Revision: Mechanical Properties of Fluids Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

A substance which begins to flow when external force is applied on it is called a fluid. (Liquids and gases are fluids.)

A state of matter that does not have a definite shape or definite volume, occupies the entire space of the container it is in, and is highly compressible is called a gas.

A state of matter that has a definite volume, takes the shape of its container, and is generally considered to be incompressible (volume does not change significantly under pressure) is called a liquid.

The measure of a fluid's resistance to flow is called viscosity.

The normal force (or thrust) exerted by a liquid at rest per unit area of the surface in contact with it is called pressure of liquid or hydrostatic pressure.

The force which produces compression is called thrust. Its S.I unit is the newton.

The pressure exerted by the atmosphere on the earth's surface is called atmospheric pressure.

The total pressure exerted by a fluid, which includes both the atmospheric pressure as well as any other additional pressure due to the fluid itself, is called absolute pressure.

The phenomenon in which the liquid pressure at a point is independent of the quantity of liquid and depends only upon the depth of the point below the liquid surface is called hydrostatic paradox.

The difference between the absolute pressure and the atmospheric pressure at a point in a liquid is called gauge pressure.

The gaseous envelope surrounding the earth is called the earth's atmosphere.

A low-pressure area is an area in the atmosphere where the pressure is lower than its surrounding areas. In this situation, the wind from the surroundings blows towards the center of low pressure.

High pressure is an area of the atmosphere where the barometric pressure is higher than its surrounding areas. In this case, the wind from the center of high pressure blows towards the surrounding low-pressure areas.

The pressure exerted by this mercury column is considered as the pressure of magnitude ‘one atmosphere’ (1 atm).

SI unit of pressure is the pascal (Pa) or Nm⁻²
One Pascal: When a force of one newton acts normally on an area of one square metre (1 m²) then the pressure acting on the surface acting on the surface is called one Pascal.

Surface tension is defined as the force acting on a unit length of an imaginary line drawn on the free surface of the liquid, the direction of the force being perpendicular to the line so drawn and acting parallel to the surface.

When a liquid is in contact with a solid, the angle between the tangent drawn to the free surface of the liquid and the surface of solid at the point of contact measured inside the liquid is called the angle of contact.

Surface tension is defined as the force per unit length acting at right angles to an imaginary line drawn on the free surface of the liquid. 

The angle between tangents drawn at the point of contact to the liquid surface and the solid surface inside the liquid is called the angle of contact for a pair of solid and liquid. It is denoted by θ.

The sphere of influence of a molecule is defined as an imaginary sphere with the molecule at its centre and a radius equal to the range of molecular attraction.

A thin film of liquid near its surface having thickness equal to the molecular range of attraction is called surface film.

The potential energy is greater for molecules at the surface film as compared to molecules well inside the liquid. This extra energy of the molecule on the surface layer of a liquid is called the surface energy of the liquid.

An imaginary sphere drawn round a molecule (taken as centre) with a radius equal to the range of molecular attraction is called the sphere of influence of that molecule.

The property of a liquid due to which its free surface tries to have minimum surface area and behaves as if it were under tension somewhat like a stretched elastic membrane is called surface tension.

The work per unit area done by the force that creates a new surface is called surface energy.

The difference of pressure between the two sides of a liquid surface, which arises in equilibrium because the pressure inside a bubble or drop is greater than outside, is called excess pressure.

The angle between the surface of the solid and the tangent drawn to the surface of the liquid at the point of contact on the side of liquid is called the angle of contact of that liquid with that solid.

OR

The angle enclosed between the tangents to the liquid surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid with the solid, is called the angle of contact.

The difference of pressure between the two sides of a liquid surface (inside and outside a drop or bubble) due to the property of surface tension, which tends to contract the drop or bubble and compress the matter enclosed, is called Excess Pressure.

The phenomenon in which a liquid rises in a capillary tube when the angle of contact is acute, or falls when the angle of contact is obtuse, due to the interplay of pressure caused by the liquid column and pressure difference due to surface tension, is called capillary ascent (or descent).

A tube with a hole of very small diameter is called a capillary tube or capillary.

The rise or fall of level of liquid in a capillary tube is called capillary action or capillarity.

The phenomenon where a liquid in a capillary tube either ascends or descends relative to the surrounding liquid when a tube of very narrow bore is dipped in it is called capillarity.

The vertical height attained by a liquid in a capillary tube at equilibrium, which is independent of the shape of the capillary provided the radius of meniscus remains the same, is called the capillary rise height (h).

