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Fluids

There are four phases of matter: solid, liquid, gas and plasma. Liquids, gases and plasmas are called fluids. A fluid can be compressible or incompressible. Fluids whose volume cannot be reduced by applying pressure are called incompressible fluids. Liquids are incompressible fluids, but gases are not.

Density

Density $\rho$ of a substance is defined as mass per unit volume:

$\rho=\dfrac{m}{V}$

where $m$ is the mass of the substance and $V$ is its volume. SI unit of density is $kg/m^3$. Density of water is $1000 kg/m^3$.

Pressure

Pressure is defined as force per unit area. If a force of magnitude, $F$ is applied on a surface of area $A$, then the pressure $P$ exerted on the surface by the force is:

$P=\dfrac{F}{A}$

pressure
SI unit of pressure is $N/m^2$ or pascal (Pa).

Pressure in fluids

In fluids, pressure at a given level(depth) is due to the weight of the fluid above that level. When you go from the surface of a liquid to a depth $h$, the pressure of the fluid increases by an amount:

$\Delta P=\rho g h$

where $\rho$ is the density of the fluid.
pressure in fluids
The proof of the above equation is as follows.
Let us consider a cylindrical container of area of cross section, $A$ as in the figure above. So, the volume of the fluid column of height, $h$ is $V=Ah$.
Let $m$ is the mass of the fluid in the column of height $h$, so the weight of the column is $W=mg$. In terms of density, we can write, $m=\rho V=\rho Ah$; and therefore, $W=\rho Ah g$. The weight, $W$ is the force that acts in the cross section of the container at the depth $h$ and creates the pressure there. Therefore,

$\Delta P=W/A=\rho Ahg/A=\rho g h$.

You see from the equation that pressure in a fluid is independent of the area of cross section $A$ of the container. This tells you that the pressure is independent of the shape of the container, only the depth is what matters. Pressure increases as depth increases in fluids. At a given depth, pressure is same at all points. An object in a fluid experiences pressure from all directions.

Atmospheric pressure

On earth, we feel pressure that is due to the weight of the atmospheric gases above the earth's surface. At the sea level, atmospheric pressure is,

$P_{atm}=1.013\times 10^5 N/m^2$, We call this $1\:atm$.

Absolute and gauge pressure

When you measure pressure by a gauge such as tire pressure gauge, what you measure is the pressure difference between the system (tire) and the atmospheric pressure. Such a pressure is called gauge pressure. Absolute pressure adds the atmospheric pressure to the gauge pressure. Therefore,

Absolute pressure = atmospheric pressure + gauge pressure

Note that all the pressure differences are gauge pressure as it does not include the atmospheric pressure.

Pascal's principle

In a confined fluid, if an external pressure is applied, then the pressure at every point within the fluid increases by that amount. This is called Pascal's principle.
pascal's principle

Pascal principle is applied in hydraulic brakes and in hydraulic lifts. Hydraulic lifts are used to lift a heavy objects with little force.

hydraulic lift

In a hydraulic lift, there is a confined fluid with two pistons, a smaller and a larger one on either side. A force, $F_1$ applied at the smaller piston increases the pressure at every point within the fluid. The pressure of the fluid below the smaller piston is therefore, $P_1=F_1/A_1$, where $A_1$ is the area of cross section of the smaller piston. And the pressure of the fluid, just below the larger piston is $P_2=F_2/A_2$. If we assume the bottom of the two pistons are at the same level, then we will have $P_1=P_2$,

(or)

$\dfrac{F_{1}}{A_{1}}= \dfrac{F_{2}}{A_{2}}$

(or)

$F_2= \dfrac{A_2}{A_1}F_1$

The force at the larger piston is determined by the factor, $A_2/A_1$. By increasing the area of cross section (diameter) of the larger piston or decreasing the area of the smaller piston, we can increase the force at the larger piston to lift heavy objects. Note that the pressure $P_1$ and $P_2$ are the gauge pressure that result from the applied force $F_1$ and they do not include the atmospheric pressure.

Pressure measurement

There are several instruments to measure pressure, such as manometer, aneroid gauge and barometer. Barometer is used to measure the atmospheric pressure.

Open tube manometer

It is a U-shaped tube with one end (on the left side) open to the atmosphere and the other end (on the right side) is connected to the system with unknown pressure (i.e., the system that you want to measure its pressure). The tube is filled with some fluid. Depends on the pressure, $P$ of the system, there is a height difference between the fluid level in the left and the right side of the tube.
open tube manometer
The left top surface of the fluid is at the atmospheric pressure and the right top surface of the fluid is at the unknown system's pressure. Further, the pressure within the fluid at a depth $h$ on the left side of the tube is same as that of the pressure at the top of the fluid in the right side of the tube as they are in the same level. So, the pressure difference between the top of the fluid and at depth $h$ on the left side of the tube is

$\Delta P=P-P_{atm}=\rho gh$

(or)

$P=P_{atm}+\rho g h$

where $\rho$ is the density of the fluid and $h$ is change in the fluid level height between the right and left side of the tube.

