Exhaustive Notes on Fluid Mechanics and Bernoulli's Principles

Foundations of Fluid Mechanics

Fluid mechanics is positioned as a critical branch of both Physics and Engineering, dedicated to exploring the behavior of matter in specific states: liquids, gases, and plasmas. This study is divided into two primary disciplines: statics, which examines fluids at rest, and dynamics, which analyzes fluids in motion. The term "fluid" serves as a collective category for any substance that flows, encompassing both liquids and gases. While both are fluids, they are distinguished by their behavior under confinement: a liquid possesses a definite volume but adapts its shape to its container, whereas a gas lacks both a definite volume and shape, expanding to fill any space it occupies.

Defining States of Matter and Fluid Characteristics

The traditional classification of matter into solids, liquids, and gases is noted as somewhat artificial. Scientifically, the distinction depends more accurately on the timescale required for a substance to change its shape in response to an external force. A solid maintains a definite volume and shape. A liquid retains a definite volume but has no definite shape. An unconfined gas has neither. At a molecular level, a fluid consists of a collection of randomly arranged molecules held together by weak cohesive forces and the forces exerted by the boundaries of their container. Within this framework, fluid statics explores concepts such as pressure variation with depth and buoyancy (explaining why objects float or sink), while fluid dynamics investigates flow velocity, streamlines, and the conservation of energy within moving systems.

Forces and Pressure in Static Fluids

A defining characteristic of fluids is their inability to sustain shearing stresses or tensile stresses. Consequently, the only stress a static fluid can exert on a submerged object is a compressive stress. The force exerted by a static fluid is always directed perpendicular to the surface of the object. Pressure, denoted as $P$, is defined as the ratio of the magnitude of this force to the surface area over which it acts:

$P \equiv \frac{F}{A}$

Pressure is a scalar quantity, specifically because it is proportional to the magnitude of the force vector. In instances where pressure varies across a surface, it is calculated by evaluating the differential force $dF$ on a differential surface area $dA$:

$dF = P \, dA$

The standard unit for pressure is the pascal (Pa), where:

$1 \, \text{Pa} = 1 \, \text{N/m}^2$

While pressure itself is a scalar, the resulting force is a vector, with its direction always being perpendicular to the area of interest.

Density and Material Composition

Density, represented by the Greek letter rho ($\rho$), is the mass per unit volume of a substance:

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

Density is temperature-dependent because volume fluctuates with temperature transitions. Significant differences in density between gases, liquids, and solids indicate that the average molecular spacing in gases is much greater than in condensed phases.

Table 14.1: Densities of Common Substances

The following densities are measured at standard temperature ($0^{\circ}\text{C}$) and atmospheric pressure:

• Air: $1.29 \, \text{kg/m}^3$
• Air (at $20^{\circ}\text{C}$): $1.20 \, \text{kg/m}^3$
• Aluminum: $2.70 \times 10^3 \, \text{kg/m}^3$
• Benzene: $0.879 \times 10^3 \, \text{kg/m}^3$
• Brass: $8.4 \times 10^3 \, \text{kg/m}^3$
• Copper: $8.92 \times 10^3 \, \text{kg/m}^3$
• Ethyl alcohol: $0.806 \times 10^3 \, \text{kg/m}^3$
• Fresh water: $1.00 \times 10^3 \, \text{kg/m}^3$
• Glycerin: $1.26 \times 10^3 \, \text{kg/m}^3$
• Gold: $19.3 \times 10^3 \, \text{kg/m}^3$
• Helium gas: $1.79 \times 10^{-1} \, \text{kg/m}^3$
• Hydrogen gas: $8.99 \times 10^{-2} \, \text{kg/m}^3$
• Ice: $0.917 \times 10^3 \, \text{kg/m}^3$
• Iron: $7.86 \times 10^3 \, \text{kg/m}^3$
• Lead: $11.3 \times 10^3 \, \text{kg/m}^3$
• Mercury: $13.6 \times 10^3 \, \text{kg/m}^3$
• Nitrogen gas: $1.25 \, \text{kg/m}^3$
• Oak: $0.710 \times 10^3 \, \text{kg/m}^3$
• Osmium: $22.6 \times 10^3 \, \text{kg/m}^3$
• Oxygen gas: $1.43 \, \text{kg/m}^3$
• Pine: $0.373 \times 10^3 \, \text{kg/m}^3$
• Platinum: $21.4 \times 10^3 \, \text{kg/m}^3$
• Seawater: $1.03 \times 10^3 \, \text{kg/m}^3$
• Silver: $10.5 \times 10^3 \, \text{kg/m}^3$
• Tin: $7.30 \times 10^3 \, \text{kg/m}^3$
• Uranium: $19.1 \times 10^3 \, \text{kg/m}^3$

