Consider fluid flow between two infinite horizontal plates which are parallel (the gap between them being $$50$$ $$mm$$). The top plate is sliding parallel to the stationary bottom plate at a speed of $$3$$ $$m/s.$$ The flow between the plates is solely due to the motion of the top plate. The force per unit area (magnitude) required to maintain the bottom plate stationary is _______________ $$N/{m^2}.$$

Viscosity of the fluid $$\mu = 0.44\,\,kg/m$$-$$s$$ and density $$\rho = 888$$ $$kg/{m^3}.$$

The volume tric flow rate (per unit depth) between two streamlines having stream functions $${\psi _1}$$ & $${\psi _2}$$ is

The large vessel shown in the figure contains oil and water. A body is submerged at the interface of oil and water such that $$45$$ percent of its volume is in oil while the rest is in water. The density of the body is _______________ $$kg/{m^3}.$$

The specific gravity of oil is $$0.7$$ and density of water is $$1000$$ $$kg/{m^3}.$$

Acceleration due to gravity $$g = 10\,m/{s^2}$$

Consider a frictionless, massless and leak-proof plug blocking a rectangular hole of dimensions $$2R \times L$$ at the bottom of an open tank as shown in the figure. The head of the plug has the shape of a semi-cylinder of radius $$R.$$ The tank is filled with a liquid of density $$\rho $$ up to the tip of the plug. The gravitational acceleration is $$g.$$ Neglect the effect of the atmospheric pressure.

The force $$F$$ required to hold the plug in its position is