Mechanical Properties of Fluids Study Notes

Mechanical Properties of Fluids

Introduction to Fluids

Fluids are substances that can flow, a property that encompasses both liquids and gases. This ability to flow distinguishes fluids from solids in a fundamental way. Fluids are ubiquitous: the Earth is enveloped in air, and two-thirds of its surface is covered by water. Water is essential for mammalian existence, and all processes in living beings, including plants, are mediated by fluids.

Comparison: Solids, Liquids, and Gases

• Shape: Unlike solids, fluids have no definite shape of their own. They take the shape of their container.
• Volume: Solids and liquids have a fixed volume at a given temperature and pressure. Gases, however, expand to fill the entire volume of their container.
• Compressibility: The volume of any phase depends on the stress or pressure acting on it. For solids and liquids, the change in volume due to external pressure is very small; they have much lower compressibility compared to gases.
• Shear Stress: A key property of fluids is that they offer very little resistance to shear stress. While shear stress changes the shape of a solid while keeping its volume fixed, the shape of a fluid changes upon the application of even a very small shear stress. The shearing stress of fluids is roughly a million times smaller than that of solids.

Pressure

Pressure is defined as the impact of force over a specific coverage area. Everyday experiences, such as a sharp needle piercing skin while a blunt spoon with the same force does not, illustrate that a smaller area of contact results in a greater impact.

Fluid Pressure

When an object is submerged in a fluid at rest, the fluid exerts a force normal (perpendicular) to the object's surface. If there were a component of force parallel to the surface, the fluid would flow parallel to it (by Newton's third law). Since the fluid is at rest, the force must be perpendicular.

Using an idealized pressure-measuring device—an evacuated chamber with a spring-calibrated piston—the inward force $F$ exerted by the fluid on a piston of area $A$ allows for the definition of average pressure $P_{av}$:

$P_{av} = \frac{F}{A}$

In a limiting sense, where the area $\triangle A$ becomes arbitrarily small:

$P = \text{lim}_{\triangle A \rightarrow 0} \frac{\triangle F}{\triangle A}$

Characteristics and Units of Pressure

• Quantity Type: Pressure is a scalar quantity. It is the component of the force normal to the area, not the vector force, that is used in calculations.
• Dimensions: $[ML^{-1}T^{-2}]$
• SI Unit: Pascal ($Pa$), named after Blaise Pascal ($1623-1662$). $1 Pa = 1 N/m^2$.
• Atmospheric Pressure ($atm$): The pressure exerted by the atmosphere at sea level. $1 atm = 1.013 \times 10^5 Pa$.
• Other Units:
  • $1 torr = 133 Pa$ (equivalent to $1 mm Hg$).
  • $1 bar = 10^5 Pa$.

Density and Specific Gravity

Density ($\rho$) is the mass ($m$) per unit volume ($V$):

$\rho = \frac{m}{V}$

• Dimensions: $[ML^{-3}]$
• SI Unit: $kg/m^3$
• Water Density: At $4^{\text{o}}C$ ($277 K$), the density of water is $1.0 \times 10^3 kg/m^3$.
• Relative Density (Specific Gravity): The ratio of a substance's density to the density of water at $4^{\text{o}}C$. It is a dimensionless, positive scalar.

Example 9.1: Two femurs (thigh bones), each of area $10 cm^2$, support a $40 kg$ mass.

• Total area $A = 2 \times 10 \times 10^{-4} m^2 = 20 \times 10^{-4} m^2$.
• Force $F = 40 kg \times 10 m/s^2 = 400 N$.
• $P_{av} = \frac{400 N}{20 \times 10^{-4} m^2} = 2 \times 10^5 N/m^2$.

Pascal’s Law

Blaise Pascal observed that the pressure in a fluid at rest is the same at all points if they are at the same height.

Proof of Pascal's Law

Consider a small right-angled prismatic element $ABC-DEF$ in a fluid at rest. The element is small enough that gravity affects all points equally. The fluid exerts normal forces $F_a$, $F_b$, and $F_c$ on the faces of area $A_a$, $A_b$, and $A_c$. By equilibrium: $F_b \text{sin}(\theta) = F_c$$F_b \text{cos}(\theta) = F_a$

By geometry: $A_b \text{sin}(\theta) = A_c$$A_b \text{cos}(\theta) = A_a$

Thus: $\frac{F_b}{A_b} = \frac{F_c}{A_c} = \frac{F_a}{A_a}$

This implies $P_b = P_c = P_a$. Pressure is independent of direction and orientation of the surface.