Variation of Pressure with Depth

In a fluid at rest, all portions must be in static equilibrium. This necessitates that all points at the same depth share the same pressure, regardless of the container's shape. To derive the relationship between pressure and depth, consider a sample of liquid in a cylinder with cross-sectional area $A$ extending from depth $d$ to $d + h$. Three external forces act on this volume: a downward force at the top ($P_0 A$), an upward force at the bottom ($PA$), and the downward gravitational force ($Mg$). Assuming the liquid is incompressible (density remains constant), the mass is given by:

$M = \rho V = \rho A h$

Summing the forces in the vertical direction facilitates the equilibrium condition:

$\sum F_y = PA - P_0 A - Mg = 0$

By substituting $M = \rho Ah$ and solving for $P$:

$P = P_0 + \rho gh$

This indicates that the pressure at depth $h$ is greater than the pressure at a higher point by an amount $\rho gh$. If the surface is open to the atmosphere, $P_0$ represents atmospheric pressure:

$P_0 = 1.00 \, \text{atm} = 1.013 \times 10^5 \, \text{Pa}$

Pascal’s Law and Hydraulics

Pascal’s Law, named after Blaise Pascal, states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container. This principle is fundamental to hydraulic systems (such as car lifts, hydraulic brakes, and forklifts), where a small input force ($F_1$) over a small area ($A_1$) can generate a large output force ($F_2$) over a larger area ($A_2$):

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

Consistent with the Conservation of Energy, the volume of liquid moved on the input side must equal the volume moved on the output side ($A_1 \Delta x_1 = A_2 \Delta x_2$). This leads to the conclusion that work remains constant:

$W_1 = W_2 \implies F_1 \Delta x_1 = F_2 \Delta x_2$

Tools for Pressure Measurement

Two primary devices are used to measure pressure: the Barometer and the Manometer. The mercury barometer, invented by Torricelli, consists of a tube filled with mercury inverted in a dish. The height ($h$) of the mercury column is used to calculate atmospheric pressure:

$P_0 = \rho_{Hg} g h$

Standard atmospheric pressure is approximately $0.760 \, \text{m}$ of mercury. The open-tube manometer measures the pressure of a gas in a vessel. One end is open to the atmosphere and the other is connected to the vessel. The pressure at the bottom of the U-tube is calculated as:

$P = P_0 + \rho gh$

Absolute pressure is the total pressure $P$, while gauge pressure is defined as the difference between absolute and atmospheric pressure:

$P_{gauge} = P - P_0 = \rho gh$

Archimedes’s Principle and Buoyancy

Archimedes (c. 287–212 BC), a Greek mathematician and engineer, discovered the nature of the buoyant force. Archimedes’s Principle states that the magnitude of the buoyant force ($B$) on an object is equal to the weight of the fluid displaced by that object. The buoyant force arises because the pressure at the bottom of a submerged object is greater than the pressure at the top. For a cube of height $h$:

$B = (P_{bot} - P_{top}) A = (\rho_{fluid} gh) A = \rho_{fluid} g V_{disp}$

Since $M = \rho_{fluid} V_{disp}$, the force is equivalent to $Mg$, the weight of the displaced fluid.

For a totally submerged object, the net force is determined by the difference between the buoyant force and gravity:

$F_{net} = B - F_g = (\rho_{fluid} - \rho_{obj}) g V_{obj}$

• If \rho_{obj} < \rho_{fluid}, the object accelerates upward.
• If \rho_{obj} > \rho_{fluid}, the object sinks.
• If $\rho_{obj} = \rho_{fluid}$, the object is in equilibrium.

For a floating object (which is only partially submerged), the fraction of the volume below the surface is determined by the density ratio:

$\frac{V_{disp}}{V_{obj}} = \frac{\rho_{obj}}{\rho_{fluid}}$

Practical examples include icebergs, where roughly $89\%$ of the volume is below the seawater surface because of the density of ice ($917 \, \text{kg/m}^3$) relative to seawater ($1.03 \times 10^3 \, \text{kg/m}^3$).

