$A$ flat plate of area $0.1 \,m^2$ is placed on a flat surface and is separated from it by a film of oil $10^{-5} \,m$ thick whose coefficient of viscosity is $1.5 \,N \cdot s \cdot m^{-2}$. The force required to cause the plate to slide on the surface at a constant speed of $1 \,mm \cdot s^{-1}$ is ............ $N$.

Explain the velocity gradient and the coefficient of viscosity,and provide their units.

$A$ plastic circular disc of radius $R$ is placed on a thin oil film,spread over a flat horizontal surface. The torque required to spin the disc about its central vertical axis with a constant angular velocity is proportional to

$A$ flat plate of area $10 \, cm^2$ is separated from a large plate by a layer of glycerine $1 \, mm$ thick. If the coefficient of viscosity of glycerine is $20 \, \text{poise}$, the force required to keep the plate moving with a velocity of $1 \, cm/s$ is .......... $dyne$.

Water is flowing steadily in a river. $A$ and $B$ are two layers of water at heights $40 \ cm$ and $90 \ cm$ from the bottom. The velocity of layer $A$ is $12 \ cm/s$. If the velocity of the water at the bottom is $0$,what is the velocity of layer $B$ (in $cm/s$)?

$A$ thin square plate of side $2\ m$ is moving at the interface of two very viscous liquids of viscosities $\eta_1 = 1 \text{ poise}$ and $\eta_2 = 4 \text{ poise}$ respectively, as shown in the figure. Assume a linear velocity distribution in each fluid. The liquids are contained between two fixed plates. $h_1 + h_2 = 3\ m$. $A$ force $F$ is required to move the square plate with a uniform velocity $10\ m/s$ horizontally. The value of the minimum applied force will be ........ $N$.