Shown below are three cylindrical gates which restrain water in $$2$$-$$D$$ channel. Which gate experiences the maximum vertical component, the minimum vertical component and the maximum horizontal component of the hydrostatic force?

The velocity profile across a boundary layer on a flat plate may be approximated as linear $$${V_x}\left( {x,y} \right) = {{{V_0}y} \over {\delta \left( x \right)}}$$$

Where $${{V_0}}$$ is the velocity far away and $$\delta \left( x \right)$$ is the boundary layer thickness at a distance $$x$$ from the leading edge, as shown below.

(a)$$\,\,\,\,\,\,$$ Use an appropriate control volume to determine the rate of mass influx into the
$$\,\,\,\,\,\,$$$$\,\,$$$$\,\,\,\,\,\,$$boundary layer up to $$x.$$

(b)$$\,\,\,\,\,\,$$ Obtain the momentum thickness into the boundary layer up to $$x.$$

(c)$$\,\,\,\,\,\,$$ In which direction (up or down) does the shear stress act on the face $$AB$$ of
$$\,\,\,\,\,\,$$$$\,\,\,\,\,\,\,$$the fluid element shown near the plate?

Shown below are three tanks, tank $$1$$ without an orifice tube and tanks $$2$$ and $$3$$ with orifice tubes as shown. Neglecting losses and assuming the diameter of orifice to be much less than that of the tank, write expressions for the exit velocity in each of the three tanks.

The radiative heat transfer rate per unit area $$\left( {W/{m^2}} \right)$$ between two plane parallel gray surfaces (emissivity $$=0.9$$) maintained at $$400K$$ and $$300K$$ is ($${{\sigma _b} = }$$ Stephen Boltzmann constant $$5.67 \times {10^{ - 8}}\,\,W/{m^2}{K^4}$$)