Mercury barometer

Mercury barometer is used to measure the atmospheric pressure. It is a long tube filled with mercury and is inverted on a beaker containing mercury. There is a vacuum created at the top of the inverted tube. But the force due to the atmospheric pressure, pushes the mercury upwards in the tube. The mercury raises to a level such that the weight of the column of mercury is balanced by the force of the atmospheric pressure.
mercury barometer
The pressure at the top of the mercury column is zero as there is a vacuum above that. Inside the column, the pressure at the level of the dotted line is same as the pressure of the mercury surface in the beaker that is at the atmospheric pressure. Therefore the pressure difference between the top and at the level shown as dotted line of the column is

$P_{atm}-0=\rho gh$

Or,

$P_{atm}=\rho gh$

where $\rho$ is the density of mercury.
At the atmospheric pressure at the sea level (i.e., at $1\:atm$), the mercury in a barometer raises to a height of $76\:cm$. Mercury is preferred in the barometer due to its higher density. If other liquids are used, we need to have a much longer tube as it raises to a level much higher than mercury. For example, if water is used instead of mercury, it raises to $10.3 m$ height at the atmospheric pressure, so we need a tube longer than $10.3 m$. So, with mercury we can have a compact barometer.

Buoyancy

Objects such as wood float in water and objects weigh less in water and can be easily lifted. Why?. It is due to a phenomenon called buoyancy.

An object immersed in a fluid, experiences higher pressure at its bottom than that at its top. So the magnitude of downward force $F_1$ exerted by the fluid on the top is smaller than the magnitude of the upward force $F_2$ at its bottom. As a result, there is a net upward force acting on the object. This net upward force is called the buoyant force, $F_B$

buoyancy
The buoyant force is

$F_B=F_2-F_1$

Archimedes principle

Archimedes in 250 B.C suggested that the buoyant force on an object in a fluid is equal to the weight of the fluid displaced by the object.

i.e., $F_B=\rho_{f} V_{disp}g=m_f g$

where $\rho_F$ is the density of the fluid, $m_f$ is the mass of the fluid displaced by the object and $V_{disp}$ is the volume of the object.
Archimedes principle is valid whether the object is submerged or floating in the fluid. If the object is submerged in the fluid, then $V_{disp}$ is the volume of the object.
submerged object
But if the object floats, then $V_{disp}$ is the volume of the object within the fluid. If an object has less density than a fluid, then the object floats in that fluid. For an object to float, the buoyant force on the object should balance its weight. Therefore, for a floating object,

$F_B=mg$

where $m$ is the mass of the object. If the buoyant force is less than the weight of the object, then the object sinks.
floating object

Apparent weight of an object

In fluids, an object weighs less than its actual weight due to the buoyant force. The weight of an object in a fluid is called its apparent weight.

Apparent weight of an object in a fluid is

$w'=mg-F_B$

Fluid dynamics or hydrodynamics

When fluid flows, the flow can be smooth, called streamline or laminar flow or erratic (not smooth), called turbulent flow. In a laminar flow, each path of the particle (streamline) does not cross each other and the neighboring layers of the fluid slide by each other smoothly. In a turbulent flow, whirlpool-like circles forms in the fluid at higher speeds. Laminar flow is easy to study, and here we will focus on the fluid flow that is laminar.

The amount of fluid flows per unit time across a cross sectional area is called the flow rate of the fluid. There are two types of flow rates: mass flow rate and volume flow rate.
flow rate
Mass flow rate is the mass of the fluid passing through a given cross sectional area per unit time,

Mass flow rate $= \dfrac{\Delta m} {\Delta t}$

Volume flow rate is the volume of the fluid that passes through a given cross sectional area per unit time,

Volume flow rate $= \dfrac{\Delta V} {\Delta t}$
flow rate
If a fluid flows with velocity $v$ through a tube of area of cross section $A$ , then the volume flow rate of the fluid is

Volumes flow rate$=Av$

You can prove this as follows. Consider a uniform diameter tube of area of cross section $A$. Next, choose a cross section anywhere in the tube. Now, let us consider a time $\Delta t$, and during which, the liquid travels from the chosen cross section to a distance $\Delta l$. During the time the volume of water leaves the cross section is $A\Delta l$. Therefore, the volume flow rate is $A\Delta l/\Delta t$. Since, $\Delta l/\Delta t$ is the speed of the fluid, we get the volume flow rate as $Av$.

When a incompressible fluid flow through a tube of varying cross section (diameter), its volume flow rate is constant.
i.e., if an incompressible fluid flows through a tube that contains two sections with different area of cross sections, then

$A_1v_1=A_2v_2$

This is called equation of continuity.

Bernoulli's principle

It states that where the velocity of a fluid is high its pressure is low and where the velocity is low, the fluid pressure is high.
Bernoulli equation
Bernoulli showed that for a given fluid, $P+\dfrac{1}{2}\rho v^2+\rho g y$ is a constant.

i.e., $P_1 + \dfrac{1}{2} \rho v_1^2 + \rho\, g\, y_1$ $=P_2+\dfrac{1}{2}\rho v_2^2+\rho\, g\, y_2$.

This equation is called Bernoulli's equation.