Bernoulli's Equation Cannot Be Applied When The Flow Is

Bernoulli's equation, a cornerstone of fluid dynamics, elegantly describes the relationship between pressure, velocity, and elevation in a moving fluid. It's a powerful tool for analyzing a wide range of fluid flow scenarios, from the lift generated by an airplane wing to the flow rate through a pipe. However, like all models, Bernoulli's equation operates under a specific set of assumptions. When these assumptions are violated, the equation's accuracy diminishes, and its application becomes inappropriate. Understanding the limitations of Bernoulli's equation is crucial for correctly applying it and avoiding erroneous conclusions in fluid mechanics problems.

The Foundation of Bernoulli's Equation: Assumptions and Limitations

Bernoulli's equation is derived from the principle of conservation of energy applied to a flowing fluid. It essentially states that for an inviscid, incompressible fluid in steady flow, the total mechanical energy of the fluid remains constant along a streamline. This "total energy" is the sum of pressure energy, kinetic energy (related to velocity), and potential energy (related to elevation).

The standard form of Bernoulli's equation is:

P + (1/2)ρV² + ρgh = constant

Where:

• P = Static pressure of the fluid
• ρ = Density of the fluid
• V = Velocity of the fluid
• g = Acceleration due to gravity
• h = Elevation of the fluid

This equation is incredibly useful, but it's vital to remember the conditions under which it holds true. The primary assumptions that underpin Bernoulli's equation are:

1. Inviscid Flow: The fluid is assumed to have zero viscosity. Viscosity is the internal friction within a fluid that resists flow. In reality, all fluids possess some degree of viscosity.
2. Incompressible Flow: The density of the fluid is assumed to be constant. This is generally a good approximation for liquids, but it breaks down when dealing with gases at high speeds where density changes significantly.
3. Steady Flow: The flow parameters (velocity, pressure, density) at any given point do not change with time. This means the flow is constant and predictable.
4. Flow Along a Streamline: Bernoulli's equation applies along a single streamline. A streamline is an imaginary line that is everywhere tangent to the velocity vector of the fluid.
5. No Energy Added or Removed: The equation assumes no pumps (adding energy) or turbines (extracting energy) exist within the flow section being analyzed.
6. No Heat Transfer: The process is assumed to be adiabatic, meaning no heat is added to or removed from the fluid.
7. No Shaft Work: There are no rotating components within the flow that add or extract energy through shaft work.

When these assumptions are significantly violated, the results obtained from Bernoulli's equation can be inaccurate or completely misleading. Let's delve into specific scenarios where Bernoulli's equation is not applicable.

When Bernoulli's Equation Fails: Scenarios and Explanations

1. Viscous Flow (High Viscosity)

Explanation: Viscosity introduces frictional forces within the fluid. These forces dissipate energy as heat, violating the assumption of constant total mechanical energy. The higher the viscosity, the more significant the energy losses due to friction.

Examples:

• Flow of Honey or Molasses: These thick, viscous liquids experience significant frictional losses as they flow. Applying Bernoulli's equation to predict pressure drops or velocity changes in these flows would lead to substantial errors.
• Lubrication Systems: The flow of oil in engine lubrication systems is governed by viscosity. While Bernoulli's principle might offer some insights, a more comprehensive analysis considering viscous effects is crucial for accurate design.
• Flow in Narrow Channels: As the channel becomes narrower, the effect of viscosity becomes more pronounced due to the increased shear stress near the walls. Microfluidic devices, for instance, require careful consideration of viscous effects, rendering Bernoulli's equation insufficient.

Alternatives: For viscous flows, the Navier-Stokes equations, which incorporate viscous terms, provide a more accurate description of the fluid motion. However, these equations are considerably more complex to solve. Computational Fluid Dynamics (CFD) simulations are often employed to model viscous flows accurately.

2. Turbulent Flow

Explanation: Turbulent flow is characterized by chaotic, irregular fluctuations in velocity and pressure. These fluctuations create eddies and vortices that dissipate energy, again violating the assumption of constant total mechanical energy. While Bernoulli's equation strictly applies to steady flow, it can be used as a rough approximation in some turbulent flows if one considers the average velocity and pressure. However, this approach ignores the significant energy losses associated with turbulence.

Examples:

• Flow Around a Bluff Body: When a fluid flows around a blunt object (like a cylinder or a sphere), the flow separates, creating a turbulent wake behind the object. Bernoulli's equation is not suitable for analyzing the flow within the wake due to the intense turbulence and energy dissipation.
• Flow in a Rough Pipe: The roughness of the pipe wall induces turbulence, leading to energy losses. The Darcy-Weisbach equation, which incorporates a friction factor to account for these losses, is more appropriate for calculating pressure drops in rough pipes.
• High-Speed Flow in a Pipe with Obstructions: Obstructions in a pipe, such as valves or sudden changes in diameter, can trigger turbulence. The energy losses due to this turbulence cannot be accurately predicted using Bernoulli's equation alone.

Alternatives: For turbulent flows, Reynolds-Averaged Navier-Stokes (RANS) equations are often used. These equations average the flow variables over time, allowing for a more manageable solution. More advanced turbulence models, such as Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), can provide even more accurate results but are computationally expensive.

3. Compressible Flow (High-Speed Gas Flow)

Explanation: Bernoulli's equation assumes constant density. This is a reasonable approximation for liquids and gases at low speeds. However, when the speed of a gas approaches or exceeds the speed of sound, the density of the gas changes significantly due to compression. This density change invalidates the incompressibility assumption.

