Bernoulli's Equation for Ideal Fluid Flow

Hello! Welcome to your fifth lesson in the "Continuum and Fluid Mechanics" module.

In our last lesson, we derived the continuity equation from the fundamental principle of mass conservation. We found that for an incompressible fluid, the velocity field $V$ must be divergence-free, i.e., $\nabla \cdot V = 0$ . This was our first major governing equation for fluid flow.

Today, we will derive the second major governing equation for an ideal fluid by applying the principle of conservation of energy (or, more fundamentally, Newton's second law). This will lead us to one of the most famous and useful equations in all of physics: Bernoulli's equation. This lesson is designed to meet the learning outcome: Derive and apply Bernoulli's equation for ideal, incompressible, steady fluid flow.

Our path will be as follows:

We will perform a rigorous derivation of Bernoulli's equation by applying Newton's second law to a fluid particle moving along a streamline.
We will dissect the equation, understanding the physical meaning of each term.
We will apply the equation to classic engineering and physics problems, often in combination with the continuity equation.
Finally, we will explore the critical assumptions and limitations of the equation, particularly the subtlety of applying it across different streamlines, which will touch on concepts relevant to the advanced problems you enjoy.

1. Derivation from Newton's Second Law

While Bernoulli's equation can be derived from the work-energy theorem, a more fundamental approach, which connects directly to the underlying dynamics, is to integrate Euler's equation of motion for an ideal (inviscid) fluid. This aligns with your preference for rigorous, first-principles derivations.

The following textbook chapter provides a complete and formal derivation. We will walk through the key logical steps together.

Elementary Fluid - Dynamics—The Bernoulli Equation

This chapter, 'Elementary Fluid Dynamics—The Bernoulli Equation,' from a standard fluid mechanics textbook, will be our primary guide. It starts with Newton's second law and meticulously builds up to the final equation and its applications.

Please read the following sections: Section 3.1: Newton’s Second Law (pp. 94-95): Focus on how fluid motion is described using streamline coordinates (s, n) and the derivation of streamwise acceleration, a_s = V rac{\partial V}{\partial s}. Section 3.2: F = ma along a Streamline (pp. 96-99): Carefully follow the free-body diagram analysis of a fluid particle. The key is to see how the pressure gradient and gravity forces along the streamline balance the particle's acceleration. This leads to Equation 3.5. Integration to the Bernoulli Equation (top of p. 99): Read the short paragraph that integrates Eq. 3.6 under the assumption of an incompressible fluid to arrive at the final Bernoulli equation (Eq. 3.7). Pay close attention to the assumptions being made at each step.

Let's summarize the essential logic of the derivation you just read.

Setup: We analyze a small fluid particle moving along a streamline in a steady, inviscid flow. The coordinate system is aligned with the flow: s is the direction along the streamline, and n is the direction normal to it.
Newton's Second Law ( $F = ma$ ): We apply $\sum F_{s} = m a_{s}$ to the particle.
- Acceleration ( $a_{s}$ ): For steady flow, a particle accelerates only because its velocity changes with position. Using the chain rule, the streamwise acceleration is $a_{s} = \frac{d V}{d t} = \frac{\partial V}{\partial s} \frac{d s}{d t} = V \frac{\partial V}{\partial s}$ .
- Forces ( $\sum F_{s}$ ): The two forces acting on the particle are pressure and gravity.
  - Pressure Force: A pressure gradient $\frac{\partial p}{\partial s}$ creates a net force $- \frac{\partial p}{\partial s} d V$ on the particle of volume $d V$ .
  - Gravity Force: The component of the gravitational force along the streamline is $- m g sin θ = - (ρ d V) g \frac{d z}{d s}$ .
Equation of Motion: Combining these gives the equation of motion along the streamline:
$- \frac{\partial p}{\partial s} - ρ g \frac{d z}{d s} = ρ V \frac{\partial V}{\partial s}$
This is a one-dimensional form of Euler's equation for an ideal fluid.
Integration: For a steady flow, the partial derivatives with respect to $s$ can be treated as total derivatives. We can also use the identity $V \frac{d V}{d s} = \frac{d}{d s} (\frac{1}{2} V^{2})$ . Rearranging the equation gives:
$\frac{d p}{d s} + ρ \frac{d}{d s} (\frac{1}{2} V^{2}) + ρ g \frac{d z}{d s} = 0$
$\frac{d}{d s} (p + \frac{1}{2} ρ V^{2} + ρ g z) = 0$
Integrating this equation with respect to $s$ yields the conclusion that the quantity in the parenthesis must be constant along the streamline.