The path, straight or curved, the tangent to which at any point gives the direction of the flow of liquid at that point is called a streamline.

If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow.

When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called turbulent flow.

That flow of a liquid in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceding element passes through that point is called streamline flow.

The velocity of liquid flow up to which its flow is streamlined and above which its flow becomes turbulent is called critical velocity.

A pure number which determines the nature of flow of liquid through a pipe, defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid, is called Reynold's number.

The coefficient of viscosity of a liquid is defined as the viscous force acting tangentially per unit area of a liquid layer having a unit velocity gradient in a direction perpendicular to the direction of flow of the liquid.

The rate of change of velocity (dv) with distance (dx) measured from a stationary layer is called velocity gradient.

∴ Velocity gradient = `(dv)/dx`

The maximum constant velocity acquired by a body while falling freely through a viscous medium is called the terminal velocity V_T.

The maximum limit of velocity of a body falling in a viscous fluid, at which the net force acting on the body becomes zero, is called terminal velocity.

The constant maximum velocity acquired by a body while falling through a viscous fluid is called terminal velocity.

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

The principle which states that for a non-viscous liquid in streamline flow passing through a tube of varying cross-section, the product of the area of cross-section and the velocity of flow remains constant at every point is called the Equation of Continuity.

A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied

1. f is defined at every point on an open interval containing a.
2. \[\lim_{x\to a}f\left(x\right)\] exists.
3. \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].

Viscosity is that property of fluid, by virtue of which, the relative motion between different layers of a fluid experience a dragging force.
SI unit = N s /m²

The coefficient of viscosity can be defined as the viscous force per unit area per unit velocity gradient.

The continuity equation soys that the volume rote of flow of on incompressible fluid for a steady flow is the some throughout the flow.
A₁v₁ = A₂v₂ or, Av = constant

The branch of physics which deals with the properties of fluids at rest is called hydrostatics.

Any substance that can flow is a fluid.

The normal force (F) exerted by a fluid at rest per unit surface area (A) of contact is called the pressure (P) of the fluid.

P = \[\frac {F}{A}\]

SI unit: Pascal
Dimension: [L^-1M¹T²]

Atmospheric pressure is the pressure exerted by the weight of the column of air above a unit area.

The maximum distance from a molecule up to which the molecular force is effective is called the range of molecular force.

Any two mo.lecules attract each other. This force between molecules is called intermolecular force.

The force of attraction between the molecules of the same substance is called cohesive force or force of cohesion.

The force of attraction between the molecules of different substances is called the adhesive force or force of adhesion.

Surface tension T is defined as the tangential force acting per unit length on both sides of an imaginary line drawn· on the free surface of liquid.

Mathematically. T = \[\frac {F}{L}\]
Sl unit: N/m
Dimension:  [L⁰M¹T^-2]

The extra energy of the molecules in the surface layer is called the surface energy of the liquid.

The angle of contact is the angle between the tangent drawn to the free surface of the liquid and the solid surface at the point of contact, measured within the liquid.

The surface layer of a liquid with thickness equal to the range of intermolecular force is called the surface film.

An imaginary sphere with a molecule at its center and radius equal to the molecular range is called the sphere of influence of the molecule.

The branch of Physics which deals with the study of properties of fluids in motion is called hydrodynamics.

The phenomenon of rise or fall of a liquid in a capillary tube when dipped in the liquid is called capillarity.

The velocity beyond which a streamline flow becomes turbulent is called the critical velocity.

The rote of change of veiocity (dv) with distance (dx) measured from a stationary layer is coiled velocity gradient (dv/dx).

1 atm = 1.01 × 10⁵ Pa = 1.01 bar = 760 torr

P_gauge = P_absolute​ − P_{atmospheric​}

Case 1: θ < 90° (Concave Meniscus)

ΔP = 0 (no curvature, no excess pressure)

For a convex or concave surface and a liquid drop: ΔP = \[\frac {2T}{R}\] (one liquid surface)

ΔP = \[\frac {4T}{R}\] (two liquid surfaces — inner and outer)

ΔP = \[\frac {2T}{R}\] (only one liquid surface)

For a drop on a solid surface: cosθ = \[\frac {T_2-T_1}{​T_3}\]​​,

where T₁ = solid-liquid,

T₂ = solid-air, 

T₃ = liquid-air surface tension

For two different liquids in the same tube:

\[\frac{h_1}{h_2}=\frac{\rho_2T_1}{\rho_1T_2}\]

