Ch 09 - Mechanical Properties of Fluids11th Physics

The Fundamental Distinction

• Solids can resist significant shear stress and maintain a fixed shape.
• Fluids offer very little resistance to shear stress. Their shape changes continuously upon the application of even a very small shear stress (about a million times smaller than that of solids).
• This inherent inability to resist shear stress is the primary reason why fluids flow and assume the shape of their containers.

Variables & Units:

• P = Pressure (N/m² or Pascal, Pa)
• F = Normal force exerted by the fluid (N)
• A = Area of contact (m²)

Conditions/Notes:

• Dimensional Formula: [ML⁻¹T⁻²]
• Pressure is a Scalar quantity. Even though force is a vector, pressure relates to the component of force strictly normal to the area, acting equally in all directions within a fluid at rest.

Pressure Conversions

• 1 atm = Standard atmospheric pressure at sea level = $1.013 \times 10^5$ Pa.
• 1 bar = $10^5$ Pa (often used in meteorology; closely approximates 1 atm).
• 1 torr = Pressure equivalent to exactly 1 mm of an Hg column = $133$ Pa.
• 1 atm = $760$ mm of Hg = $760$ torr.

Variables & Units:

• P₂ = Pressure at lower depth (Pa)
• P₁ = Pressure at upper depth (Pa)
• \rho = Density of the fluid (kg/m³)
• g = Acceleration due to gravity (m/s²)
• h = Vertical height difference between the two points (m)

Conditions/Notes:

• Valid strictly for fluids where density ($\rho$) remains uniform (incompressible liquids).

The Distinction

• Absolute Pressure ($P$): The total, actual pressure measured at a given depth.
  Formula: $P = P_a + \rho gh$
• Gauge Pressure ($P_g$): The excess pressure above the local atmospheric pressure. It is the value typically measured by common pressure gauges (like tyre pressure or blood pressure cuffs).
  Formula: $P_g = P - P_a = \rho gh$

Pascal's Law Comprises Two Aspects:

1. Static Fluid Equilibrium: The pressure in a fluid at rest is exactly the same at all points if they are at the same horizontal level.
2. Pressure Transmission: Whenever external pressure is applied to any part of a fluid contained in a closed vessel, it is transmitted undiminished and equally in all directions to every point of the fluid and the walls of the container.

Variables & Units:

• F₂ = Upward force generated on the large piston (N)
• F₁ = Downward force applied to the small piston (N)
• A₂ = Cross-sectional area of the large piston (m²)
• A₁ = Cross-sectional area of the small piston (m²)

Conditions/Notes:

• Since pressure is transmitted undiminished ($P = F_1/A_1 = F_2/A_2$), a small force over a small area creates a massive lifting force over a large area.
• The factor $(A_2 / A_1)$ is the Mechanical Advantage.

The Hydrostatic Paradox

• The liquid level will be exactly the same in all interconnected vessels, regardless of their shape, volume capacity, or base area.
• Reasoning: Liquid pressure depends only on the vertical depth ($h$) and density ($\rho$), and is completely independent of the total weight of the liquid or the container's geometry. For the system to achieve static equilibrium, the pressure at the common bottom interconnecting pipe must be identical throughout, meaning $h$ must be identical across all vessels.

Engineering Advantages of Mercury

1. High Density: Hg has a massive density ($13.6 \times 10^3 \text{ kg/m}^3$), which keeps the barometer height manageable (approx 76 cm at sea level). Water would require an unwieldy glass tube over 10 meters tall!
2. Low Vapour Pressure: The space above the mercury column is nearly a perfect vacuum. Mercury's vapor pressure is negligible at room temperature, making the atmospheric reading highly accurate.
3. Non-Wetting: It does not wet the glass walls, ensuring a clear, convex meniscus that moves freely without sticking to the tube.
4. Visibility: Its opaque, shiny metallic surface makes it easily observable.

Variables & Units:

• A = Cross-sectional area of the pipe (m²)
• v = Speed of the fluid (m/s)

Conditions/Notes:

• The product $Av$ is the Volume Flux or flow rate ($m^3/s$).
• Represents the Conservation of Mass.
• Implies that fluid speeds up in narrower regions (where streamlines are closely spaced) and slows down in wider regions.