Characteristics of Fluid Flow

Fluid flow is categorized into two types: laminar and turbulent. Laminar (steady) flow occurs when every particle follows a smooth, non-crossing path, and any particle reaching a specific point has a consistent velocity. Turbulent flow is irregular, exhibiting whirlpool-like regions, and occurs when fluid speeds exceed a critical threshold. Viscosity represents internal friction within a fluid, where layers resist moving relative to one another, converting kinetic energy into internal energy.

To simplify the study of complex fluids, the model of an "Ideal Fluid" is used, based on four assumptions:

1. Non-viscous flow: Internal friction is neglected.
2. Steady flow: Particle velocity at any location is constant over time.
3. Incompressible flow: Density remains constant throughout.
4. Irrotational flow: The fluid has no angular momentum about any point.

The Equation of Continuity and Streamlines

A streamline is the specific path taken by a particle in steady flow, where velocity is always tangent to the path. A bundle of streamlines forms a "tube of flow." For an incompressible fluid moving through a pipe of non-uniform area, the mass entering must equal the mass leaving in any time interval ($m_1 = m_2$). This leads to the Equation of Continuity:

$A_1 v_1 = A_2 v_2$

The product $Av$ is known as the volume flux or flow rate and remains constant. Consequently, fluid speed is higher in constricted areas (small $A$) and lower in wider areas (large $A$).

Bernoulli’s Equation

Daniel Bernoulli (1700–1782) derived the relationship between fluid speed, pressure, and elevation. Bernoulli’s Equation is an expression of the Conservation of Energy for fluids. It states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline:

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

Or between two points:

$P_1 + \frac{1}{2} \rho v_1^2 + \rho gy_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gy_2$

For static fluids ($v=0$), this simplifies back to the pressure-depth relation: $P_1 - P_2 = \rho gh$. A key implication for moving fluids (including gases) is that as the speed of a fluid increases, its internal pressure decreases.

Dynamic Applications: Lift, Drag, and Spin

1. Airplane Wings: The curvature of the wing causes air to move faster above the wing than below it. According to the Bernoulli effect, the pressure above the wing is lower, creating an upward force called "lift." Lift is influenced by speed, wing area, curvature, and the angle of attack.
2. Golf Balls: Rapid backspin on a golf ball, combined with the friction-increasing dimples, creates lift that offsets the range lost to air resistance, allowing the ball to travel further.
3. Atomizers: A stream of air moving over an open tube immersed in liquid reduces the pressure inside the tube. This causes the liquid to rise and be dispersed as a fine spray.

Quantitative Analysis Examples

Storage Tank Leak: A tank open to the atmosphere ($P = P_0$) has a hole $16.0 \, \text{m}$ below the water level. The flow rate is $2.50 \times 10^{-3} \, \text{m}^3/\text{min}$.

• The exit speed is calculated using $v_2 = \sqrt{2gy_1} = \sqrt{2(9.80 \, \text{m/s}^2)(16.0 \, \text{m})} = 17.7 \, \text{m/s}$.
• Using the flow rate ($4.17 \times 10^{-5} \, \text{m}^3/\text{s}$) and the area-velocity relation ($A_2 v_2 = Q$), the hole diameter is found to be $1.73 \, \text{mm}$.

Fire Extinguisher: To achieve a water jet speed of $30.0 \, \text{m/s}$ when the water level is $0.500 \, \text{m}$ below the nozzle, the required gauge pressure is found via Bernoulli’s equation:

$P_{gauge} = \frac{1}{2} (1000 \, \text{kg/m}^3) (30.0 \, \text{m/s})^2 + (1000 \, \text{kg/m}^3) (9.80 \, \text{m/s}^2) (0.500 \, \text{m})$

$P_{gauge} = 455 \, \text{kPa}$.

Questions & Discussion

Question: Why do airplane pilots prefer to take off with the airplane facing into the wind?

Response: When taking off into the wind, the airspeed over the wings is effectively increased. This higher relative airspeed generates a larger lifting force earlier in the takeoff roll, which enables the pilot to take off safely using a shorter length of runway.