For the same liquid in tubes of different radii:

h₁​r₁​ = h₂​r₂​

Principle: Pressure due to liquid column = Pressure difference due to surface tension

h = \[\frac {2T}{Rdg}\] = \[\frac {2T cos θ}{rdg}\]

where r = radius of capillary tube and θ = angle of contact.

\[R_n=\frac{v_c\rho D}{\eta}\]

where:

• v_c​ = critical velocity of the liquid
• ρ = density of the liquid
• D = diameter of the pipe
• η = coefficient of viscosity of the liquid

v = \[\frac{2}{9}\cdot\frac{r^2(\rho-\sigma)g}{\eta}\]

where:

• v = terminal velocity
• r = radius of the body
• ρ = density of the body
• σ = density of the fluid
• g = acceleration due to gravity
• η = coefficient of viscosity of the fluid

For a non-viscous liquid in streamline flow passing through a tube of varying cross-section:

av = constant

or equivalently:

a ∝ \[\frac {1}{v}\]

where: 

• a = area of cross-section of the tube
• v = velocity of flow of the liquid

h = \[\frac {2T cosθ}{rρg}\]

\[\mathbf{v}_{\mathrm{c}}=\frac{R_{\mathrm{n}}\eta}{\rho d}\]

where,
v_c = critical velocity of the fluid
R_n= Reynolds number
η = coefficient of viscosity
ρ = density of fluid
d = diameter of tube 

\[\eta=\frac{2}{9}\frac{r^{2}\left(\rho-\sigma\right)g}{\mathrm{v}}\]

p = p₀ + pgh

Where:

• p_0​ = Atmospheric pressure
• ρ = Density of liquid
• g = Acceleration due to gravity
• h = Depth

"In a liquid at the same level, the pressure will be the same at all points. If not, then due to pressure difference, the liquid cannot be at rest."

T = `F/L`

where F = Force (N), L = Length (m)

= SI unit of T = `N/m`

Surface tension can also be written as

T = `W/A`

where W = Work (J), A = Area (m²)

= SI unit of T = `J/m^2`

We know

1J=1N×1m

So,

`J/m^2 = (N * m)/m^2 = N/m`

Both units are the same

`1N/m equiv 1J/m^2`

Statement:

"The viscous force Fᵥ acting on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere r, its velocity (v) through the fluid, and the coefficient of viscosity (η) of the fluid."

Formula:

(b) There is no gravitational force acting downwards. However, when the starting velocity is 20 m/s, the viscous force, which is directly proportional to velocity, becomes maximum and tends to accelerate the ball upwards.

\[\text{ When the ball falls under gravity, }\]

\[\text{ neglecting the density of air: } \]

\[\text{ Mass of the sphere = m }\]

\[\text{ Radius = r }\]

\[\text{ Viscous drag coeff . }= \eta\]

\[\text{Terminal velocity is given by}: \]

\[\text{ mg  }= 6\pi\eta r v_T \]

\[ \Rightarrow \frac{6\pi\eta r v_T}{m} = g . . . (1)\]

\[\text{ Now, at terminal velocity, the acceleration of the ball due to the viscous force is given by: } \]

\[a = \frac{6\pi\eta r v_T}{m}\]

\[\text{ Comparing equations (1) and (2), we find that : } \]

\[ \text{ a = g }\]

Thus, we see that the initial acceleration of the ball will be 9.8 ms - 2  .

(c) The velocity of the ball will decrease with time because of the upward viscous drag. As the force of viscosity is directly proportional to the velocity of the ball, the acceleration due to the viscous force will also decrease.

(d) When all the kinetic energy of the ball is radiated as heat due to the viscous force, the ball comes to rest. 

Statement:

"According to this theorem, the total energy (pressure energy, potential energy and kinetic energy) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe."

Mathematical Form:

For unit volume of liquid flowing through a pipe:

\[P+\rho gh+\frac{1}{2}\rho v^2\] = constant

where:

• P = pressure energy per unit volume
• ρ = density of the fluid
• g = acceleration due to gravity
• h = height of the fluid (potential energy term)
• v = velocity of the fluid (kinetic energy term)

Applications of Bernoulli's Theorem:

• Speed of efflux
• Venturi tube
• Lifting up of aeroplane
• Working of an atomizer
• Blowing off of roofs by stormy wind

The law states that, "The viscous force (F_v) acting on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere (r), its velocity (v) through the fluid, and the coefficient of viscosity (η) of the fluid". 

This is Bernoulli's equation. It states that the work done per unit volume of a fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow.

Mathematically.
p + \[\frac {1}{2}\]ρv² + ρgh = constant

Pascal's law states that the pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every point of the fluid and also on the walls of the container, provided the effect of gravity is neglected. 

