Fluid Mechanics Questions

Consider a water tank as shown in the figure. It has one wall at $x=L$ and is very wide in the $z$ direction. When filled with a liquid of surface tension $S$ and density $\rho$, the liquid surface makes an angle ${\theta }_0$ (${\theta }_0\ll 1$) with the $x$ -axis at $x=L$. If $y(x)$ is the height of the surface, then the equation for $y(x)$ is:(Take $\theta (x)\approx \sin \theta (x)\approx \tan \theta (x)=\frac{dy}{dx}$ and $g$ as acceleration due to gravity)

A balloon is made of a material of surface tension $S$ and its inflation outlet (from where gas is filled in it) has small area $A$. It is filled with a gas of density $\rho$ and takes a spherical shape of radius $R$. When the gas is allowed to flow freely out of it, its radius $r$ changes from $R$ to $0$ (zero) in time $T$. If the speed $v(r)$ of gas coming out of the balloon depends on $r$ as $r^{\alpha }$ and $T\propto S^a{A}^{\beta }{\rho }^{\gamma }{R}^{\delta }$ then:

The apparent depth of water in cylindrical water tank of diameter $2R\text{cm}$ is reducing at the rate of $x\text{cm/minute}$ when water is being drained out at a constant rate. The amount of water drained in $\text{c.c.}$ per minute is ($n_1=\text{refractive index of air}$, $n_2=\text{refractive index of water}$):

A given shaped glass tube having uniform cross section is filled with water and is mounted on a rotatable shaft as shown in figure. If the tube is rotated with a constant angular velocity $\omega$ then:

Assertion: Specific gravity of a fluid is a dimensionless quantity.Reason: It is the ratio of density of fluid to the density of water.

Assertion: For Reynold number $R_e>2000$, the flow of fluid is turbulent.Reason: Inertial forces are dominant compared to the viscous forces at such high Reynold numbers.

A rectangular block of mass $m$ and area of cross-section $A$ floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium, it undergoes oscillation with a time period $T$. Then:

The cylindrical tube of a spray pump has radius $R$, one end of which has $n$ fine holes, each of radius $r$. If the speed of the liquid in the tube is $V$, the speed of the ejection of the liquid through the holes is:

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as $[{\eta }^x{\rho }^y{r}^z]$, where $\eta ,\rho$ and $r$ are the coefficient of viscosity, density of liquid and radius of the tube respectively, then the values of $x,y$ and $z$ are:

Water rises to a height $h$ in a capillary tube. If the length of capillary tube above the surface of water is made less than $h$, then:

Two non-mixing liquids of densities $\rho$ and $n\rho (n>1)$ are put in a container. The height of each liquid is $h$. A solid cylinder of length $L$ and density $d$ is put in this container. The cylinder floats with its axis vertical and length $pL(p<1)$ in the denser liquid. The density $d$ is equal to: