Fluid Mechanics

Fluids

Fluids are those substances which can flow when an external force is applied on it.

Liquids and gases are fluids.

Fluids do not have finite shape but takes the shape of the containing vessel,

The total normal force exerted by liquid at rest on a given surface is called thrust of liquid.

The SI unit of thrust is newton.

In fluid mechanics the following properties of fluid would be considered

(i) When the fluid is at rest – hydrostatics

(ii) When the fluid is in motion – Fluid Mechanics

Pressure Exerted by the Liquid

The normal force exerted by a liquid per unit area of the surface in contact is called pressure of liquid or hydrostatic pressure.

Pressure exerted by a liquid column

p = hρg

Where, h = height of liquid column, ρ = density of liquid

and g = acceleration due to gravity

Mean pressure on the walls of a vessel containing liquid upto height h is (hρg / 2).

Pascal’s Law

The increase in pressure at a point in the enclosed liquid in equilibrium is transmitted equally in all directions in liquid and to the Walls of the container.

The working of hydraulic lift, hydraulic press and hydraulic brakes are based on Pascal’s law.

Atmospheric Pressure

The pressure exerted by the atmosphere on earth is atmospheric pressure.

It is about 100000 N/m².

It is equivalent to a weight of 10 tones on 1 m².

At sea level, atmospheric pressure is equal to 76 cm of mercury column. Then, atmospheric pressure

= hdg = 76 x 13.6 x 980 dyne/cm²

[The atmospheric pressure does not crush our body because the pressure of the blood flowing through our circulatory system] balanced this pressure.]

Atmospheric pressure is also measured in torr and bar.

1 torr = 1 mm of mercury column

1 bar = l0⁵ Pa

Aneroid barometer is used to measure atmospheric pressure.

Buoyancy

When a body is partially or fully immersed in a fluid an upward force acts on it, which is called buoyant force or simply buoyancy.

The buoyant force acts at the centre of gravity of the liquid displaced] by the immersed part of the body and this point is called the centre buoyancy.

Archimedes’ Principle

When a body is partially or fully immersed in a liquid, it loses some of its weight. and it is equal to the weight of the liquid displaced by the immersed part of the body.

If T is the observed weight of a body of density σ when it is fully immersed in a liquid of density p, then real weight of the body

w = T / ( 1 – p / σ)

Laws of Floatation

A body will float in a liquid, if the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body.

If W is the weight of the body and w is the buoyant force, then

(a) If W > w, then body will sink to the bottom of the liquid.

(b) IfW < w, then body will float partially submerged in the liquid.
(c) If W = w, then body will float in liquid if its whole volume is just immersed in the liquid,

The floating body will be in stable equilibrium if meta-centre (centre of buoyancy) lies vertically above the centre of gravity of the body.

The floating body will be in unstable equilibrium if meta-centre (centre of buoyancy) lies vertically below the centre of gravity of the body.

The floating body will be in neutral equilibrium if meta-centre (centre of buoyancy) coincides with the centre of gravity of the body.

Density and Relative Density

Density of a substance is defined as the ratio of its mass to its volume.

Density of a liquid = Mass / Volume

Density of water = 1 g/cm³ or l0³ kg/m³

It is scalar quantity and its dimensional formula is [ML^-3].

Relative density of a substance is defined as the ratio of its density to the density of water at 4°C,

Relative density = Density of substance / Density of water at 4°C

= Weight of substance in air / Loss of weight in water

Relative density also known as specific gravity has no unit, no dimensions.

For a solid body, density of body = density of substance

While for a hollow body, density of body is lesser than that of Substance.

When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while, that of lowest density at the top and interfaces will be plane.

