The continuity equation is about keeping track of mass, and it works a bit differently for different types of flow. Let's break it down:

Incompressible Flow

• This is for fluids like water, where the density stays the same.
• The continuity equation becomes simpler:
  $A_1v_1 = A_2v_2$
  Here, $A$ stands for cross-sectional area (how wide something is), and $v$ stands for velocity (how fast it moves).

Compressible Flow

• In this case, the density can change, especially with gases.
• The equation is a bit more complicated to include these changes:
  $\frac{\partial (\rho)}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$
  Here, $\rho$ is density and $\vec{v}$ is velocity.

Key Takeaway

• For incompressible flow, the density is constant, which makes the math easier.
• For compressible flow, the density changes, making the math more complex, but it helps us understand gases, especially when they move really fast.

Both types of flow keep mass the same, but they tell different stories with their equations!