that the cross-sectional area of the pipe and the velocity of the fluid are inversely proportional - that is, fluids flow faster through narrower pipes. We can see this by the fact that the streamlines are forced closed together whenever the pipe narrows. Next time you watch water flowing from a faucet note how the water stream narrows as the water falls. This reduction in cross-sectional area is required by the Continuity Equation since the water is increasing in speed as it falls. Ideal liquids also obey a special statement of conservation of energy first developed by Daniel Bernoulli in 1738 which is often expressed as |