Density of a Mixture of Substances

When two liquids of mass m₁ and m₂ having density p₁ and p₂ are mixed together then density of mixture is

p = m₁ + m₂ / (m₁ /p₁ ) + (m₂ + p₂)

= p₁p₂ (m₁ + m₂) / (m₁p₂ + m₂p₁)

When two liquids of same mass m but of different densities p₁ and p₂ are mixed together then density of mixture is

p = 2p₁p₂ / p₁ + p₂

When two liquids of same volume V but of different densities p₁ and p₂ are mixed together then density of mixture is

p = p₁ + p₂ / 2

Density of a liquid varies with pressure

p = p_o [ 1 + Δp / K]

where, p_o = initial density of the liquid, K = bulk modulus of elasticity of the liquid and Δp = change in pressure.

Viscosity

The property of a fluid by virtue of which an internal frictional force acts between its different layers which opposes their relative motion is called viscosity.

These internal frictional force is called viscous force.

Viscous forces are intermolecular forces acting between the molecules of different layers of liquid moving with different velocities.


where, (dv/dx) = rate of change of velocity with distance called velocity gradient, A = area of cross-section and = coefficient of viscosity.

SI unit of η is Nsm^-2 or pascal-second or decapoise. Its dimensional formula is [ML^-1T^-1].

The knowledge of the coefficient of viscosity of different oils and its variation with temperature helps us to select a suitable lubricant for a given machine.

Viscosity is due to transport of momentum. The value of viscosity (and compressibility) for ideal liquid is zero.

The viscosity of air and of some liquids is utilised for damping the n.ving parts of some instruments.

The knowledge of viscosity of some organic liquids is used in determining the molecular weight and shape of large organic moleculars like proteins and cellulose.

Variation of Viscosity

The viscosity of liquids decreases with increase in temperature


where, η₀ and η_t is are coefficient of viscosities at 0°C t°C, α and β are constants.

The viscosity of gases increases with increase in temperatures as

η ∝ √T

The viscosity of liquids increases with increase in pressure but the viscosity of water decreases with increase in pressure.

The viscosity of gases do not changes with pressure.

Poiseuille’s Formula

The rate of flow (v) of liquid through a horizontal pipe for steady flow is given by


where, p = pressure difference across the two ends of the tube. r = radius of the tube, n = coefficient of viscosity and 1 = length of th tube.

The Rate of Flow of Liquid

Rate of flow of liquid through a tube is given by

v = (P/R)

where, R = (8 ηl/πr⁴), called liquid resistance and p = liquid pressure.

(i) When two tubes are connected in series

• Resultant pressure difference p = p₁ + p₂
• Rate of flow of liquid (v) is same through both tubes.
• Equivalent liquid resistance, R = R₁ + R₂

(ii) When two tubes are connected in parallel

1. Pressure difference (p) is same across both tubes.
2. Rate of flow of liquid v = v₁ + v₂
3. Equivalent liquid resistance (1/R) = (1/R₁) + (1/R₂)

Stoke’s Law

When a small spherical body falls in a long liquid column, then after sometime it falls with a constant velocity, called terminal velocity. When a small spherical body falls in a liquid column with terminal velocity then viscous force acting on it is

F = 6πηrv

where, r = radius of the body, V = terminal velocity and η = coefficient of viscosity.

This is called Stoke’s law.


where,

• ρ = density of body,
• σ = density of liquid,
• η = coefficient of viscosity of liquid and,
• g = acceleration due to gravity

1. If ρ > ρ₀, the body falls downwards.
2. If ρ < ρ₀, the body moves upwards with the constant velocity.
3. If po << ρ, v = (2r²ρg/9η)

Importance of Stoke’s Law

1. This law is used in the determination of electronic charge by Millikan in his oil drop experiment.
2. This law helps a man coming down with the help of parachute.
3. This law account for the formation of clouds.

Flow of Liquid

1. Streamline Flow The flow of liquid in which each of its particle follows the same path as followed by the proceeding particles, is called streamline flow.
2. Laminar Flow The steady flow of liquid over a horizontal surface in the form of layers of different velocities, is called laminar flow.
3. Turbulent Flow The flow of liquid with a velocity greater than its critical velocity is disordered and called turbulent flow.