Archimedes' Principle

• The buoyant force (upward thrust) acting on a body totally or partially immersed in a fluid is exactly equal in magnitude to the weight of the fluid displaced by the body.
• $F_B = V_{immersed} \cdot ho_{fluid} \cdot g$

Sink or Float Condition

• Sinks: If the object's overall density is strictly greater than the fluid's density ($ho_{object} > ho_{fluid}$), the downward weight exceeds the maximum upthrust.
• Floats: If the object's density is less than or equal to the fluid's density ($ho_{object} \le ho_{fluid}$), it will float either partially or fully submerged.

Variables & Units:

• P = Static Pressure (Pa)
• $\frac{1}{2}\rho v^2$ = Kinetic energy per unit volume (J/m³)
• \rho g h = Potential energy per unit volume (J/m³)

Conditions/Notes:

• This is a statement of the Conservation of Energy applied to fluid dynamics.
• It mathematically proves that an increase in fluid velocity ($v$) results in a simultaneous decrease in pressure ($P$) or potential energy.

Variables & Units:

• v = Speed of efflux (m/s)
• g = Acceleration due to gravity (m/s²)
• h = Vertical depth of the hole below the free liquid surface (m)

Conditions/Notes:

• The speed of the liquid emerging is identical to the velocity a body would acquire if falling freely under gravity from the same height $h$.
• Valid only when the tank is open to the atmosphere and the tank's surface area is vastly larger than the hole ($A_{tank} \gg A_{hole}$), meaning surface velocity is approx 0.

The Flow Distinction

• Streamline Flow: The velocity of each passing fluid particle at a given point remains completely constant in time. Fluid layers slide smoothly parallel to each other. By definition, no two streamlines ever intersect or cross.
• Turbulent Flow: Fluid velocity varies rapidly and randomly with time. It produces chaotic eddies and vortices, dissipating significant kinetic energy as heat.

Critical Velocity

• The absolute maximum velocity of a fluid up to which its flow remains strictly streamlined. Beyond this threshold speed, the flow loses steadiness and transitions into turbulent flow.

Variables & Units:

• \eta = Coefficient of Viscosity (Pa·s or N·s/m² or Poiseuille, Pl)
• F/A = Shearing Stress (N/m²)
• v/l = Strain rate or Velocity Gradient (s⁻¹)
• F = Viscous drag force between layers (N)

Conditions/Notes:

• Dimensions are [ML⁻¹T⁻¹].
• It mathematically defines how thick or resistant a fluid is to shear stress over time.

Boundary Conditions & Failures

1. Viscosity: Bernoulli assumes the fluid is completely non-viscous (zero internal friction). In reality, viscous drag dissipates kinetic energy into heat, violating the pure conservation of mechanical energy.
2. Compressibility: It assumes the fluid is completely incompressible. It fails for gases undergoing significant pressure/volume changes where elastic potential energy plays a role.
3. Turbulence: It holds strictly for steady, streamline flow. It completely breaks down in turbulent or non-steady flows where velocity and pressure constantly fluctuate.

Variables & Units:

• F = Viscous retarding force (N)
• \eta = Coefficient of viscosity of the fluid (Pa·s)
• a = Radius of the spherical object (m)
• v = Velocity of the object (m/s)

Conditions/Notes:

• The force is directly proportional to velocity, meaning as the object speeds up, the drag drastically increases until it balances gravity.

Variables & Units:

• v_t = Terminal velocity (m/s)
• a = Radius of the sphere (m)
• \rho = Density of the solid sphere (kg/m³)
• \sigma = Density of the surrounding fluid (kg/m³)
• g = Acceleration due to gravity (m/s²)
• \eta = Viscosity of the fluid (Pa·s)

Conditions/Notes:

• Terminal velocity is reached when Net Force = 0 (Weight = Buoyant Force + Viscous Drag).
• Note that $v_t \propto a^2$. A larger raindrop falls exponentially faster than a smaller one.

Variables & Units:

• R_e = Reynolds Number (Dimensionless)
• \rho = Density of the fluid (kg/m³)
• v = Speed of flow (m/s)
• d = Characteristic dimension / diameter of pipe (m)
• \eta = Viscosity of the fluid (Pa·s)

Boundary Conditions / Ranges:

• Re < 1000: Flow is Laminar / Streamlined.
• 1000 < Re < 2000: Unsteady / Transition flow.
• Re > 2000: Flow is Turbulent.
• Note: It represents the physical ratio of Inertial Forces to Viscous Forces.

The Mechanism of Artery Collapse

1. Plaque Buildup: Cholesterol plaque narrows the cross-sectional area of an artery.
2. Continuity Equation ($Av = \text{const}$): To maintain blood flow, the blood's velocity $v$ must drastically increase as it forces its way through the narrow constriction.
3. Bernoulli's Principle: The high kinetic energy inside the constriction results in a massive drop in internal static pressure.
4. Collapse: The external pressure from surrounding tissues now exceeds the internal blood pressure, causing the artery to physically collapse.
5. The heart pushes harder to open the artery, but the rushing blood just drops the pressure again, leading to a repeated cycle that eventually results in a heart attack.

The Magnus Effect

• A spinning ball drags adjacent layers of viscous air along with its surface. _start]
• High Velocity Side: On the side where the ball's spin aligns with the oncoming air stream, the relative velocity of the air is higher. _start]
• Low Velocity Side: On the opposite side, the spin opposes the oncoming air, making the relative velocity lower. _start]
• According to Bernoulli's principle, higher air velocity corresponds to lower pressure. _start]This induced pressure difference across the ball generates a net lateral force (dynamic lift), curving its flight path through the air.

Variables & Units:

• S = Surface Tension (N/m or J/m²)
• F = Force acting perpendicular to an imaginary line on the surface (N)
• l = Length of the line (m)
• U = Surface Energy (J)
• A = Surface Area (m²)

Conditions/Notes:

• Surface tension is a property of the interface between two media (e.g., liquid-air), not just the liquid alone.
• Typically decreases as temperature increases.

Aerofoil Mechanics

• The wing is engineered so its upper surface is highly curved while the bottom surface is relatively flat. _start]
• As the plane propels forward, this specific geometry forces the air streamlines to crowd together above the wing, resulting in a higher flow speed on top compared to the bottom. _start]
• By Bernoulli's Principle, the higher fluid speed above the wing creates a lower pressure zone. _start]The relatively higher normal pressure below pushes the wing upward, generating the net Dynamic Lift required to overcome the aircraft's weight.

Viscosity Definition

Viscosity is the internal friction or resistance to relative motion (shear stress) between adjacent layers of a fluid. _start]

Temperature Dependence

• Liquids: Viscosity DECREASES with an increase in temperature. _start]
  (Reasoning: Added heat gives liquid molecules enough kinetic energy to overcome cohesive intermolecular bonds, allowing layers to slide easily).
• Gases: Viscosity INCREASES with an increase in temperature. _start]
  *(Reasoning: Higher temperature increases the random thermal motion and collision frequency of gas molecules, creating chaotic momentum-transfer friction between adjacent flowing layers). *

Variables & Units:

• $\Delta P$ = Excess pressure ($P_{inside} - P_{outside}$) (Pa)
• S = Surface tension of the liquid (N/m)
• r = Radius of the drop/bubble (m)

Conditions/Notes:

• Liquid Drop / Air bubble in liquid: Has only ONE liquid-gas interface, hence $2S/r$.
• Soap Bubble in air: Has TWO interfaces (inner and outer surfaces), so the surface tension acts twice, yielding $4S/r$.

Variables & Units:

• h = Height of capillary rise/fall (m)
• S = Surface tension (N/m)
• $\theta$ = Angle of contact
• r = Radius of the capillary tube (m)
• \rho = Density of the liquid (kg/m³)
• g = Gravity (m/s²)

Boundary Conditions:

• If $\theta < 90^\circ$ (Acute), $\cos \theta$ is positive $\Rightarrow$ Liquid rises (capillary ascent, e.g., water in glass).
• If $\theta > 90^\circ$ (Obtuse), $\cos \theta$ is negative $\Rightarrow$ Liquid falls/depresses (capillary descent, e.g., mercury in glass).

The Origin of Surface Energy

• Molecules in the Bulk: A molecule deep inside the liquid is pulled equally in all directions by intermolecular cohesive forces, resulting in a net force of zero and a highly negative (stable) potential energy.
• Molecules on the Surface: A molecule on the surface has neighbors below and beside it, but none above. It experiences a net downward inward pull.
• Energy Storage: To bring a molecule from the bulk to the surface, mechanical work must be done against this inward cohesive pull. This work is stored as extra potential energy (Surface Energy).

Because mechanical systems naturally seek the lowest possible energy state, liquids spontaneously try to minimize this excess surface energy by minimizing their total surface area (forming spherical drops).