Examples:

• Flow Through a Supersonic Nozzle: In supersonic nozzles, the gas expands and accelerates to speeds greater than the speed of sound. The density of the gas decreases significantly as it expands. Bernoulli's equation cannot accurately predict the pressure and velocity changes in this scenario.
• Airflow Around an Aircraft at High Speeds: As an aircraft approaches the speed of sound, the air compresses near the leading edges of the wings and fuselage. These compressibility effects must be considered for accurate aerodynamic analysis.
• Flow in High-Speed Compressors: Compressors are designed to increase the pressure and density of a gas. Bernoulli's equation is not suitable for analyzing the flow within a compressor because the density changes are a fundamental part of the process.

Alternatives: For compressible flows, the Euler equations or the Navier-Stokes equations (with the inclusion of an equation of state to relate pressure, density, and temperature) are required. These equations account for the changes in density and temperature that occur at high speeds.

4. Unsteady Flow

Explanation: Bernoulli's equation is derived under the assumption of steady flow, meaning the flow parameters at any given point do not change with time. In unsteady flows, the velocity, pressure, and density can vary with time, invalidating this assumption.

Examples:

• Water Hammer: Water hammer is a phenomenon that occurs when a valve in a pipeline is suddenly closed, creating a pressure surge that propagates through the pipe. This is an unsteady flow situation, and Bernoulli's equation cannot be used to analyze the pressure wave.
• Pulsating Flow in a Pump: The flow rate from a reciprocating pump is not constant but pulsates with each stroke. This unsteady flow requires a more sophisticated analysis than Bernoulli's equation can provide.
• Sloshing in a Tank: The motion of liquid in a partially filled tank that is being accelerated or decelerated is an unsteady flow phenomenon. The free surface of the liquid will oscillate, and the pressure distribution will change with time.

Alternatives: Analyzing unsteady flows typically involves solving the unsteady form of the Navier-Stokes equations. This is often done using numerical methods, such as finite element analysis or finite volume methods.

5. Flow with Pumps or Turbines

Explanation: Bernoulli's equation assumes that no energy is added to or removed from the fluid along the streamline. Pumps add energy to the fluid, increasing its pressure or velocity. Turbines extract energy from the fluid, decreasing its pressure or velocity.

Examples:

• Flow Through a Pump: A pump increases the pressure of a fluid, adding energy to the system. Bernoulli's equation cannot be applied directly across a pump because it does not account for the energy input.
• Flow Through a Turbine: A turbine extracts energy from a fluid, converting it into mechanical work. Bernoulli's equation cannot be applied directly across a turbine because it does not account for the energy extraction.

Alternatives: When dealing with flows involving pumps or turbines, a modified form of Bernoulli's equation that includes a term for the head added by the pump (Hp) or the head removed by the turbine (Ht) is used:

P₁/ρg + V₁²/2g + z₁ + Hp = P₂/ρg + V₂²/2g + z₂ + Ht

Where:

• P₁, V₁, z₁ are the pressure, velocity, and elevation at point 1
• P₂, V₂, z₂ are the pressure, velocity, and elevation at point 2
• Hp is the head added by the pump
• Ht is the head removed by the turbine

6. Flow with Significant Heat Transfer

Explanation: Bernoulli's equation assumes an adiabatic process, meaning no heat is exchanged between the fluid and its surroundings. If significant heat transfer occurs, the internal energy of the fluid changes, and Bernoulli's equation is no longer valid.

Examples:

• Flow Through a Heat Exchanger: A heat exchanger is designed to transfer heat between two fluids. The temperature and internal energy of the fluids change significantly as they flow through the heat exchanger. Bernoulli's equation cannot be used to analyze the flow in this situation.
• Combustion Processes: In combustion processes, chemical energy is converted into thermal energy, causing a significant increase in temperature. Bernoulli's equation is not applicable because the process is highly non-adiabatic.

Alternatives: Analyzing flows with significant heat transfer requires the use of the energy equation, which accounts for the changes in internal energy due to heat transfer.

7. Flow with Shock Waves

Explanation: Shock waves are abrupt changes in pressure, density, and temperature that occur in supersonic flows. These changes are highly irreversible and involve significant energy dissipation. Bernoulli's equation, which assumes a reversible process, is not valid across a shock wave.

Examples:

• Flow Over an Aircraft Exceeding the Speed of Sound: When an aircraft flies at supersonic speeds, shock waves form around the aircraft. These shock waves cause a sudden increase in pressure and temperature.
• Explosions: Explosions generate shock waves that propagate outward from the point of detonation. These shock waves can cause significant damage due to the sudden pressure increase.

Alternatives: Analyzing flows with shock waves requires the use of the Rankine-Hugoniot relations, which relate the conditions upstream and downstream of the shock wave.

Conclusion

Bernoulli's equation is a powerful and versatile tool for analyzing fluid flow, but it's essential to understand its limitations. Applying it blindly without considering the underlying assumptions can lead to inaccurate or misleading results. When dealing with viscous, turbulent, compressible, or unsteady flows, or flows involving pumps, turbines, heat transfer, or shock waves, more sophisticated analysis techniques are required. Recognizing when Bernoulli's equation is not applicable and choosing the appropriate alternative methods is crucial for accurate and reliable fluid mechanics analysis. Mastering the conditions under which Bernoulli's equation holds true, and recognizing its limitations, is a fundamental step toward developing a deeper understanding of fluid dynamics and its applications in various engineering disciplines. The key is to always assess the validity of the assumptions before applying the equation, and to consider the potential impact of any deviations from these assumptions on the accuracy of the results. By doing so, engineers and scientists can effectively leverage the power of fluid dynamics to solve a wide range of real-world problems.