Short Notes: Mechanical Properties of Fluids
9.1 Pressure
Definition: Pressure is the normal force exerted per unit area on a surface by a fluid or solid. It is a scalar quantity and in fluids acts equally in all directions at a point.
Formula: P = F/A
Units: SI unit: pascal (Pa) where 1 Pa = 1 N m⁻². Other common units: bar, atm, cm of Hg.
| Concept | Expression / Value |
|---|---|
| Hydrostatic pressure at depth h | P = P₀ + ρ g h |
| Atmospheric pressure (standard) | P₀ = 1.013 × 10⁵ Pa = 76 cm Hg |
| Gauge pressure | P_gauge = P - P₀ |
Explanation: For a fluid at rest, pressure increases linearly with depth because weight of the fluid column above contributes to the pressure. Pressure at a point in a static fluid is the same in all directions (isotropic).
Example (quick): Pressure at 10 m depth in fresh water (ρ = 1000 kg m⁻³):
\(P = P_0 + \rho g h = 1.013\times10^{5}\ \text{Pa} + 1000\times9.8\times10 = 1.013\times10^{5} + 9.8\times10^{4} = 1.993\times10^{5}\ \text{Pa}.\)
9.2 Pascal's Law and Applications
Pascal's Law: A change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of its container.
Hydraulic machine principle: If small piston of area A₁ is connected to a larger piston of area A₂ through an enclosed fluid and a force F₁ is applied on A₁, the force on A₂ is F₂ such that
F₁ / A₁ = F₂ / A₂
Mechanical advantage: F₂ / F₁ = A₂ / A₁. Hydraulic lifts and brakes use this principle to amplify force.
- Hydraulic press: useful to lift heavy loads by applying a relatively small force on a smaller piston.
- Hydraulic brakes: pressure from the master cylinder is transmitted to wheel cylinders to apply brake shoes or pads.
- Limitations: fluid must be incompressible and system must be sealed; real systems have losses due to friction and leakage.
9.3 Archimedes' Principle
Statement: A body wholly or partly immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the body.
| Concept | Formula |
|---|---|
| Buoyant force | F_b = ρ_fluid × V_displaced × g |
| Apparent weight | W_app = W - F_b = m g - ρ_fluid V g |
| Floating condition | W = F_b or ρ_body V_body g = ρ_fluid V_immersed g |
| Fraction submerged for floating body | V_immersed / V_total = ρ_body / ρ_fluid |
Explanation: If the buoyant force equals the weight of the body, it floats with a portion submerged determined by relative densities. If weight exceeds buoyant force, the body sinks. Applications include ship buoyancy, hydrometers, and density measurements.
Example: A wooden block of density ρ_b = 600 kg m⁻³ floats in water (ρ_f = 1000 kg m⁻³). Fraction submerged = 600/1000 = 0.6 (i.e., 60% submerged).
9.4 Continuity Equation
For an incompressible fluid: Mass conservation gives volume flow rate constant along a streamline or pipe.
- A₁ v₁ = A₂ v₂ where A is cross-sectional area and v is fluid speed.
- General (compressible): ρ₁ A₁ v₁ = ρ₂ A₂ v₂.
- Volume flow rate (Q): Q = A v = constant for steady incompressible flow.
Application: If a pipe narrows (A decreases), the fluid speed increases (v increases) to keep Q constant. This principle underlies flow in cardiovascular system, nozzles, and blood flow problems.
9.5 Bernoulli's Equation
Statement (for steady, non-viscous, incompressible flow along a streamline):
P + ½ ρ v² + ρ g h = constant
| Form | Equation / Result |
|---|---|
| General Bernoulli | P + \tfrac{1}{2}\rho v^{2} + \rho g h = \text{constant} |
| Horizontal flow (h constant) | P + \tfrac{1}{2}\rho v^{2} = \text{constant} |
| Torricelli's theorem (efflux from a hole) | v = \sqrt{2 g h} where h is depth of fluid surface above hole |
Explanation: Bernoulli's equation expresses conservation of mechanical energy for a fluid element: pressure energy + kinetic energy per unit volume + potential energy per unit volume remain constant along a streamline if viscous effects and external work are negligible.