This gives us Bernoulli's Equation:
$p + \frac{1}{2} ρ V^{2} + ρ g z = constant along a streamline$

An Alternative View: Work and Energy

For a different perspective, the same equation can be derived by applying the work-energy theorem to a fluid element. The following video provides a clear walkthrough of this method.

Physics: Fluid Dynamics: Fluid Flow (1.6 of 7) Bernoulli's Equation Derived

This video by Michel van Biezen derives Bernoulli's equation from the work-energy principle. It's a great way to reinforce your understanding by seeing the same result emerge from a different physical argument.

Watch the full video. Notice how the net work done by pressure forces (W_{net} = W_1 - W_2) is equated to the change in the fluid's kinetic and potential energy (\Delta KE + \Delta PE).

2. Physical Interpretation and Assumptions

Bernoulli's equation is a statement of energy conservation for an ideal fluid. Each term has units of energy per unit volume (or pressure):

$p$ : Static Pressure. This is the pressure you would measure if you were moving along with the fluid.
$\frac{1}{2} ρ V^{2}$ : Dynamic Pressure. This represents the kinetic energy per unit volume of the fluid.
$ρ g z$ : Hydrostatic Pressure. This represents the potential energy per unit volume due to elevation.

The sum of these three is the Total Pressure, which remains constant along a streamline in an ideal flow.

Crucial Assumptions

As the textbook reading emphasized, Bernoulli's equation is powerful but rests on a strict set of assumptions. Misapplying it is a common error.

Steady Flow: The flow properties (velocity, pressure, density) at any given point do not change with time.
Inviscid Flow: There are no frictional forces. This is the "ideal fluid" assumption. In real flows, viscosity causes energy losses.
Incompressible Flow: The density $ρ$ is constant. This is a good approximation for liquids and for gases at low speeds (typically Mach number < 0.3).
Along a Streamline: The equation, as derived, states that the Bernoulli constant is the same at two points on the same streamline. We will explore when it can be applied more broadly later.
No external work or heat transfer: No pumps, turbines, or heating elements are present between the two points of application.

3. Applications with the Continuity Equation

In many practical problems, the fluid is confined within pipes or ducts of varying cross-sectional area. In these cases, we must use both Bernoulli's equation (conservation of energy) and the incompressible continuity equation (conservation of mass) simultaneously.

Continuity Equation Recap: $A_{1} V_{1} = A_{2} V_{2}$

Application: The Venturi Meter

A Venturi meter is a device used to measure the flow rate in a pipe. It consists of a constricted section (the "throat") that accelerates the fluid.

Bernoulli's Principle in a Venturi Meter — Caption: A Venturi meter demonstrates Bernoulli's principle. As the fluid flows from the wider section (1) to the narrower throat (2), its velocity increases ($V_2 > V_1$) according to the continuity equation. Bernoulli's equation then dictates that the static pressure must decrease ($p_2 < p_1$), which is shown by the lower fluid level in the second vertical tube.

Let's analyze it:

Apply the continuity equation between points 1 and 2: $V_{1} = \frac{A _{2}}{A _{1}} V_{2}$ .
Apply Bernoulli's equation between points 1 and 2 (assuming a horizontal pipe, so $z_{1} = z_{2}$ ):
$p_{1} + \frac{1}{2} ρ V_{1}^{2} = p_{2} + \frac{1}{2} ρ V_{2}^{2}$
Rearrange to find the pressure difference:
$p_{1} - p_{2} = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2})$
Substitute $V_{1}$ from the continuity equation:
$p_{1} - p_{2} = \frac{1}{2} ρ (V_{2}^{2} - (\frac{A _{2}}{A _{1}})^{2} V_{2}^{2}) = \frac{1}{2} ρ V_{2}^{2} (1 - (\frac{A _{2}}{A _{1}})^{2})$
By measuring the pressure difference $p_{1} - p_{2}$ , one can solve for the velocity $V_{2}$ and thus the volume flow rate $Q = A_{2} V_{2}$ . This is a cornerstone of many engineering applications.

Test your understanding!

