Concept:
The general form of the continuity equation in cartesian coordinates:
\(\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho \omega } \right) = 0\)
For steady flow:
\(\frac{{\partial \rho }}{{\partial t}} = 0\)
\(\frac{\partial }{{\partial x}}\left( {\rho u} \right) + \frac{\partial }{{\partial y}}\left( {\rho v} \right) + \frac{\partial }{{\partial z}}\left( {\rho \omega } \right) = 0\)
If the fluid is incompressible, then ρ is constant
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
\(\nabla .\vec V = 0\)
Calculation:
\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)
\(\frac{\partial }{{\partial x}}({a_1}x + {a_2}y + {a_3}z) + \frac{\partial }{{\partial y}}({b_1}x + {b_2}y + {b_3}z) + \frac{\partial }{{\partial z}}({c_1}x + {c_2}y + {c_3}z) = 0\)
\(\begin{array}{l} \Rightarrow {a_1} + {b_2} + {c_3} = 0 \Rightarrow 2 - 4 + {b_2} = 0 \Rightarrow {b_2} = 2 \end{array}\)