Class 11 > Unit # 07: Fluid Dynamics > Bernoulli's Principle & Equation

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Fluid Dynamics - Bernoulli's Principle and Equation

Q. State Bernoulli’s Principle and derive Bernoulli’s Equation.

Bernoulli’s Principle

"Bernoulli's principle states that where the velocity of a fluid is high, the pressure is low; and where the velocity is low, the pressure is high."

Bernoulli’s Equation

Based on the principle of conservation of energy, total mechanical energy in a steady, streamline fluid flow remains constant. The derivation relies on the following key assumptions:

1. The flow must be steady and streamline.
2. The fluid is incompressible, ensuring constant density ($\rho$) at all points.
3. There are no viscous forces, and friction is negligible (ideal fluid).

Consider a pipe whose cross-sectional diameter and vertical elevation change as a fluid of density $\rho$ flows from point 1 to point 2.

Figure: flow of an incompressible fluid through an enclosed tube or pipe of varying cross-sectional area.

1. Work Done on the Fluid

The work done by a force $F$ to displace the fluid by an infinitesimal distance $\Delta x$ is given by $W = F \Delta x$.

• At point 1 (inlet): $W_1 = F_1 \Delta x_1 = P_1 A_1 \Delta x_1 = P_1 V$
• At point 2 (outlet): $W_2 = -F_2 \Delta x_2 = -P_2 A_2 \Delta x_2 = -P_2 V$ (negative because the outflowing force opposes motion)

The total net work done on the fluid element as it moves from point 1 to 2 is:

$W = P_1 V - P_2 V = (P_1 - P_2) V$

Since volume $V = \frac{m}{\rho}$, we can express the work done in terms of mass $m$ and density $\rho$:

$W = (P_1 - P_2) \frac{m}{\rho}$

2. Change in Kinetic Energy

As the fluid flows from point 1 (velocity $v_1$) to point 2 (velocity $v_2$), the change in kinetic energy ($\Delta K.E.$) is:

$\Delta K.E. = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2$

3. Change in Potential Energy

As the fluid moves from an elevation $y_1$ to $y_2$, the change in gravitational potential energy ($\Delta P.E.$) is:

$\Delta P.E. = m g y_2 - m g y_1$

Energy Conservation (Work-Energy Theorem)

The total work done on the moving fluid equals the sum of the changes in its kinetic and potential energies:

$W = \Delta K.E. + \Delta P.E.$

Substituting our expressions for $W$, $\Delta K.E.$, and $\Delta P.E.$:

$(P_1 - P_2) \frac{m}{\rho} = \left(\frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2\right) + (m g y_2 - m g y_1)$

Dividing the entire equation by mass $m$ and multiplying by density $\rho$ yields:

$P_1 - P_2 = \frac{1}{2} \rho v_2^2 - \frac{1}{2} \rho v_1^2 + \rho g y_2 - \rho g y_1$

Rearranging terms to group all parameters of point 1 on one side and point 2 on the other:

$P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2$

Conclusion:

Since points 1 and 2 were chosen arbitrarily along the streamline, we can write Bernoulli's equation in its general compact form as:

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

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