Fluid Mechanics

It is an important area of physics that explores how liquids and gases behave, both when they're still and when they're moving. It includes key ideas like pressure, buoyancy, viscosity, surface tension, and laws such as Pascal’s and Bernoulli’s. It also looks at different types of flow, like smooth (laminar) and chaotic (turbulent). These concepts help us understand everything from how airplanes fly and pumps work to how water flows in rivers and blood moves through our bodies.

1.0Definition of Fluid

Fluid Mechanics studies the behavior of fluids—liquids and gases—at rest and in motion. A fluid is a substance that continuously deforms under even a small shear stress and cannot resist shear force when at rest.

2.0Density of Liquid

Density ($\rho$)of any substance is defined as the mass per unit volume

$Density=\frac{Mass}{Volume}\Rightarrow \rho =\frac{m}{v}$

Relative Density 

• Relative Density (RD), also known as Specific Gravity, is the ratio of the density of a substance to the density of water at 4°C.
• It is a dimensionless quantity, meaning it has no units, since it's a pure ratio.
• In the CGS system, the density of water at 4°C is 1g/cm³, so the numerical value of RD equals the density in g/cm³.
• In the SI system, the density of water at 4°C is 1000 kg/m³.
• RD helps compare how heavy a substance is relative to water, without needing unit conversion.

3.0Pressure in a Fluid

• Pressure in a Fluid is the force exerted perpendicular to any surface in contact with a fluid at rest (liquid or gas).
• Although the fluid is at rest, its molecules are in constant motion, colliding with surfaces and creating pressure.
• At any point inside the fluid, the forces on an imaginary surface are equal and opposite to maintain equilibrium.
• Pressure P is defined as the normal force dF丄​ per unit area dA: $P=\frac{d{F}_{\perp }}{dA}$
• If pressure is uniform over a finite surface area A, then, $P=\frac{d{F}_{\perp }}{dA}$
• The SI unit of pressure is the Pascal (Pa), where:1 Pa=1N/m2
• Another common unit in meteorology is the Bar, with:1 Bar=105 Pa

Note: Fluid pressure acts perpendicular to any surface regardless of its orientation. Since pressure has no specific direction, it is a scalar quantity, unlike force, which is a vector with direction.

Atmospheric Pressure ($P_0$): The pressure exerted by Earth’s atmosphere, varying with weather and altitude. Standard atmospheric pressure at sea level is 1.013 × 10⁵ Pa.

Absolute pressure and Gauge Pressure: Pressure above atmospheric pressure is called gauge pressure, while total pressure is absolute pressure.

Gauge pressure = Absolute Pressure – Atmospheric Pressure

Note: Absolute pressure is always greater than or equal to zero. While gauge pressure can be negative also.

4.0Variation of Pressure

1. Variation of Pressure at two points in horizontal plane

$P_A=P_B$

The pressure is the same at two points in the same horizontal level.

2. Variation of pressure at two points in different height depth

If two points in a fluid differ in depth by h , the pressure difference between them is given by:

$P_A=P_B+\rho gh$

3. Variation of pressure at two points when fluid is in horizontal acceleration

$P_A-P_B=l\rho a$

$\frac{h_A-h_B}{l}=\frac{a}{g}$

$tan\theta =\frac{a}{g}$

In a horizontally accelerating fluid, the free surface tilts at an angle θ such that  $Tan\theta =\frac{a}{g}$​, and pressure varies along the direction of acceleration.

4. Variation of pressure at two points when fluid is in vertical acceleration

In a fluid accelerating vertically upward, the pressure difference between two points separated by height h is:

$(P_A-P_B)=\rho (g+a)h$

Special cases :

(1) If a is (–) ve i.e. the vessel is accelerating downward then, $(P_A-P_B)=\rho (g-a)h$

(2) If a is greater than g then fluid occupies the upper part of the container.

5. Variation of pressure when fluid is under both horizontal and vertical acceleration

$a_x=acos\varphi ,a_y=asin\varphi$

• Horizontal Pressure Difference (between two points A and B, separated by distance l ) $\Delta P=l\rho a_x=l\rho acos\varphi$
• Vertical Pressure Difference (between two points separated by height h )$\Delta P=\rho \left(g+a_y\right)h=\rho \left(g+a\sin \varphi \right)h$
• Free Surface Inclination: The angle θ between the fluid's free surface and the horizontal is given by$\tan \theta =\frac{a\cos \varphi }{g+a\sin \varphi }$

6. Variation of pressure in a rotating fluid

• When a liquid rotates with angular velocity ω, the free surface forms a paraboloid of revolution described by $y=\frac{{\omega }^2{x}^2}{2g}$
• The slope of the surface is, $\frac{dy}{dx}=tan\theta =\frac{{\omega }^{2}x}{g}$
• The maximum height at the container’s edge $h_{max}$ relates to minimum height at the center $h_{min}$​ as: $h_{max}=h_{min}+\frac{{\omega }^2{R}^2}{2g}$
• Volume conservation leads to: $h_0=h_{max}-\frac{{\omega }^2{R}^2}{4g}$
• Thus, the rise in liquid level at the periphery equals the fall at the center.
• Pressure variation at any point inside the liquid.

$P_B=P_0+\frac{\rho {\omega }^2{x}^2}{2}$

5.0Pascal's Law

Pressure applied to a confined fluid is transmitted equally throughout the fluid and to the walls of its container.

Hydraulic Lift 

• A small piston with area $A_1$ applies force $F_1$​ on a confined fluid, creating pressure: $P=\frac{F_1}{A_1}$
• This pressure is transmitted equally to a larger piston with area $A_2$
• Since pressure is the same in both pistons, $\frac{F_1}{A_1}=\frac{F_2}{A_2}\Rightarrow F2=F_1\frac{A_2}{A_1}$
• Because $A_2>A_1$​​, the force $F_2$ on the larger piston is greater than $F_1$
• Hydraulic lifts multiply force by the ratio of piston areas.
• Applications include dentist chairs, car lifts, jacks, elevators, and hydraulic brakes.

Important points in Pressure

1. At the same point in a fluid, pressure is the same in all directions. In the figure,

$P_1=P_2=P_3=P_4$

2. Forces acting on a fluid in equilibrium have to be perpendicular to its surface. Because it cannot sustain the shear stress.
3. In the same liquid pressure will be the same at all points at the same level. For example, in the figure:

${\rho }_1{h}_1={\rho }_2{h}_2$

$h\propto \frac{1}{\rho }$

4. Torricelli Experiment (Barometer) :It is a device used to measure atmospheric pressure. In principle any liquid can be used to fill the barometer, but mercury is the substance of choice because its great density makes possible an instrument of reasonable size.    

$P_0=\rho gh=(13.6\times 10^3)(9.8)(0.760)=1.01\times 10^{5}N/m^2$

Mercury barometer reads the atmospheric pressure P0 directly from the height of the mercury column.

5. Manometer :It is a device used to measure the pressure of a gas inside a container. The U-shaped tube often contains mercury.

$P_1=P_2$

$P_1$ = pressure of the gas in the container (P)

$P_1$ = atmospheric pressure $(P_0)+\rho gh$

$P_0=P_0+\rho gh$

$P-P_0=\rho gh\Rightarrow \text{Gauge Pressure}$

$\rho$ is the density of the liquid used in the U-tube.  

6.0Archimedes Principle

• When an object is immersed in a fluid, it appears to weigh less due to an upward buoyant force.
• This force equals the weight of the fluid displaced by the object.
• This phenomenon is known as Archimedes’ Principle.

Magnitude of Buoyant Force $(F)=v_i{\rho }_{L}g$

$v_i$= immersed volume of solid, L=density of liquid, g=acceleration due to gravity

7.0Law of Floatation

• An object of volume V and density s​ floats in a liquid of density L with immersed volume Vi
• At equilibrium, $\text{Weight}=\text{Buoyant Force}\Rightarrow V{\rho }_{s}g=V_i{\rho }_{L}g\Rightarrow \frac{V_i}{V}=\frac{{\rho }_s}{{\rho }_{}L}$
• Percentage of Volume Immersed: $\frac{V_i}{V}\times 100=\frac{{\rho }_s}{{\rho }_L}\times 100$

Three Cases:

1. ​${\rho }_s<{\rho }_L$ The object floats partially; only a fraction is submerged.
2. ${\rho }_s={\rho }_L$ The object is fully submerged but floats at any depth.
3. ${\rho }_s>{\rho }_L$ The object is denser than the liquid and sinks.

