Question

The rate of change of energy in a moving model is ( hofrac{De}{Dt}). In the final equation, this term is reduced to (frac{partial( ho e)}{partial t}+ abla.( ho evec{V})). Which of these equations is used for this reduction?

(a) Equations of state

(b) Stress-strain equation

(c) Momentum equation

(d) Continuity equation

Answer 1

Right option is (d) Continuity equation

For explanation: Continuity equation is used as given below.

( hofrac{De}{Dt}= hofrac{partial e}{partial t}+ hovec{V}. abla e )

But,

( hofrac{partial e}{partial t}=frac{partial( ho e)}{partial t}-efrac{partial ho}{partial t})

And

( hovec{V}. abla e= abla.( ho evec{V})-e abla.( ho vec{V}))

Therefore,

( hofrac{De}{Dt}=frac{partial( ho e)}{partial t}-efrac{partial ho}{partial t}+ abla.( ho evec{V})-e abla.( ho vec{V}))

( hofrac{De}{Dt}=frac{partial( ho e)}{partial t}-e(frac{partial ho}{partial t}+ abla.( ho vec{V}))+ abla.( ho evec{V}))

Applying the continuity equation, (frac{partial ho}{partial t}+ abla.( ho vec{V})=0), and hence

( hofrac{De}{Dt}=frac{partial( ho e)}{partial t}+ abla.( ho evec{V})).