Question

Expand the substantial derivative Dρ/Dt.

(a) (frac{D ho}{Dt}=frac{d ho}{dt}+u frac{d ho}{dx}+vfrac{d ho}{dy}+wfrac{d ho}{dz})

(b) (frac{D ho}{Dt}=frac{partial ho}{partial t}+u frac{d ho}{dy}+vfrac{d ho}{dz}+wfrac{d ho}{dx})

(c) (frac{D ho}{Dt}=frac{d ho}{dz}+ufrac{partial ho}{partial y}+vfrac{partial ho}{partial z}+w frac{partial ho}{partial t})

(d) (frac{D ho}{Dt}=frac{partial ho}{partial t}+ufrac{partial ho}{partial x}+vfrac{partial ho}{partial y}+w frac{partial ho}{partial z})

Answer 1

Correct choice is (d) (frac{D ho}{Dt}=frac{partial ho}{partial t}+ufrac{partial ho}{partial x}+vfrac{partial ho}{partial y}+w frac{partial ho}{partial z})

Explanation: As the location coordinates (x, y, z) vary with time,

(frac{D ho}{Dt}=frac{partial ho}{partial t}+frac{partial ho}{partial x}frac{partial x}{partial t}+frac{partial ho}{partial y}frac{partial y}{partial t}+frac{partial ho}{partial z}frac{partial z}{partial t})

(frac{D ho}{Dt}=frac{partial ho}{partial t}+ufrac{partial ho}{partial x}+vfrac{partial ho}{partial y}+frac{partial ho}{partial t})