Critical Velocity

The critical velocity is that velocity of liquid flow, below which its fl is streamlined and above which it becomes turbulent.

Critical velocity v_c = (k_η/rρ)

where,

• K = Reynold’s number,
• η = coefficient of viscosity of liquid
• r = radius of capillary tube and ρ = density of the liquid.

Reynold’s Number


Reynold’s number is a pure number and it is equal to the ratio of inertial force per unit area to the viscous force per unit area for flowing fluid.

where, p = density of the liquid and v_c = critical velocity.

For pure water flowing in a cylindrical pipe, K is about 1000.

When 0< K< 2000, the flow of liquid is streamlined.

When 2000 < K < 3000, the flow of liquid is variable betw streamlined and turbulent.

When K > 3000, the flow of liquid is turbulent.

It has no unit and dimension.

Equation of Continuity

If a liquid is flowing in streamline flow in a pipe of non-unif cross-section area, then rate of flow of liquid across any cross-sec remains constant.

a₁v₁ = a₂v₂ av = constant

The velocity of liquid is slower where area of cross-section is larger faster where area of cross-section is smaller.


The falling stream of water becomes narrower, as the velocity of f stream of water increases and therefore its area of cross-s decreases.

Energy of a Liquid

A liquid in motion possess three types of energy

(i) Pressure Energy Pressure energy per unit mass = p/ρ

where,

p= pressure of the liquid and p = density of the liquid.
Pressure energy per unit volume = p

(ii) Kinetic Energy

• Kinetic energy per unit mass = (1/2v²)
• Kinetic energy per unit volume = 1/2ρv²

(iii) Potential Energy

• Potential energy per unit mass = gh
• Potential energy per unit volume = ρgh

Bernoulli’s Theorem

If an ideal liquid is flowing in streamlined flow then total energy, i.e., sum of pressure energy, kinetic energy and potential energy per unit
volume of the liquid remains constant at every cross-section of the tube.


Mathematically

It can be expressed as


where, (p/ρg) = pressure head, (v²/2g) = velocity head and h = gravitational head.

For horizontal flow of liquid,

• where, pis called static pressure and (1/2 ρv²) is called dynamic pressure.
• The erefore in horizontal flow of liquid, if p increases, v decreases and Nce-versa.
• The theorem is applicable to ideal liquid, i.e., a liquid which is non-viscous incompressible and irrotational.

Applications of Bernoulli’s Theorem

1. The action of carburetor, paintgun, scent sprayer atomiser insect sprayer is based on Bernoulli’s theorem.
2. The action of Bunsen’s burner, gas burner, oil stove exhaust pump is also based on Bernoulli’s theorem.
3. Motion of a spinning ball (Magnus effect) is based on Bernoulli theorem.
4. Blowing of roofs by wind storms, attraction between two close parallel moving boats, fluttering of a flag etc are also based Bernoulli’s theorem.

Venturimeter

It is a device used for measuring the rate of flow of liquid t pipes. Its working is based on Bernoulli’s theorem.


Rate of flow of liquid,


where, a₁ and a₂ are area of cross-sections of tube at bra and narrower part and h is difference of liquid columns in ver tubes.

Torricelli’s Theorem

Velocity of efflux (the velocity with which the liquid flows out orifice or narrow hole) is equal to the velocity acquired by a falling body through the same vertical distance equal to the dep orifice below the free surface of liquid.


Velocity of efflux, v = √2gh

where, h = depth of orifice below the free surface of liquid.

Horizontal range, S = √4h(H — h)

where, H = height of liquid column.

Horizontal range is maximum, equal to height of the liquid column H, when orifice is at half of the height of liquid column.

Important Points

• In a pipe the inner layer .central layer) have maximum velocity and the layer in contact with pipe have least velocity.
• Velocity profile of liquid flow in a pipe is parabolic.
• Solid friction is independent of area of surfaces in contact while viscous force depends on area of liquid layers.
• A lubricant is chosen ac:ording to the nature of machinary. In heavy machines lubricating oils of high viscosity are used and in light machines low viscosity oils are used.
• The cause of viscosity 1 liquids is the cohesive forces among their molecules while cause of viscosity in gases is diffusion.