• Pressure exerted by a liquid column depends on height and density of the liquid column.
• It is independent of the shape of the containing vessel or total mass of the liquid.
• Atmospheric pressure is maximum at the surface of the earth and decreases as we move up into the atmosphere.

• Hydraulic Press — Two cylinders (C & D) filled with liquid; small force applied on piston P₁ (smaller area A₁) is converted into a very large upward force on piston P₂ (larger area A₂), since A₂ > A₁.
• Hydraulic Lift — Works on Pascal's Law to lift or support heavy objects such as cars and trucks using liquid pressure.
• Hydraulic Brakes — Small force on the brake pedal is instantly transmitted equally through brake fluid to all cylinders, producing a large thrust on the wheels to stop the vehicle.

• A highly soluble impurity increases surface tension, while a partially soluble impurity (e.g., detergent) decreases it; a waterproofing agent increases it.
• Surface tension decreases with increase in temperature, given by T = T₀(1 − αθ), where T₀​ is surface tension at 0°C and α is the temperature coefficient.
• When a soap bubble is charged (positively or negatively), force acts outward on the surface, increasing its radius — thus electrification always decreases surface tension.

• Surface tension depends only on the nature of liquid and is independent of area of surface or length of line considered.
• Surface tension of a liquid decreases with rise of temperature; it is zero at boiling point and vanishes at critical temperature.
• Due to surface tension, a drop or bubble tends to contract, which increases internal pressure — this difference between inside and outside pressure is called excess pressure.
• For a drop and bubble in liquid: Δp = \[\frac {2T}{R}\]​; for a bubble in air: Δp = \[\frac {4T}{R}\]​(two free surfaces).

• When T₂ > T₁, cos θ is +ve, angle is acute — liquid partially wets the solid. (e.g., Kerosene on glass)
• When T₂ < T₁, cos θ is −ve, angle is obtuse — liquid does not wet the solid. (e.g., Mercury on glass)
• When T₂ − T₁ = T₃, cos θ = 1, θ = 0° — liquid completely wets and spreads over the solid. (e.g., Pure water on clean glass)
• When T₂ − T₁ > T₃, cos θ > 1, which is impossible — no drop forms, liquid simply spreads; equilibrium is not possible.

• Pascal’s law states that pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every part of the fluid and to the walls of the container.
• It applies only to fluids at rest (static conditions).
• The transmitted pressure acts equally in all directions.
• Mathematically, pressure remains constant throughout the fluid:
  \[\frac {F_1}{A_1}\] = \[\frac {F_2}{A_2}\]​​
• It is the working principle of hydraulic devices such as hydraulic lifts and hydraulic brakes.

• Soluble impurities, such as common salt, increase the surface tension of water, whereas detergents and phenol decrease it.
• Insoluble impurities reduce surface tension by decreasing cohesive forces, affecting droplet shape and meniscus formation.
• In most liquids, surface tension decreases with increasing temperature and vanishes at the critical temperature.

• For a plane surface, pressure just below the surface equals atmospheric pressure.
• For a convex surface, the pressure inside the liquid is greater than outside.
• For a concave surface, pressure inside the liquid is less than outside.

• Capillary rise occurs when liquid wets the tube (e.g., water in a glass).
• Capillary fall occurs when a liquid does not wet the tube (e.g., mercury in a glass tube).
• Rise or fall depends on surface tension and angle of contact.
• The narrower the tube, the greater the rise or fall.
• Capillarity plays an important role in natural processes like water rising in plants and oil rising in a lamp wick.

• In steady flow, properties such as pressure and velocity at a point remain constant over time.
• A flow line is the path followed by a particle in a moving fluid.
• A streamline is a curve whose tangent at any point gives the direction of fluid velocity.
• A flow tube is a bundle of streamlines; fluid does not cross its boundaries in steady flow.
• Laminar flow is smooth and orderly, while turbulent flow is irregular and chaotic.

• Speed of efflux: Liquid flowing out of a hole at depth hhh moves at the same speed as a body falling freely through a height hhh.
• Venturimeter: When a fluid passes through a narrow section, its speed increases and pressure decreases.
• Aeroplane lift: Faster airflow above the wings creates lower pressure, producing an upward lift force.
• Atomiser: High-speed air creates low pressure, causing liquid to rise and spray as fine droplets.
• Storm effect on roofs: Fast wind over a roof lowers pressure above it, and higher pressure below can lift the roof.

• They do not oppose deformation; they get permanently deformed.
• They have the ability to flow.
• They can take the shape of the container.