Variables & Units:

• $\theta$ = Angle of the free liquid surface tilted from the horizontal (Degrees or Radians)
• a = Horizontal acceleration of the container (m/s²)
• g = Acceleration due to gravity (m/s²)

Conditions/Notes:

• The fluid element attains equilibrium under the action of gravity and pseudo force ($-ma$).
• The free surface of the liquid always orients itself strictly perpendicular to the direction of net effective gravity.

Variables & Units:

• P_g = Gauge pressure (Pa)
• \rho = Density of the fluid (kg/m³)
• h = Depth below the free surface (m)
• g = Acceleration due to gravity (m/s²)
• a = Vertical acceleration of the container (m/s²)

Conditions/Notes:

• If the container accelerates upwards, effective gravity increases ($g_{eff} = g + a$).
• If the container accelerates downwards, effective gravity decreases ($g_{eff} = g - a$).
• For a freely falling container ($a = g$), the gauge pressure everywhere in the fluid is zero (weightlessness condition).

1. Temperature

• Surface tension decreases with an increase in temperature.
• It completely vanishes at the boiling point (or critical temperature) because the intermolecular cohesive forces are entirely overcome by the high thermal kinetic energy of the molecules.

2. Impurities

• Highly Soluble Impurities (e.g., sodium chloride/salt in water) generally increase the surface tension.
• Slightly Soluble or Insoluble Impurities (e.g., soap, detergent, or phenol in water) selectively accumulate at the surface and significantly decrease the surface tension.

Variables & Units:

• Q = Volume flow rate / Volume flux (m³/s)
• P = Pressure difference across the two ends of the tube (Pa)
• r = Radius of the tube (m)
• \eta = Coefficient of viscosity (Pa·s)
• l = Length of the tube (m)

Conditions/Notes:

• Flow must be streamlined/laminar.
• Notice that flow rate is highly sensitive to the radius ($Q \propto r^4$). A small decrease in artery radius (like from cholesterol plaque) drastically restricts blood flow, forcing the heart to increase pressure $P$ to maintain $Q$.

Variables & Units:

• $\Delta U$ = Change in Surface Energy / Work Done (Joules, J)
• T = Surface Tension of the liquid (N/m)
• N = Number of smaller droplets
• r = Radius of each smaller droplet (m)
• R = Radius of the original large drop (m)

Conditions/Notes:

• Volume remains conserved: $\frac{4}{3}\pi R^3 = N \times \frac{4}{3}\pi r^3 \implies R^3 = N r^3$.
• Splitting: Increases total surface area ($\Delta A > 0$), meaning energy is absorbed from the liquid, leading to a drop in temperature (cooling effect).
• Merging: Decreases total surface area ($\Delta A < 0$), meaning energy is released, leading to a rise in temperature (heating effect).

Angle of Contact Definition

The angle formed between the tangent to the liquid surface at the point of contact and the solid surface, measured strictly inside the liquid.

The Intermolecular Battle

• Acute Angle ($\theta < 90^\circ$): Occurs when Adhesion (liquid-solid attraction) is stronger than Cohesion (liquid-liquid attraction). The liquid wets the surface, spreading out. The resulting meniscus in a tube is concave. (Example: Water on clean glass).
• Obtuse Angle ($\theta > 90^\circ$): Occurs when Cohesion is stronger than Adhesion. The liquid does not wet the surface, tending to bead up into spherical droplets. The resulting meniscus is convex. (Example: Mercury on glass, or water droplets on a waxy lotus leaf).

The Energy Densities:

1. $P$ (Static Pressure): Represents the Pressure Energy per unit volume. It is the work done by fluid pressure to move a fluid element into a region.
2. $\frac{1}{2}\rho v^2$: Represents the Kinetic Energy per unit volume (also known as dynamic pressure).
3. $\rho g h$: Represents the Gravitational Potential Energy per unit volume.

Note: In any ideal, streamline flow, these three energy densities can mutually exchange and convert into one another, but their total sum is universally conserved along a streamline.

Manipulating the Contact Angle

• Detergents & Soaps (Wetting Agents): They dramatically decrease the angle of contact (making it highly acute). By lowering surface tension, water is allowed to easily spread out, wet the surface entirely, and penetrate the microscopic pores of fabrics to remove trapped dirt.
• Waterproofing Agents (e.g., Waxes, Teflon): They actively increase the angle of contact, forcing it to become obtuse ($\theta > 90^\circ$). Because the cohesion of the water becomes much stronger than its adhesion to the treated surface, the water beads up into tight spherical droplets and rolls off, completely failing to wet the fabric.