Fluid Mechanics

A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma = \dfrac53$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a1}$, the radius of the bubble is found to be $r_1$ and the temperature of the enclosed gas is $T_1$. When the atmospheric pressure is $P_{a2}$, the radius of the bubble and the temperature of the enclosed gas are $r_2$ and $T_2$, respectively.

Which of the following statement(s) is(are) correct?

The terminal velocity ($v_t$) of the spherical rain drop depends on the radius (r) of the spherical rain drop as :

The velocity of upper layer of water in a river is $36\ \mathrm{kmh^{-1}}$. Shearing stress between horizontal layers of water is $10^{-3}\ \mathrm{Nm^{-2}}$. Depth of the river is ______ m. (Co-efficient of viscosity of water is $10^{-2}$ Pa.s)

An ideal fluid of density 800 kg m$^{-3}$, flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from $a$ to $\dfrac a2$. The pressure difference between the wide and narrow sections of pipe is 4100 Pa. At wider section, the velocity of fluid is $\dfrac{\sqrt x}6$ ms$^{-1}$ for $x=$ ______. (Given $g=10$ ms$^{-2}$)

If $\rho$ is the density and $\eta$ is coefficient of viscosity of fluid which flows with a speed $v$ in the pipe of diameter d, the correct formula for Reynolds number $R_e$ is :

The velocity of a small ball of mass 'm' and density $d_1$, when dropped in a container filled with glycerine, becomes constant after some time. If the density of glycerine is $d_2$, then the viscous force acting on the ball, will be :

The area of cross-section of a large tank is 0.5 m$^2$. It has a narrow opening near the bottom having area of cross-section 1 cm$^2$. A load of 25 kg is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is 40 cm above the bottom, will be ______ cms$^{-1}$. [Take $g=10$ ms$^{-2}$]

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Product of Pressure (P) and time (t) has the same dimension as that of coeffcient of viscosity.

Reason R : Coefficient of viscosity $=\dfrac{\text{Force}}{\text{Velocity gradient}}$

Choose the correct answer from the options given below :

A water drop of diameter 2 cm is broken into 64 equal droplets. The surface tension of water is 0.075 N/m. In this process the gain in surface energy will be :

A water drop of radius 1 $\mu$m falls in a situation where the effect of buoyant force is negligible. Co-efficient of viscosity of air is $1.8\times10^{-5}$ Nsm$^{-2}$ and its density is negligible as compared to that of water $10^6$ gm$^{-3}$. Terminal velocity of the water drop is :

(Take acceleration due to gravity $=10$ ms$^{-2}$)

A liquid of density 750 kgm$^{-3}$ flows smoothly through a horizontal pipe that tapers in cross-sectional area from $A_1=1.2\times10^{-2}$m$^2$ to $A_2=\dfrac{A_1}2$. The pressure difference between the wide and narrow sections of the pipe is 4500 Pa. The rate of flow of liquid is ______ $\times10^{-3}$ m$^3$s$^{-1}$.

A small spherical ball of radius 0.1 mm and density $10^4$ kg m$^{-3}$ falls freely under gravity through a distance $h$ before entering a tank of water. If, after entering the water the velocity of ball does not change and it continue to fall with same constant velocity inside water, then the value of $h$ will be ______ m

(Given $g=10$ ms$^{-2}$, viscosity of water $=1.0\times10^{-5}$ N-sm$^{-2}$).

An air bubble of negligible weight having radius r rises steadily through a solution of density $\sigma$ at speed $v$. The coefficient of viscosity of the solution is given by :

The excess pressure inside a liquid drop is 500 Nm$^{-2}$. If the radius of the drop is 2 mm, the surface tension of liquid is $x\times10^{-3}$ Nm$^{-1}$. The value of $x$ is ______.

A drop of liquid of density $\rho$ is floating half immersed in a liquid of density $\sigma$ and surface tension $7.5\times10^{-4}$ Ncm$^{-1}$. The radius of drop in cm will be : ($g=10$ ms$^{-2}$)

A water drop of radius 1 cm is broken into 729 equal droplets. If surface tension of water is 75 dyne/cm, then the gain in surface energy upto first decimal place will be :

(Given $\pi=3.14$)

Two cylindrical vessels of equal cross-sectional area 16 cm$^2$ contain water upto heights 100 cm and 150 cm respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water = $10^3$ kg/m$^3$ and g = 10 ms$^{-2}$] :

A spherical soap bubble of radius 3 cm is formed inside another spherical soap bubble of radius 6 cm. If the internal pressure of the smaller bubble of radius 3 cm in the above system is equal to the internal pressure of the another single soap bubble of radius r cm. The value of r is ______.

The diameter of an air bubble which was initially 2 mm, rises steadily through a solution of density 1750 kg m$^{-3}$ at the rate of 0.35 cms$^{-1}$. The coefficient of viscosity of the solution is ______ poise (in nearest integer). (the density of air is negligible).

A pressure-pump has a horizontal tube of cross sectional area 10 cm$^2$ for the outflow of water at a speed of 20 m/s. The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is :

[given: density of water = 1000 kg/m$^3$]