Surface tension is the property of any liquid by virtue of which tries to minimize its free surface area.

Surface tension of a liquid is measured as the force acting per length on an imaginary line drawn tangentially on the free surface the liquid.

Surface tension S = Force/Length = F/l = Work done/Change in area

Its SI unit is Nm^-1 or Jm^-2 and its dimensional formula is [MT^-2].

It is a scalar quantity. Surface tension is a molecular phenomenon which is due to cohesive force and root cause of the force is electrical in nature.

Surface tension of a liquid depends only on the nature of liquid and independent of the surface area of film or length of the line .

Small liquid drops are spherical due to the property of surface tension.

Adhesive Force

The force of attraction acting between the molecules of different substances is called adhesive force, e.g., the force of attracts acting between the molecules of paper and ink, water and glass, etc.

Cohesive Force

The force of attraction acting between the molecules of same substan is called cohesive force. e.g., the force of attraction acting between molecules of water, glass, etc.

Cohesive forces and adhesive forces are van der Waals’ forces.

These forces varies inversely as the seventh power of distance between the molecules.

Molecular Range

The maximum distance upto which a molecule can exert a force of attraction on other molecules is called molecular range.

Molecular range is different for different substances. In solids and liquids it is of the order of 10^-9 m.

If the distance between the molecules is greater than 10-9 m, the force of attraction between them is negligible.

Surface Energy

If we increase the free surface area of a liquid then work has to be done against the force of surface tension. This work done is stored in liquid stu-face as potential energy,

This additional potential energy per unit area of free surface of liquid is called surface energy.
Surface energy (E) = S x &ΔM
where. S = surface tension and ΔA = increase in surface area.

(i) Work Done in Blowing a Liquid Drop If a liquid drop is blown up from a radius r₁ to r₂ then work done for that is
W = S . 4π (r₂² – r₁²)

(ii) Work Done in Blowing a Soap Bubble As a soap bubble has two free surfaces, hence work done in blowing a soap bubble so as to increase its radius from r¹ to r² is given by
W = S.8π(r₂² – r₁²)

(iii) Work Done in Splitting a Bigger Drop into n Smaller Droplets
If a liquid drop of radius R is split up into n smaller droplets, all of same size. then radius of each droplet

r = R. (n)^-1/3
Work done, W = 4π(nr² – R²)
= 4πSR² (n^1/3 – 1)

(iv) Coalescance of Drops If n small liquid drops of radius reach combine together so as to form a single bigger drop of radius R=n^1/3.r, then in the process energy is released. Release of energy is given by

ΔU = S.4π(nr² – R²)

= 4πSπn(1 – n^1/3)

Angle of Contact


The angle subtended between the tangents drawn at liquid surface at solid surface inside the liquid at the point of contact, is called of contact (9).

Angle of contact depends upon the nature of the liquid and solid contact and the medium which exists above the free surface of liquid.

When wax is coated on a glass capillary tube, it becomes water-proof.
The angle of contact increases and becomes obtuse. Water does not in it. Rather it falls in the tube by virtue of obtuse angle of contact.

If θ is acute angle, i.e; θ <90°, then liquid meniscus will be concave upwards.

• If θ is 90°, then liquid meniscus will be plane.
• If θ is obtuse, i.e; θ >90°, then liquid meniscus will be convex upwards.
• If angle of contact is acute angle, i.e; θ <90°, then liquid will wet the surface.
• If angle of contact is obtuse angle, ie; θ > 90°, then liquid will not wet the surface.

Angle of contact increases with increase in temperature of Angle of contact decreases on adding soluble impurity to a liquid.

Angle of contact for pure water and glass is zero. For ordinary water and glass is 8°. For mercury and glass is 140°. For pure water silver is 90°. For alcohol and clean glass θ = 0°.