Water ( $ρ = 1000 kg/m^{3}$ ) flows through a horizontal pipe that narrows from a diameter of 10 cm to 5 cm. If the velocity in the 10 cm section is 1 m/s, what is the pressure difference between the two sections?

Show answer

First, let's find the areas:
$A_{1} = π (D_{1} /2)^{2} = π (0.05 m)^{2} = 0.0025 π m^{2}$
$A_{2} = π (D_{2} /2)^{2} = π (0.025 m)^{2} = 0.000625 π m^{2}$
Note that $A_{2} / A_{1} = (D_{2} / D_{1})^{2} = (5/10)^{2} = 1/4$ .

Next, use the continuity equation to find $V_{2}$ :
$A_{1} V_{1} = A_{2} V_{2} ⟹ V_{2} = V_{1} \frac{A _{1}}{A _{2}} = (1 m/s) \times 4 = 4 m/s$ .

Now, apply Bernoulli's equation ( $z_{1} = z_{2}$ ):
$p_{1} + \frac{1}{2} ρ V_{1}^{2} = p_{2} + \frac{1}{2} ρ V_{2}^{2}$
$p_{1} - p_{2} = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2})$
$p_{1} - p_{2} = \frac{1}{2} (1000 kg/m^{3}) ((4 m/s)^{2} - (1 m/s)^{2})$
$p_{1} - p_{2} = 500 (16 - 1) = 500 \times 15 = 7500 Pa$ or $7.5 kPa$ .

The pressure in the narrower section is 7.5 kPa lower than in the wider section.

4. Advanced Topic: Rotational Flow and Applicability

A key restriction is that Bernoulli's equation applies along a streamline. When can we apply it between points on different streamlines? The answer is: when the flow is irrotational.

Irrotational Flow: The fluid particles do not rotate as they move. In such a flow, the Bernoulli constant is the same for all streamlines. Examples include uniform flow or flow from a large reservoir.
Rotational Flow: The fluid particles rotate (have vorticity). In this case, the Bernoulli constant is different for each streamline. You cannot apply the equation between points on different streamlines. A classic example is a vortex or a rotating body of fluid.

This is a subtle but critical point often tested in advanced physics problems.

RESOLVED-22 [PHYSICS JEE ADVANCED] BERNOULLI'S EQUATION FOR TWO DIFFERENT STREAMLINES | VORTICITY |

This video, aimed at the JEE Advanced level, tackles a problem where naively applying Bernoulli's equation fails. It explains the crucial difference between irrotational and rotational flow and why Bernoulli's principle is not universally applicable between any two points.

Watch from 7:03 to 15:53. The presenter uses a 'toothpick' analogy to give an intuitive feel for rotational vs. irrotational flow. He then states the formal condition: for irrotational flow, the Bernoulli constant is the same everywhere. For rotational flow, it's only constant along a single streamline. You don't need to follow every detail of the problem's solution, but focus on this conceptual distinction.

The key takeaway from the video is that for a flow to be irrotational, the net "spin" of a fluid element must be zero. In the rotating fluid problem discussed, a fluid element clearly rotates as it revolves around the center, so the flow is rotational. Therefore, $p + \frac{1}{2} ρ V^{2}$ is not constant between a point near the center and a point near the edge. This is a perfect example of the care required when using this seemingly simple equation.

Conclusion

Today we have derived, interpreted, and applied one of the central equations of fluid dynamics.

Key Takeaways:

Bernoulli's equation, $p + \frac{1}{2} ρ V^{2} + ρ g z = constant$ , is a mathematical statement of energy conservation for an ideal fluid flow.
It can be rigorously derived by integrating Newton's second law along a streamline for a steady, incompressible, inviscid flow.
The terms represent static, dynamic, and hydrostatic pressure.
In practical applications, it is often used with the continuity equation ( $A_{1} V_{1} = A_{2} V_{2}$ ) to solve for unknown pressures, velocities, or flow rates.
The equation has strict limitations. It is only valid along a single streamline, unless the flow can be proven to be irrotational, in which case it holds globally.

Preview of the Next Lesson:
We have now established the governing equations for an ideal fluid (continuity and Bernoulli/Euler). But what about real fluids, where friction (viscosity) is present? In our next lesson, we will lift the "inviscid" assumption and derive the much more general and powerful Navier-Stokes equations. This will require us to introduce the concept of viscous stress and incorporate it into the momentum balance we started today.

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