Short Answer:

The continuity equation expresses conservation of mass in a flow. For a steady, incompressible fluid (constant density and no time change), it reduces to the simple condition that the divergence of velocity is zero:

In integral terms this means the net volume flux through any closed surface is zero, and for a pipe it gives the familiar relation  .

Detailed Explanation :

Continuity equation for steady incompressible flow

Goal and basic idea
The continuity equation comes from the law of conservation of mass: mass cannot be created or destroyed. For fluids this law says that the rate at which mass enters a control volume minus the rate at which mass leaves must equal the rate of accumulation of mass inside. For steady flow there is no time variation of mass inside the control volume (no accumulation). For incompressible fluids density   is constant. Combining these two facts leads to a simple, useful form of the continuity equation.

Integral (control-volume) form

Start with an arbitrary fixed control volume   bounded by surface   with outward unit normal  . Mass flux through a surface element   is  . Conservation of mass gives the integral balance:

First term is rate of mass accumulation inside  . For steady flow the mass inside does not change with time, so  . That leaves

For incompressible fluid   is constant and can be taken outside the integral (and nonzero), so we obtain

This statement means the net volume flux (volume per unit time) out of any closed surface is zero. In words: what goes in must come out.

Differential form (local)

To get the local differential form, apply the divergence theorem to the integral form:

Because the control volume   is arbitrary, the integrand must be zero everywhere:

This is the standard differential form for steady incompressible flow. In Cartesian coordinates with  ,

This equation says the sum of the rates of change of the three velocity components with respect to their coordinates is zero — a local statement that no net volume is being created or destroyed at any point.

One-dimensional special case

For simple 1-D flow along the x-axis (e.g., flow through a pipe where velocity is uniform over cross sections), the differential form reduces to  , meaning   is constant along x for incompressible steady flow in that idealized case. In practical pipe flow the common engineering relation follows from integral form applied between two cross sections:

because   cancels for incompressible fluid. This is the familiar continuity relation used in many calculations.

Derivation from a small differential box (intuitive)

Consider a small rectangular box of sides  . Mass flow into the face at   is  ; mass flow out at   is  . Do the same for y and z faces. Sum the net outflow terms and set equal to rate of change of mass inside. For steady and incompressible (  constant, no accumulation) the net outflow must be zero, giving:

same result as above. This differential-box method makes clear how the divergence arises.

Physical meaning and use

means local volume is conserved: fluid elements can deform but their instantaneous volume does not change. For incompressible steady flows this is a very strong constraint that simplifies analysis and is used in solving flow fields, simplifying Navier–Stokes equations, and in computational fluid dynamics as a primary constraint for velocity fields.

Examples

• Flow through a constant-area pipe:   constant so   constant along pipe; continuity satisfied.
• Nozzle: area decreases so velocity increases so that   stays constant.
• Flow around a body: the velocity field around the body must satisfy   everywhere outside the solid.

Summary of main results
Integral (steady, incompressible):
Differential (steady, incompressible):
1-D pipe relation:

Conclusion

The continuity equation for steady incompressible flow is the mathematical expression of mass conservation under the assumptions of constant density and no time variation. In integral form it states that net volume flux through any closed surface is zero, and in local (differential) form it reduces to  . For simple pipe flows this leads to the familiar relation  . These forms are fundamental constraints used in all fluid-mechanics analyses and designs.