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The derivation is based on the concept of Law of conservation of mass. Statement: The flow of fluid in a continuous flow across a section is always a constant.

*Continuity
Equation*

**The
derivation is based on the concept of Law of conservation of mass.**

*Statement: The flow of fluid in a continuous flow
across a section is always a constant.*

Consider an enlarging section in a fluid flow of fluid density*?*** .** Consider
two sections 1 and 2 as shown in Fig. Let the sectional properties be as under

** A_{1}**and

** dt **is the time taken for the fluid
to cover a distance

*m _{1} = Density at section 1 *

*?*_{1}** x A_{1} x dx**

Mass rate
of fluid flowing across section 1-1 is given by

*m _{1}/dt
= (Density at sec 1 *

*?*_{1}** x A_{1} x dx/dt = **

Similarly
Mass rate of fluid flowing across section 2-2 is given by

*m _{2}/dt
= *

From law
of conservation of mass, mass can neither be created nor destroyed.

Hence
from Eqs. 1 and 2, we get

*?*_{1}*A _{1}
*

If the
density of the fluid is constant, then the equation reduces to

*A _{1}*

The above equation is discharge continuity equation in one
dimensional form for an steady, incompressible fluid flow

*Continuity
Equation in three dimensional or differential forms*

Consider a parallelepiped ** ABCDEFGH** in a fluid
flow of density

Similarly
mass rate of fluid flow leaving the section EFGH along *x* direction is
given by

Net gain
in mass rate of the fluid along the *x* axis is given by the difference
between the mass rate of flow entering and leaving the control volume. i.e. Eq.
1 – Eq. 2

This expression is known as the general Equation of Continuity
in three dimensional form or differential form.

If the
fluid is incompressible then the density is constant and hence

The
continuity equation in two-dimensional form for compressible and incompressible
flows are respectively as below

Tags : Civil - Mechanics Of Fluids - Fluid Kinematics And Dynamics

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