Why Incompressible Fluids Are Considered in the Bernoulli Equation

When discussing fluid flow, particularly in the context of Bernoulli's theorem, the assumption of incompressible flow is often employed. This article delves into the reasoning behind this practice, exploring the physics involved and the implications of this simplifying assumption.

Overview of Bernoulli's Theorem

Bernoulli's theorem is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation in a moving fluid. The theorem is expressed mathematically as:

P/ρg z v2/2g constant

where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, z is the elevation, and v is the velocity of the fluid. The key assumption in this equation is that the fluid is incompressible, meaning the density ρ is constant throughout the flow.

Assumption of Constant Density

The need for the fluid to be incompressible arises from the force balance Fma applied to a unit volume of fluid. When solving the equation, assuming a constant density allows for a simpler and more tractable solution. This allows the elegant and often straightforward results of Bernoulli's equation.

If the density is not assumed constant, the equation becomes more complex and is typically solved numerically. The increase in complexity makes it difficult to derive and apply the results in a practical manner.

Conservation of Energy and Kinetic Energy

The fundamental basis of Bernoulli's equation lies in the conservation of energy. When a small element of the fluid moves through a pressure difference, work is done on it. This work is converted into kinetic energy, and the relationship between kinetic energy (Ke) and work done can be directly established.

For compressible fluids, the situation becomes more complex as the fluid's density can change, leading to additional forms of energy (such as thermal and potential energy) playing a role. This complexity makes it more challenging to attribute the changes in kinetic energy solely to work done through a pressure difference.

Applicability of Incompressible Flow Assumption

While the assumption of incompressible flow simplifies the analysis, it is not without its limitations. Research shows that for flows with speeds significantly less than the speed of sound, the incompressible flow assumption is highly accurate and practical.

Specifically, for most practical applications involving water or air, the speed of the flowing fluid is much less than the speed of sound. In such cases, the deviations from incompressible flow assumptions are negligible, and the simplifying approach is well justified.

Conclusion

In conclusion, the assumption of incompressible fluids in the Bernoulli equation is a practical and highly effective tool in fluid dynamics. While there are instances where compressible fluids must be considered, for the vast majority of applications, the simplicity and accuracy of the incompressible flow assumption make it an invaluable tool in both theoretical and practical analyses.

Keywords: Bernoulli's Theorem, Incompressible Fluid, Flow Analysis, Fluid Dynamics, Pressure and Velocity Relationship