Question

The velocity field of an incompressible flow is given by \(V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k\), where a₁ = 2 and c₃ = -4.find the value of b₂

Answer 1

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:

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

\(\frac{\partial }{{\partial x}}({a_1}x + {a_2}y + {a_3}z) + \frac{\partial }{{\partial y}}({b_1}x + {b_2}y + {b_3}z) + \frac{\partial }{{\partial z}}({c_1}x + {c_2}y + {c_3}z) = 0\)

\(\begin{array}{l} \Rightarrow {a_1} + {b_2} + {c_3} = 0 \Rightarrow 2 - 4 + {b_2} = 0 \Rightarrow {b_2} = 2 \end{array}\)