Variation of Pressure with Depth

For a fluid at rest, consider a cylindrical element of height $h$ and area $A$. The vertical forces must balance the weight $mg$ of the fluid element.

If $P_1$ is the pressure at height $h$ and $P_2$ is the pressure at the bottom: $(P_2 - P_1)A = mg$

Since $m = \rho V = \rho h A$: $P_2 - P_1 = \rho g h$

Absolute and Gauge Pressure

If point $1$ is at the surface of the liquid open to the atmosphere ($P_1 = P_a$), and $P_2 = P$ at depth $h$:

$P = P_a + \rho g h$

• Absolute Pressure ($P$): The total pressure at a depth.
• Gauge Pressure ($P - P_a$): The excess pressure above atmospheric pressure ($\rho g h$).

Hydrostatic Paradox

The pressure at a certain depth depends only on the vertical height of the fluid column, the density, and gravity—not the shape of the container or the amount of liquid. If three vessels of different shapes are connected at the bottom and filled, the liquid level will be the same in all three because the pressure at the bottom must be equal.

Example 9.2: Pressure on a swimmer $10 m$ below the lake surface.

• $P = P_a + \rho g h$
• $P = 1.01 \times 10^5 Pa + (1000 kg/m^3 \times 10 m/s^2 \times 10 m) = 2.01 \times 10^5 Pa \thickapprox 2 atm$.

Pressure Measurement

Mercury Barometer

Invented by Evangelista Torricelli ($1608-1647$). A long glass tube filled with mercury is inverted into a trough of mercury. The space above the mercury in the tube is a vacuum (neglecting mercury vapor). Atmospheric pressure $P_a$ at the trough surface balances the pressure of the mercury column of height $h$:

$P_a = \rho g h$

At sea level, $h \thickapprox 76 cm$.

Open Tube Manometer

A U-tube containing a liquid (low density for small pressure differences, high density for large ones). One end is open to the atmosphere, and the other is connected to the system. The gauge pressure is proportional to the height difference $h$ between the two arms:

$P - P_a = \rho g h$

Hydraulic Machines

Pascals' Law states that external pressure applied to any part of an enclosed fluid is transmitted undiminished in all directions. This is the basis for hydraulic systems.

Hydraulic Lift

Two pistons of areas $A_1$ (small) and $A_2$ (large) are connected. A force $F_1$ on $A_1$ creates pressure $P = F_1/A_1$. This same pressure acts on $A_2$, producing an upward force $F_2$:

$F_2 = P \times A_2 = \frac{F_1 A_2}{A_1}$

• Mechanical Advantage: The ratio $\frac{A_2}{A_1}$. A small force transforms into a large force.

Hydraulic Brakes

A small force on a brake pedal moves a master piston, transmitting pressure through oil to larger pistons at the wheels, which then push brake shoes against the lining. This ensures equal braking effort on all four wheels.

Fluid Dynamics and Streamline Flow

Fluid dynamics is the study of fluids in motion.

Steady Flow

Flow is steady (streamline) if the velocity of every passing fluid particle at a given point remains constant in time. Particles follow a smooth path called a streamline, defined as a curve whose tangent at any point is in the direction of fluid velocity. Streamlines never cross.

Equation of Continuity

For an incompressible fluid, the mass of fluid entering a pipe must equal the mass leaving it. For areas $A_1, A_2$ and velocities $v_1, v_2$:

$A_1 v_1 = A_2 v_2$

• Volume Flux (Flow Rate): The product $Av$ remains constant.
• Where streamlines are crowded (smaller area), velocity increases.

Turbulent Flow

Beyond a critical speed, steady flow becomes turbulent, characterized by whirlpool-like regions called white water rapids.

Bernoulli’s Principle

Developed by Daniel Bernoulli in $1738$, this principle relates pressure, velocity, and height for steady, incompressible, non-viscous flow. It is a statement of the conservation of energy.

Bernoulli’s Equation

Along a streamline:

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

• $P$: Pressure energy per unit volume.
• $\frac{1}{2} \rho v^2$: Kinetic energy per unit volume.
• $\rho g h$: Potential energy per unit volume.