Apparent Weight in a Liquid

• When a body is fully immersed in a liquid, it experiences loss in weight due to upthrust (buoyant force)
• Apparent weight: $W_{app}=W_{actual}-Upthrust=Vg({\rho }_s-{\rho }_L)$

If the liquid is water:

• Relative Density (R.D) of the body: $\text{R.D}=\frac{Weightinair}{Lossinweightinwater}=\frac{{\rho }_s}{{\rho }_w}$

Buoyant Force in an Accelerating Fluid:

• When a body is immersed in a fluid inside an accelerating lift, buoyant force changes based on effective gravity: $F=V{\rho }_L{g}_{eff}$, where $g_{eff}=g-a$

Cases:

1. Lift accelerating upward: $g_{eff}=g+a\Rightarrow \text{Buoyant force increases}$
2. Lift accelerating downward $g_{eff}=g-a\Rightarrow \text{Buoyant force decreases}$
3. Free fall $(a=g):g_{eff}=0\Rightarrow \text{No buoyant force}\to$ Objects appear weightless, and bubbles do not rise.

8.0Fluid Flow

Conditions of Ideal Fluid Flow

1. Fluid is incompressible: density remains constant over time and position.
2. Fluid is non-viscous: no dissipative forces between fluid layers.
3. Flow is irrotational: fluid particles have zero angular velocity relative to each other.
4. Flow is steady (streamlined): flow properties do not change with time.

Steady (Streamline) Flow:

• Velocity and density at any point remain constant with time.
• Velocity and density may vary with position but not with time.

Streamlines:

• Curves tangent to the fluid velocity direction at every point.
• Streamline density is proportional to velocity magnitude.
• Streamlines never cross (only one velocity direction at a point).

Flow Rates:

• Mass flow rate $\frac{dm}{dt}=\rho AV$(mass per unit time).
• Volume flow rate $\frac{dV}{dt}=AV$ (volume per unit time).

9.0Equation of Continuity

Continuity Equation:

• This Represents the law of conservation of mass in fluid flow.
• In steady flow, mass entering a tube per unit time = mass leaving it.

${\rho }_1{A}_1{v}_1={\rho }_2{A}_2{v}_2(\because {\rho }_1={\rho }_2)$

$A_1{v}_1=A_2{v}_2\Rightarrow Av=Constant$

The velocity of liquid is smaller in the wider parts of a tube and larger in the narrower parts.

10.0Bernoulli's Theorem

1. In a steady, incompressible, and non-viscous flow of an ideal fluid, the total mechanical energy (sum of pressure energy, kinetic energy, and potential energy) per unit volume remains constant along a streamline.
2. Expresses the conservation of mechanical energy in fluid flow.
3. Applies to ideal fluids with these properties:

• Incompressible
• Non-viscous
• Steady flow
• Irrotational flow

Energy at any point in ideal flow includes:

• Pressure energy
• Kinetic energy
• Potential energy

$P_1+\rho g{h}_1+\frac{1}{2}\rho v_1^2=P_2+\rho g{h}_2+\frac{1}{2}\rho v_2^2\Rightarrow P+\rho gh+\frac{1}{2}\rho v^2=\text{constant}$

11.0Applications of Bernoulli's Equation

1. Venturimeter: A device used to measure the flow rate of a fluid through a pipe

The discharge or volume flow rate can be obtained as,

$\frac{dV}{dt}=A_1{v}_1=A_1\sqrt{\frac{2gh}{{\left(\frac{A_1}{A_2}\right)}^2-1}}$

2. Speed of Efflux: Refers to the speed at which a fluid exits an orifice under pressure.

$\rho gh+P_0=\frac{1}{2}\rho v^2+P_0$

$v=\sqrt{2gh}$

Torricelli’s Theorem: The speed of a liquid flowing out of an orifice is equal to the speed it would gain if it fell freely from the liquid surface to the orifice.