Angle of contact, meniscus, shape of liquid surface



Capillarity

The phenomenon called capillarity. of rise or fall of liquid column in a capillary tube is Ascent of a liquid column in a capillary tube is given by

h = (2S cos θ / rρg) – (r / 3)

If capillary is very narrow, then

h=2S cos θ / rρg

where, r = radius of capillary tube, p = density of the liquid, and
θ = angle of contact and S = surface tension of liquid.

• If θ < 90°, cos e is positive, so h is positive, i.e., liquid rises in a capillary tube.
• If θ > 90°, cos 9 is negative, so h is negative, i.e., liquid falls in a capillary tube.
• Rise of liquid in a capillary tube does not violate law of conservation of energy.

Some Practical Examples of Capillarity

1. The kerosene oil in a lantern and the melted wax in a candle, rise in the capillaries formed in the cotton wick and burns.
2. Coffee powder is easily soluble in water because water immediately wets the fine granules of coffee by the action of capillarity.
3. The water given to the fields rises in the innumerable capillaries formed in the stems of plants and trees and reaches the leaves.

Zurin’s Law

If a capillary tube of insufficient length is placed vertically in a then liquid never come out from the tube its own, as

Rh = constant ⇒ R₁h₁ = R₂h₂

where, R = radius of curvature of liquid meniscus and
h = height of liquid column.

When a tube is kept in inclined position in a liquid the vertical height remains unchanged then length of liquid column.


Liquid rises (water in glass capillary) or falls (mercury in capillary) due to property of surface tension

T = Rρgh / 2 cos θ

where, R = radius of capillary tube, h = height of liquid, p = density of liquid, e = angle of contact,

T = surface tension of liquid and 9 = acceleration due to gravity.

Excess Pressure due to Surface Tension

(i) Excess pressure inside a liquid drop = 2S / R

(ii) Excess pressure inside an air bubble in a liquid = 2S / R

(iii) Excess pressure inside a soap bubble = 4S / R

where, S = surface tension and R = radius of drop/bubble.

(iv) Work done in spraying a liquid drop of radius R into n droplets of radius r = T x increase in surface area

= 4πTR³ (1/r – 1/R)

Fall in temperature

Δθ = 3T/J (1/r – 1/R)

where. J = 4.2 J/cal.

(v) When n small drops are combined into a bigger drop, then work done is given by

W = 4πR²T (n ^1/3 – 1)

Temperature increase

Δθ = 3T/J (1/r – 1/R)

(vi) When two bubbles of radii r₁ and r₂ coalesce into a bubble of radius r isothermally, then
r² = r₁2 + r₂²

(vii) When two soap bubbles of radii ‘1 and ‘2 are in contact with each other, then radius (r) of common interface.


Factors Affecting Surface Tension

1. Surface tension of a liquid decreases with increase temperature and becomes zero at critical temperature.
2. At boiling point, surface tension of a liquid becomes zero becomes maximum at freezing point.
3. Surface tension decreases when partially soluble impurities such as soap, detergent, dettol, phenol, etc are added in water.
4. Surface tension increases when highly soluble impurities such as salt is added in water.
5. When dust particles or oil spreads over the surface of water, its surface tension decreases.

When charge is given to a soap bubble, its size increases surface tension of the liquid decreases due to electrification.

In weightlessness condition liquid does not rise in a capillary tube.

Some Phenomena Based on Surface Tension

1. Medicines used for washing wounds, as detol, have a surface tension lower than water.
2. Hot soup is more tasteful than the cold one because the surface tension of the hot soup is less than that of the cold and so spreads over a larger area of the tongue.
3. Insects and mosquitoes swim on the surface of water in ponds and lakes due to surface tension. If kerosence oil is sprayed on the water surface, the surface tension of water is lowered and the insects and mosquitoes sink in water and are dead.
4. If we deform a liquid drop by pushing it slightly, then due to surface tension it again becomes spherical.