Water is flowing on a horizontal fixed surface,such that its flow velocity varies with $y$ (vertical direction) as $v = k \left( \frac{2y^2}{a^2} - \frac{y^3}{a^3} \right)$. If the coefficient of viscosity for water is $\eta$,what will be the shear stress between the layers of water at $y = a$?

For the situation shown in the figure,water flows on the surface of a fixed plate. The velocity of water as a function of distance $y$ is given as: $u = \alpha \left[ \frac{y}{h} - 2\left( \frac{y}{h} \right)^2 \right]$. Determine the magnitude of the shear stress that the water applies at the base of the plate. The coefficient of viscosity is $\eta$.

$A$ 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 ........ $N$.

When the temperature increases,the viscosity of

$A$ thin plate is placed between two fixed plates,separating two liquids with coefficients of viscosity $\eta$ and $4\eta$,as shown in the figure. If the plate is to be pulled by applying the minimum force,then the ratio $\frac{d_2}{d_1}$ is:

$A$ liquid having a coefficient of viscosity $0.02 \, \text{decapoise}$ is filled in a container of cross-sectional area $20 \, m^2$. If the viscous drag between two adjacent layers in flowing is $1 \, N$, then the velocity gradient is ........ $s^{-1}$.