Range 

Vertical motion:

Time to fall from height $(H-h),t=\sqrt{\frac{2(H-h)}{g}}$

Horizontal motion:

Velocity of efflux: $v=\sqrt{2gh}$

So Range, $R=v\cdot t=\sqrt{2gh}\sqrt{\frac{2(H-h)}{g}}=\sqrt{2h(H-h)}$

Key Conclusions:

1. Symmetry $R_h=R_{H-h}(RangeissameforheighthandH-h)$
2. Maximum Range $\frac{dR}{dh}=0\Rightarrow H-2h=0\Rightarrow h=\frac{H}{2}$, Range is maximum at $h=\frac{H}{2}$
3. Maximum Range Value: $R_{\text{max}}=2\sqrt{\frac{H}{2}\cdot \frac{H}{2}}$

Time taken to empty a tank, $t=\frac{A}{a}\sqrt{\frac{2H}{g}}$

12.0Surface Energy

Molecular Theory of Surface Tension

• Molecules on the surface have extra energy due to unbalanced forces.
• This energy per unit surface area is called surface energy.

Surface Tension and Work Done

• A liquid film is formed on a wire frame with a movable wire of length l
• Each surface pulls the wire with force $Tl\to \text{total force from both surfaces}=2Tl$
• To keep the wire in equilibrium, apply external force: F=2Tl
• If the wire moves a small distance dx

$\text{Work done}dW=F.dx=2Tl.dx=T(2l.dx)=T.dA$

Increase in surface area: dA=2l.dx

$T=\frac{dW}{dA}$

Surface tension T = work done per unit increase in surface area.

This work is stored as potential energy of the surface.

Splitting of a Bigger Drop into Smaller Droplets

1. Volume Conservation

• Let a large drop of radius R  split into n smaller drops of radius r
• Volume remains constant: $\frac{4}{3}\pi R^3=n\cdot \frac{4}{3}\pi r^3\Rightarrow R^3=n{r}^3\Rightarrow r=\frac{R}{n^{1/3}}$

2. Surface Area and Energy

• Initial surface area: $A_i=4\pi R^2$
• Final surface area: $A_f=n\cdot 4\pi r^2$
• Change in area: $\Delta A=4\pi (n{r}^2-R^2)$
• Initial surface energy: $E_i=4\pi R^{2}T$
• Final surface energy: $E_f=4\pi Tn{r}^2$
• Work done (W) = Increase in surface energy: $W=\Delta E=4\pi T(n{r}^2-R^2)$

3. Alternate Form of Work Done

Substitute $r=\frac{R}{n^{1/3}}$

$W=4\pi R^{2}T\left(n^{1/3}-1\right)$

$W=4\pi R^{3}T\left(\frac{1}{r}-\frac{1}{R}\right)$

4. Thermal Effect

• Energy used to increase surface area → internal energy decreases.
• Temperature drops due to energy absorption.
• Using heat relation: $W=Jms\Delta \theta$
• Final expression for temperature drop: $\Delta \theta =\frac{3T}{J\rho p}\left(\frac{1}{r}-\frac{1}{R}\right)$

13.0Excess Pressure

Case	Number of Surfaces	Excess Pressure
Soap bubble (in air)	2	$\frac{4T}{r}$
Liquid drop	1	$\frac{2T}{r}$
Air bubble in liquid $(P_1>P_2)$	1	$\frac{2T}{r}$

14.0Contact Angle

Contact Angle and Surface Curvature

• The surface of a liquid near its contact with another medium is usually curved.
• The contact angle ${\theta }_C$ is the angle between the tangent to the liquid surface at the point of contact and the solid surface, measured inside the liquid.

Types of Molecular Forces

• Cohesive force: Between molecules of the same substance (e.g., water-water).
• Adhesive force: Between molecules of different substances (e.g., water-glass).

