Consider steady flow of an incompressible fluid through two long and straight pipes of diameters $${d_1}$$ and $${d_2}$$ arranged in series. Both pipes are of equal length and the flow is turbulent in both pipes. The friction factor for turbulent flow though pipes is of the form, $$f = K{\left( {{\mathop{\rm Re}\nolimits} } \right)^{ - n}},$$ where $$K$$ and $$n$$ are known positive constants and $$Re$$ is the Reynolds number. Neglecting minor losses, the ratio of the frictional pressure drop in pipe $$1$$ to that in pipe $$2,$$ $$\left( {{{\Delta {P_1}} \over {\Delta {P_2}}}} \right),$$ is given by

The velocity profile inside the boundary layer for flow over a flat plate is given as $${u \over {{U_\infty }}} = \sin \left( {{\pi \over 2}\,{y \over \delta }} \right),$$ where $${U_\infty }$$ is the free stream velocity and $$'\delta '$$ is the local boundary layer thickness. If $${\delta ^ * }$$ is the local displacement thickness, the value of $${{{\delta ^ * }} \over \delta }$$ is

Consider the two-dimensional velocity field given by
$$\overrightarrow V = \left( {5 + {a_1}x + {b_1}y} \right)\widehat i + \left( {4 + {a_2}x + {b_2}y} \right)\widehat j,$$
where $${a_1},\,\,{b_1},\,\,{a_2}$$ and $${b^2}$$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

Heat is generated uniformly in a long solid cylindrical rod ( diameter $$ = 10\,\,mm$$) at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The thermal conductivity of the rod material is $$25$$ $$W/m.K.$$ Under steady state conditions, the temperature difference between the center and the surface of the rod is ________________ $${}^ \circ C.$$