For a steady flow, the velocity field is $$\overrightarrow V = \left( { - {x^2} + 3y} \right)\widehat i + \left( {2xy} \right)\widehat j.$$ The magnitude of the acceleration of the particle at $$(1, -1)$$ is

For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter, the average velocity in the fully developed region is constant. Which one of the following statements about the average velocity in the developing region is TRUE?

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