6.3 Bernoulli's Equation

Bernoulli's equation is a cornerstone of fluid dynamics, expressing the principle of conservation of energy for an ideal fluid in motion. It states that for a fluid flowing along a streamline, the sum of its pressure energy, kinetic energy, and potential energy per unit volume remains constant.

The equation is formulated as:

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

The equation is composed of three energy terms per unit volume:

• Pressure Energy ($P$): This represents the "flow work" or the energy stored in the fluid due to the pressure it is under.
• Kinetic Energy ($\frac{1}{2}\rho v^2$): This is the energy of the fluid due to its motion.
• Potential Energy ($\rho gh$): This is the gravitational potential energy of the fluid due to its elevation.

The core idea is that energy can be transformed between these three forms, but their total sum along a single streamline does not change. For example, if a fluid's speed increases (higher kinetic energy), its pressure or height must decrease to keep the total energy constant.

Derivation of the Equation

The equation is derived from the work-energy theorem, which states that the total work done on a system is equal to the change in its mechanical energy (kinetic + potential).

Step 1: The Work-Energy Theorem

Consider a volume of fluid, $\Delta V$, moving along a pipe. The net work done on this volume of fluid equals the change in its kinetic and potential energy.

$W_{\text{net}}=\Delta KE+\Delta PE$

Step 2: Work Done by Pressure

The work is done by the pressure forces at the two ends of the fluid segment.

• Work done on the fluid at point 1 (entering): $W_1=P_1{A}_1\Delta x_1=P_1\Delta V$
• Work done by the fluid at point 2 (exiting): $W_2=-P_2{A}_2\Delta x_2=-P_2\Delta V$

The net work done on the fluid is:

$W_{\text{net}}=W_1+W_2=(P_1-P_2)\Delta V$

Step 3: Changes in Mechanical Energy

The changes in kinetic and potential energy for the mass ($m=\rho \Delta V$) of the fluid are:

• Change in Kinetic Energy: $\Delta KE=\frac{1}{2}m{v}_2^2-\frac{1}{2}m{v}_1^2$
• Change in Potential Energy: $\Delta PE=mg{h}_2-mg{h}_1$

Step 4: Combining and Final Formulation

Setting the net work equal to the total change in energy:

$(P_1-P_2)\Delta V=(\frac{1}{2}m{v}_2^2-\frac{1}{2}m{v}_1^2)+(mg{h}_2-mg{h}_1)$

Substitute $m=\rho \Delta V$ and divide the entire equation by the volume $\Delta V$:

$P_1-P_2=(\frac{1}{2}\rho v_2^2-\frac{1}{2}\rho v_1^2)+(\rho g{h}_2-\rho g{h}_1)$

Rearranging the terms to group them by their position (1 or 2):

$P_1+\frac{1}{2}\rho v_1^2+\rho g{h}_1=P_2+\frac{1}{2}\rho v_2^2+\rho g{h}_2$

This shows that the sum of the three energy terms is constant at any point along the streamline.

Possible Questions/Answer

• Q: What are the limitations of Bernoulli's equation? A: Bernoulli's equation applies only to ideal fluids (incompressible and non-viscous) under steady, streamline flow. It does not account for energy losses due to friction (viscosity) or turbulence.