The instantaneous stream-wise velocity of a turbulent flow is given as follows: $$$u\left( {x,y,z,t} \right) = \overline u \left( {x,y,z} \right) + \mu '\left( {x,y,z,t} \right)$$$

The time - average of the fluctuating velocity $$u'(x,y,z,t)$$

Consider the turbulent flow of a fluid through a circular pipe of diameter, $$D.$$ Identify the correct pair of statements.
$${\rm I}.$$ The fluid is well-mixed
$${\rm II}.$$ The fluid is unmixed
$${\rm III}.$$ $$R{e_D} < 2300$$
$${\rm IV}.$$ $$R{e_D} > 2300$$

For steady, fully developed flow inside a straight pipe of diameter $$D,$$ neglecting gravity effects, the pressure drop $$\Delta p$$ over a length $$L$$ and the wall shear stress $${\tau _w}$$ are related by

Consider steady laminar incompressible axi-symmetric fully developed viscous flow through a straight circular pipe of constant cross - sectional area at a Reynolds number of $$5.$$ The ratio of inertia force to viscous force on a fluid particle is