Chapter 9: Mechanical Properties of Fluids

Unit VII – Properties of Bulk Matter

9.1 Pressure

Pressure is the normal force per unit area exerted by a fluid:

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

• SI unit: Pascal (Pa) = N/m²
• Atmospheric pressure: \(P_0 = 1.013 \times 10^5 \text{ Pa}\)

Pressure at depth h in a fluid of density \(\rho\):

\[ P = P_0 + \rho g h \]

9.2 Pascal’s Law

Pressure applied to an enclosed fluid is transmitted unchanged to every point of the fluid and the walls of the container.

Hydraulic lift:

\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \Rightarrow F_2 = F_1 \cdot \frac{A_2}{A_1} \]

A small force on a small piston creates a large force on a large piston.

9.3 Buoyancy and Archimedes’ Principle

Any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid.

\[ F_{buoy} = \rho_{fluid} \cdot V_{displaced} \cdot g \]

Condition for floating: \(\rho_{object} \leq \rho_{fluid}\)

9.4 Viscosity

Viscosity is the internal friction of a fluid that resists flow.

Newton’s law of viscosity:

\[ F = -\eta A \frac{dv}{dy} \]

where \(\eta\) is the coefficient of viscosity (SI unit: Pa·s or N·s/m²)

Stokes’ Law: Drag force on a sphere of radius \(r\) moving with velocity \(v\):

\[ F = 6\pi\eta r v \]

Terminal velocity:

\[ v_T = \frac{2r^2(\rho - \rho_0)g}{9\eta} \]

where \(\rho\) = density of sphere, \(\rho_0\) = density of fluid.

9.5 Streamline and Turbulent Flow

• Streamline flow: Each layer of fluid flows smoothly; velocity at every point is constant in time.
• Turbulent flow: Irregular, chaotic motion with eddies and whirls.

Reynolds number determines type of flow:

\[ R_e = \frac{\rho v D}{\eta} \]

• \(R_e \lt 1000\): Streamline
• \(R_e \gt 2000\): Turbulent

Equation of Continuity (for incompressible fluid):

\[ A_1 v_1 = A_2 v_2 \]

9.6 Bernoulli’s Theorem

For steady, incompressible, non-viscous flow, the total energy per unit volume is constant along a streamline.

\[ P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} \]

9.7 Bernoulli’s Principle – Diagram

Applications of Bernoulli’s theorem:

• Aircraft wings (lift)
• Carburetor
• Spray gun, Bunsen burner
• Speed of efflux: \(v = \sqrt{2gh}\) (Torricelli’s theorem)

9.8 Surface Tension

Surface tension (T) is the force per unit length acting along the surface of a liquid:

\[ T = \frac{F}{l} \]

SI unit: N/m

Surface energy: Energy per unit area = \(T\) (numerically)

Excess Pressure

• Inside a liquid drop: \(\Delta P = \dfrac{2T}{r}\)
• Inside a soap bubble: \(\Delta P = \dfrac{4T}{r}\) (two surfaces)

Capillary Rise

\[ h = \frac{2T\cos\theta}{\rho g r} \]

where \(\theta\) = contact angle, \(r\) = radius of capillary.

Key Formulas Summary

Practice Questions

Section A – MCQ (1 mark each)

Q1. Bernoulli’s theorem applies to:

(a) Viscous flow   (b) Turbulent flow   (c) Steady, non-viscous flow   (d) All flows

Q2. A ball falls through a viscous fluid and reaches terminal velocity. At this point:

(a) Net force is maximum   (b) Acceleration is 9.8 m/s²   (c) Net force is zero   (d) Velocity is increasing

Section B – Short Answer (2–3 marks)

Q3. Water flows through a pipe of cross-section 8 cm² at a speed of 3 m/s. It enters a narrower pipe of cross-section 2 cm². Find the speed in the narrow pipe.

Section D – Competency-Based Questions

Q4. (Case Study) Water in a dam is at height 20 m above a hole at the bottom.

(i) Find the speed at which water exits the hole (Torricelli’s theorem).

(ii) If the hole has area \(10^{-4}\) m², find the volume flow rate.

(iii) State two real-world applications of Bernoulli’s theorem.

(\(g = 10 \text{ m/s}^2\))