Revision: Class 11 >> Mechanical Properties of Fluids NEET (UG) Mechanical Properties of Fluids

The difference between the hydrostatic pressure (P) and the atmospheric pressure (P₀), measured by a manometer, is called gauge pressure: P − P₀ = ρgh

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

The thrust (Normal Force) exerted by a liquid at rest on unit area of the surface in contact is called pressure.

P = \[\frac {F_⊥_​​}{A}\]

• SI Unit: pascal (Pa) = 1 Nm⁻²
  Dimensions: [ML⁻¹T⁻²]

The ratio of the density of a given fluid to the density of pure water at 4°C is called relative density. It is a unitless quantity.

The mass per unit volume of a substance is called density: ρ = M/V

• SI Unit: kg/m³, Dimensions: [ML⁻³T⁰] 

The weight per unit volume of a substance is called specific weight or weight density: W = mg/V

• SI Unit: N/m³

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.

The force exerted by the air column on unit cross-sectional area at sea level (= 1.01 × 10⁵ Pa = 1.01 bar) is called atmospheric pressure.

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

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.

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.

When a liquid flows such that particles passing through a given point have different velocities from the predecessor, such a flow is called turbulent flow.

When a liquid flows such that each particle passing through a specific point follows the exact same path at the same speed as the particle before it, this type of flow is called streamline flow or steady flow.

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

∴ Velocity gradient = `(dv)/dx`

The property of a fluid by virtue of which it opposes the relative motion between its different layers, with the force that comes into play, is called viscosity; and that force is called the viscous force.

where η is the coefficient of viscosity.

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

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.

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

OR

The energy required to increase the surface area of a liquid 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.

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.

OR

The force acting along the surface of a liquid per unit length is called surface tension.

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

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.

When cohesive forces are stronger than adhesive forces (e.g., mercury in glass), the curved liquid surface formed where cos θ is negative and the liquid level is lower is called a convex meniscus.

When adhesive forces are stronger than cohesive forces, the curved liquid surface formed when the liquid is in contact with a solid is called a concave meniscus.

The rise or fall of liquid in a narrow tube due to surface tension is called capillary action.

UNIT OF RELATIVE DENSITY: No units since it is a pure ratio.

Density of a substance is defined as “Mass per Unit volume”.

Density [d]=`"mass  of the substance"/"volume of the substance"`

d=`m/v`

RELATIVE DENSITY: “is the ratio of the density of a substance to the density of water at 4° C.”
Or
RELATIVE DENSITY “is the ratio of the mass of the substance to the mass of an equal volume of water at 4° C.”

The resultant of all the forces exerted by a fluid on a body partly or wholly dipped in it, due to hydrostatic pressure, is called upthrust or buoyancy.

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

The limiting velocity up to which flow is streamline and beyond which it changes to turbulent is called critical velocity.

v_C​ = (R_e ​× η)/(ρl)

Case 1: θ < 90° (Concave Meniscus)

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

Statement: Pascal's Law states that when pressure is applied to a confined (enclosed) fluid, it is transmitted undiminished and equally in all directions throughout the fluid and to the walls of its container.

Mathematical Expression:

Key Points:

• Pressure changes by the same value at every point inside an incompressible, confined liquid.
• Used in hydraulic machines where a small force on a small area produces a large force on a large area.
• Applications: Hydraulic lift, hydraulic brake, hydraulic press, hydraulic jack.

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

Statement: Stokes' Law describes the force of viscosity exerted on a spherical object as it moves through a fluid. The viscous drag on a spherical body of radius r, moving with velocity v, in a medium of viscosity η is given by:

Key Points:

• The negative sign in F = −ηA\[\frac {dv}{dx}\] shows that viscous force opposes the direction of motion.
• At terminal velocity, upward viscous force + buoyant force = weight of the body.
• Terminal velocity formula derived from Stokes' Law:
  v₀ = \[\frac{2gr^2(\rho-\sigma)}{9\eta}\]
• Terminal velocity is proportional to r² — larger spheres fall faster.
• The coefficient of viscosity η has dimensional formula [ML⁻¹T⁻¹].
• SI unit of viscosity: Ns/m² (or Pa·s); CGS unit: dyne·s/cm² = poise.

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`

"When a body is partly or wholly dipped in a fluid, the fluid exerts a force on the body due to hydrostatic pressure. At any small portion of the surface of the body, the force exerted by the fluid is perpendicular to the surface and is equal to the pressure at that point multiplied by the area. The resultant of all these constant forces is called upthrust or buoyancy."

(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. 

• 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.

• Flow is streamline when velocity is low (parallel, orderly layers).
• Flow turns turbulent when velocity exceeds critical velocity (disordered, mixing layers).
• At low velocity → dye filament stays parallel in pipe (laminar).
• At high velocity → dye filament breaks and spreads (turbulent).

• 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).

• Surface tension causes liquid drops and bubbles to be spherical (minimum surface area for given volume = sphere).
• Air bubble in liquid → 1 liquid-air interface → excess pressure = 2T/r.
• Soap bubble → 2 surfaces → excess pressure = 4T/r.

• Aerofoil / Airplane Wing: Air moves faster over the curved top surface → lower pressure above, higher below → upward dynamic lift.
• Spray / Atomizer: High-speed air over a tube creates low pressure → liquid rises and is expelled as droplets.
• Blowing off Roofs: High-velocity wind above roof creates low pressure → atmospheric pressure inside lifts the roof off.
• Magnus Effect (Spinning Ball): A spinning ball drags air, creating pressure difference between upper and lower sides → ball moves in a curved path.
• Venturimeter: Used to measure flow rate of liquid through pipes using pressure difference between wide and narrow sections.
• Attraction between Two Boats: Water between boats moves faster → pressure decreases → boats are pulled toward each other.