For steady, fully developed flow inside a straight pipe of diameter $$D,$$ neglecting gravity effects, the pressure drop $$\Delta p$$ over a length $$L$$ and the wall shear stress $${\tau _w}$$ are related by

Water is coming out from a tap and falls vertically downwards. At the tap opening, the stream diameter is $$20$$ $$mm$$ with uniform velocity of $$2$$ $$m/s.$$ Acceleration due to gravity is $$9.81$$ $$m/{s^2}.$$ Assuming steady, inviscid flow, constant atmospheric pressure everywhere and neglecting curvature and surface tension effects, the diameter in mm of the stream $$0.5$$ $$m$$ below the tap is approximately

A hinged gate of length $$5$$ $$m,$$ inclined at $${30^ \circ }$$ with the horizontal and with water mass on its left, is shown in the figure below. Density of water is $$1000$$ $$kg/{m^3}$$. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is

Consider one-dimensional steady state heat conduction, without heat generation, in a plane wall; with boundary conditions as shown in the figure below. The conductivity of the wall is given by $$k = {k_0} + bT;$$ where $${k_0}$$ and $$b$$ are positive constants, and $$T$$ is temperature.

As $$x$$ increases, the state temperature gradient $$(dT/dx)$$ will