Water is flowing on a horizontal fixed surface, such that its flow velocity varies with $y$ (vertical direction) as $v=k\left(\frac{2 y^2}{a^2}-\frac{y^3}{a^3}\right)$. If coefficient of viscosity for water is $\eta$, what will be shear stress between layers of water at $y=a$.
Medium
Download our app for free and get started
(a)
Newton's law of viscosity, $F=\eta A \frac{d v}{d y}$
$\text { Stress }=\frac{F}{A}=\eta\left(\frac{d v}{d y}\right)=\eta k\left(\frac{4 y}{a^2}-\frac{3 y^2}{a^3}\right)$
$\text { At } y=a, \text { stress }=\eta k\left(\frac{4}{a}-\frac{3}{a}\right)=\frac{\eta k}{a}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A closed rectangular tank is completely filled with water and is accelerated horizontally with an acceleration a towards right. Pressure is $(i)$ maximum at, and $ (ii) $ minimum atView Solution
- 2Water flows steadily through a horizontal pipe of variable cross-section. If the pressure of water is $P$ at a point where flow speed is $v$ , the pressure at another point where the flow speed is $2v$ , is (Take density of water as $\rho $ )View Solution
- 3Water flows in a stream line manner through a capillary tube of radius $a$. The pressure difference being $P$ and the rate of flow is $Q$. If the radius is reduced to $\frac{a}{4}$ and the pressure is increased to $4 P$. then the rate of flow becomes ................View Solution
- 4Two identical cylindrical vessels with their bases at same level each contains a liquid of density $\rho$. The height of the liquid in one vessel is ${h_1}$ and that in the other vessel is ${h_2}$. The area of either base is $A$. The work done by gravity in equalizing the levels when the two vessels are connected, isView Solution
- 5The area of cross-section of the wider tube shown in figure is $800$ $cm^2$. If a mass of $12$ $ kg $ is placed on the massless piston, the difference in heights $h$ in the level of water in the two tubes is ........ $cm$View Solution
- 6Consider a cylindrical tank of radius $1 m$ is filled with water. The top surface of water is at $15\,m$ from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of $5\,m$ from the bottom. A force of $5 \times 10^{5} N$ is applied an the top surface of water using a piston. The speed of efflux from the hole will be.View Solution
(given atmospheric pressure $P_{A}=1.01 \times 10^{5}\,Pa$, density of water $\rho_{ w }=1000\,kg / m ^{3}$ and gravitational acceleration $g=10\,m / s ^{2}$ )
- 7The top of a water tank is open to air and its water level is maintained. It is giving out $0.74\,m^3$ water per minute through a circular opening of $2\,cm$ radius in its wall. The depth of the centre of the opening from the level of water in the tank is close to........ $m$View Solution
- 8A piston of cross-section area $100\, cm^2$ is used in a hydraulic press to exert a force of $107\, dynes$ on the water. The cross-sectional area of the other piston which supports an object having a mass $2000 \,kg$. isView Solution
- 9A body having volume $V$ and density $\rho$ is attached to the bottom of a container as shown. Density of the liquid is $d( > \rho )$. Container has a constant upward acceleration $a.$ Tension in the string isView Solution
- 10Two uniform solid balls of same density and of radii $r$ and $2r$ are dropped in air and fall vertically downwards. The terminal velocity of the ball with radius $r$ is $1\,cm\,s^{-1}$ , then find the terminal velocity of the ball of radius $2r$ (neglect bouyant force on the balls.) ........... $cm\,s^{-1}$View Solution