Cauchy Number Equations Formulas Calculator

Fluid Mechanics Dimensionless Value

Problem:

Solve for Weber number.

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flow velocity (v)

density (ρ)

bulk modulus elasticity (B_s)

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	Cauchy number
	flow velocity
	density
	bulk modulus elasticity

References - Books:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Background

The Cauchy number (Ca) is a dimensionless number relevant in fluid dynamics and continuum mechanics. It relates the inertial forces to the elastic forces within a fluid or a flow scenario. Understanding and calculating the Cauchy number is vital in analyzing and designing applications involving fluid flow where elastic properties are significant, such as in hydraulic systems, pumps, and pipelines, as well as in natural systems like rivers and atmospheric flows.

Equation

The equation defines the Cauchy number:

Ca = ρv²/B_s

where:

Ca is the Cauchy number
ρ is the fluid density (kg/m³)
v is the flow velocity (m/s)
B_s is the bulk modulus of elasticity (Pa or N/m²)

How to Solve

To solve for the Cauchy number, follow these steps:

Identify Necessary Parameters: Obtain the fluid density (ρ), flow velocity (v), and bulk modulus of elasticity (B_s).

Ensure Unit Consistency: Before substituting values into the equation, make sure all units are consistent, typically in the SI system where density is in (kg/m³), velocity in (m/s), and bulk modulus in (Pa).

Substitute and Solve: Insert the values of ρ, v, and B_s into the Cauchy number equation and solve.

Example

Suppose a fluid with a density of 1000 kg/m³ flows at a velocity of 5 m/s and has a bulk modulus of elasticity of 2.2x10⁹, Pa.

Ca = 1000 x 5² / 2.2 x 10⁹ = 25000 / 2.2 x 10⁹ = 1.136 x 10^-5

Thus, the Cauchy number for the given scenario is 1.136x10^-5.

Fields/Degrees It Is Used In

Hydraulic Engineering: This is used to design dam pipelines and understand fluid interactions with structures.
Aerospace Engineering: In studying the effects of high-speed airflow over aircraft and spacecraft surfaces.
Marine Engineering: Evaluating the impact of water flow on ship hulls, submarines, and offshore structures.
Biomedical Engineering: In blood flow dynamics, especially in arterial systems and artificial heart designs.
Geophysics: For understanding lava flow, glacier movements, and other geothermal fluid dynamics.

Real-life Applications

Pipeline Design: Ensuring pipelines can withstand internal flow pressures without rupturing.
Dam Safety: Analyzing the pressure water exerts on dams to prevent structural failure.
Submarine Hull Design: To withstand underwater currents and pressures.
Aircraft Aerodynamics: Understanding how air flows at high speeds over aircraft bodies and wings.
Artificial Heart Valves: Designing valves that can endure the fluid dynamics of blood flow.

Common Mistakes

Ignoring Unit Consistency: Failing to convert all measures to a single unit system before computing can lead to inaccurate results.
Incorrect Parameters: Using the dynamic viscosity instead of the bulk modulus or confusing flow velocity with relative velocity.
Calculation Errors: Simple mathematical errors in squaring the velocity or dividing by the bulk modulus.
Overlooking Temperature Effects: Forgetting that fluid properties like density and bulk modulus can change with temperature.
Confusing Cauchy Number with Other Dimensionless Numbers: The Reynolds or Mach numbers measure different phenomena.

Frequently Asked Questions with Answers

What distinguishes the Cauchy number from the Reynolds number?
The Reynolds number calculates the ratio of inertial forces to viscous forces, while the Cauchy number measures the ratio of inertial forces to elastic forces in a fluid flow.
Can the Cauchy number apply to gases as well as liquids?
Yes, it can apply to any fluid with definable density and bulk modulus values.
Is the Cauchy number relevant in all types of fluid flow?
It is most relevant in scenarios where elastic forces play a significant role, such as high-speed flows or when considering structural responses to fluid forces.
How does temperature affect the Cauchy number?
Temperature can change the fluid's bulk modulus and density, thus impacting the Cauchy number. Generally, as temperature increases, density decreases, and the bulk modulus may also decrease, leading to a higher Cauchy number.
Why is it essential in engineering?
Understanding and calculating the Cauchy number allows engineers to design safer and more efficient systems interacting with fluids, considering the internal pressures and stress loads to prevent structural failures.