# Cauchy Number

on . Posted in Dimensionless Numbers

Cauchy number, abbreviated as Ca, a dimensionless number, is the ratio of inertial force to compressibility force in a flow.  When the compressibility is important the elastic forces must be considered along with inertial forces.  The Cauchy number provides a measure of the importance of viscous effects compared to inertial effects in a fluid flow.  It helps determine the behavior of the flow, such as whether it is dominated by viscous forces (low Cauchy number) or inertial forces (high Cauchy number).

### The Cauchy number helps determine whether the flow is dominated by inertia or viscosity

• When Ca < 1  -  It implies that viscous forces dominate the flow, and the flow is considered viscous or "creeping."  In such cases, the fluid behaves like a highly viscous or sticky fluid, and inertia has little influence.
• When Ca > 1  -  It implies that inertial forces dominate the flow, and the flow is considered inertial.  In these situations, the fluid behaves more like an ideal, inviscid fluid, and viscous effects are negligible.

The Cauchy number is an important parameter in fluid mechanics and is used to analyze and classify different types of fluid flows based on their relative importance of viscosity and inertia.

## Cauchy Number formula

$$\large{ Ca = \frac{v^2 \; \rho }{ B } }$$

### Cauchy Number - Solve for Ca

$$\large{ Ca = \frac{v^2 \; \rho }{ B } }$$

 velocity, v density, p bulk modulus elasticity, B

### Cauchy Number - Solve for v

$$\large{ v = \sqrt{ \frac{Ca \; B }{ p } } }$$

 Cauchy Number, Ca bulk modulus elasticity, B density, p

### Cauchy Number - Solve for p

$$\large{ p = \frac{ Ca \; B }{ v^2 } }$$

 Cauchy Number, Ca bulk modulus elasticity, B velocity, v

### Cauchy Number - Solve for B

$$\large{ B = \frac{ v^2 \; p }{ Ca } }$$

 velocity, v density, p Cauchy Number, Ca

Symbol English Metric
$$\large{ Ca }$$ = Cauchy number $$\large{dimensionless}$$
$$\large{ v }$$ = velocity of the flow $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$
$$\large{ \rho }$$  (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ B }$$ = bulk modulus elasticity $$\large{\frac{lbm}{in^2}}$$  $$\large{Pa}$$

Tags: Force Equations