## Bernoulli's Equation

The most useful relation in engineering hydraulics is really three

Daniel Bernoulli disclosed the equation used most frequently in engineering hydraulics in 1738. This equation relates the pressure, velocity and height in the steady motion of an ideal fluid. The usual form is v2/2 + p/ρ + gz = constant, where v is the velocity at a point, p the pressure, ρ the density, g the acceleration of gravity, and z the height above an arbitrary reference level. Students apply the equation without much thought, sometimes inappropriately, and have no clear idea of the conditions under which it is applicable. It appears on every general engineering examination, since it is easy to trap the unwary. Actually, it is not one relation, but three, all apparently of the same form, but applying in different situations. The three forms will be explained in this paper. The most powerful form of Bernoulli's Equation is derived from the Eulerian equations of motion under rather severe restrictions. First, the velocity must be derivable from a velocity potential. Second, external forces must be conservative--that is, derivable from a potential. Thirdly, the density must either be constant, or a function of the pressure alone. In particular, thermal differences, such as occur in natural convection, are excluded. Here, we will assume the fluid is incompressible for simplicity, but it is possible to write a similar equation for compressible fluids. Vector notation is used in the Figure to show that the gradient of a certain expression becomes zero under these assumptions, and Bernoulli's Equation follows on integration and the introduction of the condition of steady motion. It is probably clearer to do the derivation with rectangular components, and to see how the condition curl v = 0 is used. The gradient of a dot product is a rather complicated thing, incidentally.

This form of Bernoulli's Equation applies to steady irrotational flow, and the constant is really a constant throughout the volume of irrotational flow. Nothing is said about streamlines.

The second form of Bernoulli's Equation arises from the fact that in steady flow the particles of fluid move along fixed streamlines, as on rails, and are accelerated and decelerated by the forces acting tangent to the sreamlines. Under the same assumptions for the external forces and the density, but without demanding irrotational flow, we have for an equation of motion dv/dt = v(dv/ds) = -dΩ/ds - (1/ρ)dp/ds, where s is distance along the streamline. This integrates immediately to v2/2 + Ω + p/ρ = c. In this case, the constant c is for the streamline considered alone; nothing can be said about other streamlines. This form of Bernoulli's Equation is more generally applicable, but less powerful than the preceding one. It is the form most often applicable to typical engineering problems. The derivation is easy and straightforward, clearly showing the hypotheses, and also that the motion is assumed frictionless. The third form of Bernoulli's Equation is derived from the conservation of energy. Bernoulli himself took an equivalent approach, although the concept of energy was not well-developed in his time. Energy balance is a favoured method of approach in engineering, and this is the usual derivation of Bernoulli's Equation in elementary work. By the use of energy concepts, the equation can be extended usefully to compressible fluids and thermodynamic processes. In the Figure, an element of fluid is transferred from one point to another in a tube with rigid boundaries. The equation of continuity for an incompressible fluid shows that the same volume of fluid Q disappears at one point and reappears at another. The imaginary pistons move with the speed of the fluid. Capital letters are used for quantities at one point, small letters for the same quantities at the second point. The energies per unit volume, made up of kinetic, potential, and pressure terms are equated. The pressure terms can also be handled as doing work on the element of fluid, which is equivalent. The virtue of this derivation is that can be extended in various directions to give important results, and that it is easily believed by students. The rigid tube can be replaced by a surface generated by streamlines, which can be shrunk down to the neighbourhood of a single streamline, which is just the second form of Bernoulli's Equation, but here derived by energy instead of by dynamics. The energy balance method can be extended to allow for friction, by assuming a loss of energy, or 'head' when expressed in terms of potential energy, between the two sections. The height of liquid in each of the vertical tubes is p/ρ, where p is the gauge pressure. Streamlines do not run up the tubes from inside the main tubes, so they measure just pressure, not total energy. Loss of energy is shown by the decrease in the heights, along the hydraulic gradient, which corresponds to a loss of energy in the flow. The velocity is constant since we have assumed a uniform pipe, so the pressure gradient is the same as the hydraulic gradient. The loss of head per unit length is often assumed proportional to the square of the velocity, for example as fv2/2g, where f is the friction factor, since the flow is usually turbulent. As an example of the use of Bernoulli's Equation, the classic problem of the velocity of efflux through a hole in the side of a tank. We imagine a streamline beginning at the free surface, where the velocity is zero, and extending into the jet a distance h below (dotted in the Figure). The pressures at the two points are the same, atmospheric. From the second (or third) form, we get gh = v2/2, or v = (2gh)1/2, which is called Torricelli's Theorem. This does not give us the rate of efflux, however, because the area of the jet is smaller than the area of the hole in the tank. The smallest jet area occurs when the sides of the jet are parallel, which is just the point we used in applying Bernoulli. This point is called the vena contracta, and has an area half, or somewhat more, of the area of the hole. For a circular hole in a thin wall, the fraction is 0.62, and if the hole has a tube of the same diameter extending into the tank (a Borda's mouthpiece, as shown in the Figure) the fraction is practically 0.50.

The reason for the fraction 1/2 can be seen by a nice application of the conservation of momentum. The momentum carried away by the fluid moving through the vena contracta is v2ρS', where S' is the area of the vena contracta, and the reaction on the container is equal and opposite. But we can also find the force on the container from the pressure distribution over the walls. The pressure on the hole of area S is zero, but that on an equal area on the other side of the tank is ρghS. This must also be the reaction on the tank due to the escaping fluid. Bernoulli tells us that v2 = 2gh, so equating these two forces, we have ρghS = 2ghρS', or S' = S/2. Borda's mouthpiece makes the pressure the static value right up to the edge of the hole; with a plane hole, the pressure is reduced, so S' > S/2 in this case.

Suppose that you have a given amount of liquid Q that you wish to discharge from a cylindrical tank through a hole of area A at the bottom of the tank. Is there an optimum shape of tank that will do this in a minimum time? How long would it take to empty such a tank anyway? The behaviour of the free jet after leaving the aperture is also a subject of interest.  The Pitot tube measures flow velocity by converting the velocity to pressure at the stagnation point at a small entry to the manometer tube pointing into the flow. This works for air as well, with an appropriate pressure gauge, reminding us that air behaves as nearly incompressible at speeds well below the speed of sound. Another important example is the Venturi flow meter, where the fluid is made to pass through passages of different areas. The rate of flow is determined from the difference in pressures (heights of manometers) at the two sections. The difference in velocity is found from continuity, and then the difference in pressure from Bernoulli. The increase in velocity in the throat is accompanied by a decrease in pressure there. If r = A1/A2, then v1 = [2gΔh/(r2 - 1)]1/2, where the symbols are defined in the Figure.

We have now seen the three different theorems that are included under the name of Bernoulli's Equation. All are for steady flow of an incompressible, nonviscous fluid. The first is valid in irrotational flow, the second along a streamline, and the third for an energy-conserving flow in a tube. All look exactly the same when written down. They are capable of being extended to situations different from these by suitable modifications, especially the last one.

#### References

H. Lamb, Hydrodynamics, 6th ed. (Cambridge: Cambridge Univ. Press, 1953), pp. 20-25. This is an excellent reference for classical fluid mechanics.

Any engineering hydraulics book will give considerable attention to Bernoulli's Equation, and many examples.