Option 1 : \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

CT 10: Soil Mechanics

3808

10 Questions
10 Marks
10 Mins

**Explanation:**

Continuity Equation is based on the principle of conservation of mass. For a fluid flowing through a pipe at all the cross-sections, the quantity of fluid per second is constant.

The continuity equation is given as \({ρ _1}{A_1}{V_1} = \;{ρ _2}{A_2}{V_2}\)

Generalized equation of continuity.

\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} + \frac{{\partial ρ }}{{\partial t}} = 0\)

This equation can be written in vector form as,

**Case 1:** For steady flow \(\frac{{\partial ρ }}{{\partial t}} = 0\) then the above equation will become,

\(\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} = 0\)

**Case 2: **For Incompressible flow, ρ is constant, therefore the continuity equation of steady incompressible for three-dimensional flow is,

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

For two dimensional flow

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)