You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices. Expressed in terms of the two - dimensional velocity fields in the $$x-$$ $$y$$ plane, are made available.
$$P:$$ $$u = 2y,\,\,\,v = - 3x$$
$$Q:$$ $$u=3xy,$$ $$\,\,\,\,$$$$v=0$$
$$R:$$ $$u=-2x,$$ $$\,\,\,\,$$$$v=2y$$

Which flows should be recommended when the application requires the flow to be incompressible and irrotational?

The velocity profile of a fully developed laminar flow in a straight circular pipe, as shown in the figure, is given by the expression. $$$u\left( r \right) = {{ - {R^2}} \over {4\mu }}\left( {{{dp} \over {dx}}} \right)\left( {1 - {{{r^2}} \over {{R^2}}}} \right)$$$
Where $${{dp} \over {dx}}$$ is a constant.

The average velocity of fluid in the pipe is

Consider steady, incompressible and irrotational flow through a reducer in a horizontal pipe where the diameter is reduced from $$20cm$$ to $$10cm.$$ The pressure in the $$20cm$$ pipe just upstream of the reducer is $$150kPa.$$ The fluid has a vapour pressure of $$50kPa$$ and a specific weight of $$5\,\,kN/{m^3}.$$ Neglecting frictional effects, the maximum discharge (in $${m^3}/s$$) that can pass through the reducer without causing cavitation is

Consider steady-state heat conduction across the thickness in a plane composite wall as shown in fig exposed to convection conditions on both sides.

Assuming negligible contact resistance between the wall surfaces, the interface temp $$T(C)$$ of the two walls will be