15.0Capillary Rise

• When a capillary tube is dipped in a liquid, the liquid rises due to surface tension.
• The liquid forms a curved meniscus and rises until the upward surface tension force balances the weight of the liquid column.

Force Balance:

• Upward force due to surface tension:$F=2\pi rTcos\theta$
• Downward force (weight of liquid column):$W=\pi r^{2}h\rho g$
• At equilibrium: $2\pi rTcos\theta =\pi r^{2}h\rho g\Rightarrow h=\frac{2Tcos\theta }{r\rho g}$
• Capillary Rise Formula: $h=\frac{2Tcos\theta }{r\rho g}$

16.0Viscosity

• Viscosity is a fluid's property that opposes relative motion between its adjacent layers.
• It is also called internal friction or fluid friction.
• The tangential force resisting layer motion is called the viscous force.

Assumptions for Laminar Flow:

1. No-slip condition: Fluid in contact with a surface moves with the same velocity as the surface.
2. Velocity gradient:

• Velocity increases uniformly from the bottom (stationary) layer to the top (moving) layer.
• Each layer experiences a pull forward from the layer above and a drag backward from the one below.
• This internal force results in laminar (smooth) flow.

Newton's Law of Viscosity

Coefficient of viscosity : $\eta =\frac{ShearStress}{RateofchangeofShearStrain}\Rightarrow \eta =\frac{F/A}{v/l}$

Newtonian  Fluids: $F_{vis}=\eta \frac{Av}{l}$

• SI Units: $\frac{N-s}{m^2}$ or deca poise
• CGS Units: $dyne-s/c{m}^2$ or poise (1 decapoise=10 poise)
• Dimension: $[M^1{L}^{-1}{T}^{-1}]$

Newtonian fluids	Non-Newtonian fluids
$\frac{du}{dy}=\frac{v}{l}=Constant$ $F_{vis}=\eta \frac{Av}{l}$	$\frac{du}{dy}$ is not constant  $F_{vis}=\eta \frac{Adu}{dy}$ $\frac{du}{dy}\to \text{Velocity Gradient}$

17.0Stoke's Law

A sphere of radius r, moving with velocity V relative to a fluid of viscosity η experiences a viscous drag, $F=-6\pi \eta rV(F_{vis}isoppositetoV)$

18.0Terminal Velocity

Terminal velocity is the constant maximum velocity attained by an object falling through a fluid (like air or water), when the net force becomes zero.

$v_0=\frac{2}{9}\frac{r^{2}g}{\eta }({\rho }_L-{\rho }_s)$

19.0Reynolds Number

• Reynolds number $(R_e)$ is a dimensionless quantity given by: $R_e=\frac{\rho vd}{\eta }$
• Critical speed: The velocity at which flow transitions from laminar to turbulent.

20.0Poiseuille’s Formula

Poiseuille studied fluid flow through narrow tubes (capillaries) and found.

Volume flow rate V is:

1. Directly proportional to pressure difference P
2. Directly proportional to $r^4$ (fourth power of tube radius)
3. Inversely proportional to viscosity η
4. Inversely proportional to tube length l

Poiseuille’s Equation: V = Volume of liquid per second = $V=\frac{\pi P{r}^4}{8\eta L}$

Liquid Resistance (R):  $R=\frac{8\eta L}{\pi r^4}$ so $V=\frac{P}{R}$

21.0Combination of Tubes

Aspect	Series Combination	Parallel Combination
Pressure Difference	$P=P_1+P_2$	$P_1=P_2=P$
Flow Rate	$V_1=V_2=V$  (same through all tubes)	$V=V_1+V_2$
Flow Equation	$V=\frac{P}{R_1+R_2}$	$V=P\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$
Effective Resistance	$R_{\text{eff}}=R_1+R_2$	$\frac{1}{R_{\text{eff}}}=\frac{1}{R_1}+\frac{1}{R_2}$
Resistance Formula (each tube)	$R=\frac{8\eta L}{\pi r^4}$	$R=\frac{8\eta L}{\pi r^4}$
Use When	Tubes are connected end-to-end	Tubes are connected side-by-side
Connection