# Atwood Number

on . Posted in Dimensionless Numbers Tags: Density Fluid

The Atwood number, abbreviated as A, a dimensionless number, describes density difference between two adjacent fluids with a common interface.  It is used in fluid dynamics to describe the flow behavior and stability of a two-phase system with a density difference.  The Atwood number represents the ratio of the density difference between the two phases to the average density of the system.  The Atwood number is commonly used in the study of multiphase flows, such as the behavior of bubbles in a liquid, the flow of oil and water in pipelines, or the motion of liquid droplets in gas environments.  It helps characterize and predict the interfacial dynamics, mixing, and stability of such two-phase systems.

In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale.

### Atwood number categorizes fluids into different regimes

• At > 0  -  This indicates that the heavier phase is on top (ρ1 > ρ2).  In this case, the system is often referred to as a "stable" configuration since the denser phase tends to settle above the lighter phase due to gravitational forces.
• At < 0  -  This indicates that the lighter phase is on top (ρ1 < ρ2).  In this case, the system is considered "unstable" because buoyant forces tend to cause the lighter phase to rise above the denser phase.
• At = 0  -  This implies that the two phases have the same density (ρ1 = ρ2).  When the Atwood number is zero, there is no density difference, and the behavior of the system is typically uniform or neutral.

## ATwood number formula

$$\large{ A = \frac { \rho_1 \;-\; \rho_2 } { \rho_1 \;+\; \rho_2 } }$$
Symbol English Metric
$$\large{ A }$$ = Atwood number $$\large{ dimensionless }$$
$$\large{ \rho_1 }$$ (Greek symbol rho) = density of heavier fluid $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ \rho_2 }$$ (Greek symbol rho) = density of lighter fluid $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$ Tags: Density Fluid