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

Consider a fully developed steady laminar flow of an incompressible fluid with viscosity $$\mu $$ through a circular pipe of radius $$R.$$ Given that the velocity at a radial location of $$R/2$$ from the center-line of the pipe is $${U_1},$$ the shear stress at the wall is $$K\mu {U_1}/R,$$ where $$K$$ is _________________.

For a fully developed laminar flow of water (dynamic viscosity $$0.001$$ $$Pa$$-s) through a pipe of radius $$5$$ $$cm,$$ the axial pressure gradient is $$-10$$ $$Pa/m$$. The magnitude of axial velocity (in $$m/s$$) at a radial location of $$0.2$$ $$cm$$ is ____________

The head loss for a laminar incompressible flow through a horizontal circular pipe is $${h_1}$$. Pipe length and fluid remaining the same, if the average flow velocity doubles and the pipe diameter reduces to half its previous value, the head loss is $${h_2}.$$ The ratio $${h_2}/{h_1}$$ is