Continuity Equation

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In fluid mechanics and aerodynamics, the continuity equation expresses the principle of conservation of mass. It ensures that mass cannot be created or destroyed within a flow.

This equation is fundamental to all flow analysis, appearing in two forms: the integral form (for finite control volumes) and the differential form (for infinitesimal fluid elements).

1️⃣ Physical Meaning

Imagine air flowing through a duct. If more air enters than leaves, the air inside must accumulate or become denser. If there is no accumulation (steady flow), what enters must exactly equal what exits.

This balance embodies mass conservation:

Mass inflow − Mass outflow = Change of mass inside

2️⃣ Integral Form of the Continuity Equation

The integral form applies mass conservation over a finite control volume. It states:

\frac{d}{dt} \int_{CV} \rho , dV + \int_{CS} \rho \mathbf{V} \cdot d\mathbf{A} = 0

Where:

  • \rho = fluid density
  • \mathbf{V} = velocity vector
  • dV = infinitesimal volume element within the control volume
  • d\mathbf{A} = outward-pointing area vector on the control surface

Explanation:

  • The first term represents the rate of accumulation of mass inside the control volume.
  • The second term is the net mass flux out through the control surface.

For steady flow, the time derivative term is zero:

\int_{CS} \rho \mathbf{V} \cdot d\mathbf{A} = 0

This states that the net mass flow rate out of the volume is zero—what enters must leave.

3️⃣ Differential Form of the Continuity Equation

For local, pointwise analysis, we use the differential form:

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0

Where:

  • \frac{\partial \rho}{\partial t} = local rate of change of density at a point
  • \nabla \cdot (\rho \mathbf{V}) = divergence of mass flux

Explanation:

  • If density changes with time at a point, it must be balanced by net inflow or outflow of mass at that point.
  • For compressible flows, both density and velocity divergence can change.
  • In steady flow, the time derivative term vanishes.

4️⃣ Incompressible Flow Simplification

For incompressible flows (constant density), the equation simplifies because \rho is constant:

\nabla \cdot \mathbf{V} = 0

Interpretation:

  • The velocity field must have zero divergence.
  • Fluid is neither expanding nor compressing locally.
  • Common assumption for low-speed aerodynamic flows (Mach < 0.3).

5️⃣ One-Dimensional Steady Flow Example

In a simple 1D steady flow through a duct:

\rho_1 A_1 V_1 = \rho_2 A_2 V_2

Where:

  • \rho = density
  • A = cross-sectional area
  • V = velocity

For incompressible flow (constant density):

A_1 V_1 = A_2 V_2

Explanation:

  • As area decreases, velocity must increase to conserve mass.
  • This principle is used in analyzing nozzles, diffusers, and wind tunnels.

6️⃣ Applications in Aerodynamics

  • Computing mass flow rates through engines, ducts, and air intakes
  • Analyzing nozzle and diffuser designs
  • Setting boundary conditions in CFD simulations
  • Describing both compressible and incompressible flows

The continuity equation is the starting point for all aerodynamic analysis because mass must be conserved in any flow, regardless of complexity.

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