Limitations

1. Viscosity: Internal friction causes energy loss as heat.
2. Incompressibility: Assumes density does not change.
3. Turbulence: Does not apply to non-steady flows where values fluctuate.

Applications of Bernoulli’s Principle

Speed of Efflux: Torricelli’s Law

Efflux is fluid outflow. For a tank with a small hole at height $y_1$ (surface at $y_2$), the speed of efflux $v_1$ is:

$v_1 = \text{sqrt}\bigg(2gh + \frac{2(P - P_a)}{\rho}\bigg)$

If the tank is open to the atmosphere ($P = P_a$):

$v_1 = \text{sqrt}(2gh)$

This matches the speed of a freely falling body from height $h$.

Dynamic Lift

• Magnus Effect: A spinning ball drags air. On one side, ball motion and air motion align (higher velocity, lower pressure); on the other, they oppose (lower velocity, higher pressure). This pressure difference creates an upward or sideways force.
• Aerofoil (Airplane Wing): Wings are shaped so streamlines crowd together above the wing. Air speed is higher on top than below, creating a pressure difference that provides lift.

Viscosity

Viscosity is the internal friction in a fluid that offers resistance to the relative motion between layers.

Laminar Flow

In a tube, the layer in contact with the wall is stationary ($v=0$), and velocity is maximum at the axis. Layers slide over each other.

Coefficient of Viscosity (\text{̗})

Stress in a flowing fluid depends on the strain rate ($v/l$).

\text{̗} = \frac{\text{Shearing Stress}}{\text{Strain Rate}} = \frac{F/A}{v/l}

• SI Unit: Poiseiulle ($Pl$), $Pa \thickspace s$, or $N \thickspace s \thickspace m^{-2}$.
• Dimensions: $[ML^{-1}T^{-1}]$.
• Note: Viscosity of liquids decreases with temperature; viscosity of gases increases with temperature.

Stokes’ Law and Terminal Velocity

Stokes’ Law

For a sphere of radius $a$ falling through a fluid of viscosity \text{̗} at velocity $v$, the viscous drag force $F$ is:

F = 6 \text{̗} \text{̑} a v

Terminal Velocity ($v_t$)

As a sphere (like a raindrop) falls, it accelerates until the upward viscous force and buoyant force balance the downward gravitational force.

v_t = \frac{2 a^2 (\rho - \text{̓}) g}{9 \text{̗}}

• $\rho$: Density of the sphere.
• \text{̓}: Density of the fluid.

Surface Tension

Surface tension is a property of the liquid surface that causes it to behave like a stretched elastic membrane. It arises because molecules at the surface have higher potential energy than those in the bulk.

Surface Energy

Inside a liquid, a molecule is attracted by others in all directions. At the surface, it is only attracted by molecules below and beside it. To bring a molecule to the surface, work must be done against these attractive forces. A liquid tends to minimize its surface area to reach a minimum energy state.

Definition of Surface Tension ($S$)

It is the surface energy per unit area, or the force per unit length acting in the plane of the interface.

$S = \frac{F}{2l}$

• Surface tension typically decreases as temperature increases.

Angle of Contact ($\theta$)

The angle between the tangent to the liquid surface and the solid surface at the point of contact.

• Acute Angle ($\theta < 90^{\text{o}}$): Liquid wets the solid (e.g., water on glass). Occurs when adhesive forces (liquid-solid) are strong.
• Obtuse Angle ($\theta > 90^{\text{o}}$): Liquid does not wet the solid (e.g., mercury on glass). Occurs when cohesive forces (liquid-liquid) dominate.

Drops, Bubbles, and Capillary Rise

Pressure Inside a Drop and Bubble

Due to surface tension, the pressure inside ($P_i$) is higher than the pressure outside ($P_{o}$).

• Liquid Drop (one interface): $P_i - P_o = \frac{2S}{r}$
• Soap Bubble (two interfaces): $P_i - P_o = \frac{4S}{r}$

Capillary Rise

Water rises in a narrow tube (capillary) because the pressure below the curved meniscus is lower than the atmospheric pressure. The height $h$ of the rise is:

$h = \frac{2 S \text{cos}(\theta)}{\rho g a}$

• $a$: Radius of the tube.
• If $\theta$ is obtuse (e.g., mercury), height $h$ is negative, meaning the liquid level is depressed.