11.9

Bernoulli’s Equation

CONCEPTS AT A GLANCE For steady flow, the speed, pressure, and elevation of an incompressible and nonviscous fluid are related by an equation discovered by Daniel Bernoulli (1700–1782). To derive Bernoulli’s equation, we will use the work–energy theorem, as outlined in the Concepts-at-a-Glance chart in Figure 11-29. This theorem, which is introduced in Chapter 6, states that the net work Wnc done on an object by external nonconservative forces is equal to the change in the total mechanical energy of the object (see Equation 6.8). As mentioned earlier, the pressure within a fluid is caused by collisional forces, which are nonconservative. Therefore, when a fluid is accelerated because of a difference in pressures, work is being done by nonconservative forces (), and this work changes the total mechanical energy of the fluid from an initial value of E0 to a final value of Ef. The total mechanical energy is not conserved. We will now see how the work–energy theorem leads directly to Bernoulli’s equation.

Figure 11-29 CONCEPTS AT A GLANCE The work–energy theorem leads to Bernoulli’s equation when the net work Wnc done by external nonconservative forces is not zero. Bernoulli’s equation reveals that the pressure associated with moving air, such as this tornado in Miami (May 12, 1997), is lower than that of stationary air, such as that in the buildings. (Reuters)

To begin with, let us make two observations about a moving fluid. First, whenever a fluid is flowing in a horizontal pipe and encounters a region of reduced cross-sectional area, the pressure of the fluid drops, as the pressure gauges in Figure 11-30a indicate. The reason for this follows from Newton’s second law. When moving from the wider region 2 to the narrower region 1, the fluid speeds up or accelerates, consistent with the conservation of mass (as expressed by the equation of continuity). According to the second law, the accelerating fluid must be subjected to an unbalanced force. But there can be an unbalanced force only if the pressure in region 2 exceeds the pressure in region 1. We will see that the difference in pressures is given by Bernoulli’s equation. The second observation is that if the fluid moves to a higher elevation, the pressure at the lower level is greater than the pressure at the higher level, as in Figure 11-30b. The basis for this observation is our previous study of static fluids, and Bernoulli’s equation will confirm it, provided that the cross-sectional area of the pipe does not change.

Figure 11-30 (a) In this horizontal pipe, the pressure in region 2 is greater than that in region 1. The difference in pressures leads to the net force that accelerates the fluid to the right. (b) When the fluid changes elevation, the pressure at the bottom is greater than that at the top, assuming the cross-sectional area of the pipe is constant.

To derive Bernoulli’s equation, consider Figure 11-31a. This drawing shows a fluid element of mass m, upstream in region 2 of a pipe. Both the cross-sectional area and the elevation are different at different places along the pipe. The speed, pressure, and elevation in this region are v2, P2, and y2, respectively. Downstream in region 1 these variables have the values v1, P1, and y1. As Chapter 6 discusses, an object moving under the influence of gravity has a total mechanical energy E that is the sum of the kinetic energy KE and the gravitational potential energy PE: When work Wnc is done on the fluid element by external nonconservative forces, the total mechanical energy changes. According to the work–energy theorem, the work equals the change in the total mechanical energy:

| (6.8)

Figure 11-31 (a) A fluid element (dark blue) moving through a pipe whose cross-sectional area and elevation change. (b) The fluid element experiences a force on its top surface due to the fluid above it, and a force on its bottom surface due to the fluid below it.

Figure 11-31b helps us understand how the work Wnc arises. On the top surface of the fluid element, the surrounding fluid exerts a pressure P. This pressure gives rise to a force of magnitude , where A is the cross-sectional area. On the bottom surface, the surrounding fluid exerts a slightly greater pressure, , where is the pressure difference between the ends of the element. As a result, the force on the bottom surface has a magnitude of . The magnitude of the net force pushing the fluid element up the pipe is . When the fluid element moves through its own length s, the work done is the product of the magnitude of the net force and the distance: . The quantity As is the volume V of the element, so the work is . The total work done on the fluid element in moving it from region 2 to region 1 is the sum of the small increments of work done as the element moves along the pipe. This sum amounts to , where is the pressure difference between the two regions. With this expression for Wnc, the work–energy theorem becomes

By dividing both sides of this result by the volume V, recognizing that m/V is the density of the fluid, and rearranging terms, we obtain Bernoulli’s equation.

| Bernoulli’s Equation
| In the steady flow of a nonviscous, incompressible fluid of density , the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by

| (11.11)

Since the points 1 and 2 were selected arbitrarily, the term has a constant value at all positions in the flow. For this reason, Bernoulli’s equation is sometimes expressed as .
Equation 11.11 can be regarded as an extension of the earlier result that specifies how the pressure varies with depth in a static fluid (), the terms and accounting for the effects of fluid speed. Bernoulli’s equation reduces to the result for static fluids when the speed of the fluid is the same everywhere (), as it is when the cross-sectional area remains constant. Under such conditions, Bernoulli’s equation is . After rearrangement, this result becomes

which is the result (Equation 11.4) for static fluids.

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