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Fluid flow is a complex phenomenon. But we can obtain some useful properties for steady or streamline flows using the conservation of energy.

Consider a fluid moving in a pipe of varying cross-sectional area. Let the pipe be at varying heights as shown in Fig. 1. We now suppose that an incompressible fluid is flowing through the pipe in a steady flow. Its velocity must change as a consequence of equation of continuity. A force is required to produce this acceleration, which is caused by the fluid surrounding it, the pressure must be different in different regions. Bernoulli’s equation is a general expression that relates the pressure difference between two points in a pipe to both velocity changes (kinetic energy change) and elevation (height) changes (potential energy change). The Swiss Physicist Daniel Bernoulli developed this relationship in 1738.

Fig. 1 The flow of an ideal fluid in a pipe of varying cross section. The fluid in a section of length v$_1$∆t moves to the section of length v$_2$∆t in time ∆t.

Consider the flow at two regions 1 (i.e., BC) and 2 (i.e., DE). Consider the fluid initially lying between B and D. In an infinitesimal time interval ∆t, this fluid would have moved. Suppose v1 is the speed at B and v$_2$  at D, then fluid initially at B has moved a distance v$_1$∆t to C (v$_1$∆t is small enough to assume constant cross-section along BC). 

In the same interval ∆t the fluid initially at D moves to E, a distance equal to v$_2$∆t. Pressures P$_1$  and P$_2$  act as shown on the plane faces of areas A$_1$  and A$_2$  binding the two regions. The work done on the fluid at left end (BC) is $W_1 = P_1A_1 (v_1 \Delta t)$ = P$_1$∆V. Since the same volume ∆V passes through both the regions (from the equation of continuity) the work done by the fluid at the other end (DE) is $W_2 = P_2A_2 (v_2 \Delta t)$ = P$_2$∆V or, the work done on the fluid is –P$_2$ ∆V. So the total work done on the fluid is

W$_1$  – W$_2$ = (P$_1$ − P$_2$)∆V

Part of this work goes into changing the kinetic energy of the fluid, and part goes into  changing the gravitational potential energy. If the density of the fluid is ρ and ∆m = ρ$A_1v_1$ ∆t = ρ∆V is the mass passing through the pipe in time ∆t, then change in gravitational potential energy is

∆U = ρg∆V(h$_2$ − h$_1$) 

The change in its kinetic energy is

∆K = $\frac{1}{2}ρ\Delta V(v_2^2-v_1^2)$

We can employ the work – energy theorem to this volume of the fluid and this yields

(P$_1$ − P$_2$)∆V = $\frac{1}{2}ρ\Delta V(v_2^2-v_1^2)$ + ρg∆V(h$_2$ − h$_1$) 

We now divide each term by ∆V to obtain

P$_1$ − P$_2$ = $\frac{1}{2}ρ(v_2^2-v_1^2)$ + ρg(h$_2$ − h$_1$) 

We can rearrange the above terms to obtain

$P_1+\frac{1}{2}\rho v_1^2+\rho gh_1=P_2+\frac{1}{2}\rho v_2^2 +\rho gh_2$

This is Bernoulli’s equation. Since 1 and 2 refer to any two locations along the pipeline, we may write the expression in general as

$P+\frac{1}{2}\rho v^2+\rho gh$ = constant

In words, the Bernoulli’s relation may be stated as follows: As we move along a streamline the sum of the pressure (P), the kinetic energy per unit volume $\frac{\rho v^2}{2}$ and the potential energy per unit volume (ρgh) remains a constant.

Note that in applying the energy conservation principle, there is an assumption that no energy is lost due to friction. But in fact, when fluids flow, some energy does get lost due to internal friction. This arises due to the fact that in a fluid flow, the different layers of the fluid flow with different velocities. These layers exert frictional forces on each other resulting in a loss of energy. This property of the fluid is called viscosity and is discussed in more detail in a later section. The lost kinetic energy of the fluid gets converted into heat energy. Thus, Bernoulli’s equation ideally applies to fluids with zero viscosity or non-viscous fluids. 

Another restriction on application of Bernoulli theorem is that the fluids must be incompressible, as the elastic energy of the fluid is also not taken into consideration. In practice, it has a large number of useful applications and can help explain a wide variety of phenomena for low viscosity incompressible fluids. Bernoulli’s equation also does not hold for non-steady or turbulent flows, because in that situation velocity and pressure are constantly fluctuating in time.

When a fluid is at rest i.e., its velocity is zero everywhere, Bernoulli’s equation becomes

$P_1+\rho gh_1=P_2+\rho gh_2$

$P_2-P_1=\rho g(h_2-h_1)$

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