Applications: Venturi meters, pitot tubes (measuring air speed), explaining lift qualitatively (airplane wing), and efflux speeds from open tanks.
9.6 Viscosity
Definition: Viscosity is a measure of internal friction in a fluid; it quantifies the resistance to relative motion between adjacent layers of fluid.
| Concept | Expression |
|---|---|
| Newton's law of viscosity (shear force) | F = η A \dfrac{dv}{dx} where η is dynamic viscosity, dv/dx is velocity gradient perpendicular to flow |
| Stokes' law (sphere in viscous fluid) | F = 6 π η r v |
| Terminal velocity of a small sphere | v_t = \dfrac{2 r^{2}(\rho_p - \rho_f) g}{9 η} |
| Poiseuille's law (laminar flow in a circular pipe) | Q = \dfrac{\pi \Delta P r^{4}}{8 η L} where Q is volume flow rate, ΔP pressure difference across length L |
| Reynolds number | Re = \dfrac{ρ v D}{η}. Flow is approximately laminar for Re < 2000 and turbulent for Re > 3000 (intermediate region may be transitional). |
Derivation sketch for terminal velocity (qualitative):
When a small sphere falls through a viscous fluid and reaches steady (terminal) speed, downward weight is balanced by upward buoyant force and viscous drag.
Balance of forces:
\(m g - ρ_f V g - 6\pi η r v_t = 0\)
Express mass m as ρ_p V and V for a sphere as \(\tfrac{4}{3}\pi r^{3}\), solve for v_t to obtain
\(v_t = \dfrac{2 r^{2}(\rho_p - \rho_f) g}{9 η}.\)
Notes: Poiseuille's law applies for steady, laminar, incompressible flow in a long, straight, circular pipe and shows the strong dependence of flow rate on radius (r⁴). Reynolds number determines flow regime.
9.7 Surface Tension
Definition: Surface tension is the property of a liquid surface that makes it behave like a stretched elastic membrane. It arises because molecules at the surface experience an inward net force.
| Concept | Expression |
|---|---|
| Surface tension (force per unit length) | S = F / L (SI unit: N m⁻¹) |
| Surface energy | ΔU = S × ΔA |
| Excess pressure inside a liquid drop | ΔP = 2 S / r |
| Excess pressure inside a soap bubble (two surfaces) | ΔP = 4 S / r |
| Capillary rise (or fall) | h = \dfrac{2 S \cos\theta}{ρ g r} where θ is contact angle, r is capillary radius |
| Special case: water wets glass (θ ≈ 0°) | h = \dfrac{2 S}{ρ g r} |
Explanation and applications: Surface tension causes droplets to be spherical, insects to walk on water, capillary action in thin tubes and plant xylem, and determines shapes of menisci. The contact angle θ describes wetting: θ < 90° indicates wetting, θ > 90° non-wetting.
Concise formula summary (for quick revision):
- P = F/A
- P = P₀ + ρ g h
- Pascal: F₁/A₁ = F₂/A₂
- Archimedes: F_b = ρ_f V_displaced g
- Continuity: A₁ v₁ = A₂ v₂
- Bernoulli: P + ½ ρ v² + ρ g h = constant
- Stokes drag: F = 6 π η r v
- Poiseuille: Q = π ΔP r⁴ / (8 η L)
- Surface tension: S = F / L, ΔP_drop = 2 S / r, h_capillary = 2 S cosθ / (ρ g r)
- Reynolds number: Re = ρ v D / η (laminar < 2000, turbulent > 3000)
FAQs on Short Notes: Mechanical Properties of Fluids
| 1. What are the key mechanical properties of fluids? |  |
| 2. How does viscosity affect fluid flow? | |
| 3. What is the significance of surface tension in fluids? | |
| 4. Can fluids be considered incompressible? When is this assumption valid? | |
| 5. How does temperature influence the mechanical properties of fluids? | |
