Fluid Mechanics

Fluid Mechanics

$Learning Objectives$

Define density, specific gravity, gauge pressure, and absolute pressure in fluid statics.
Calculate hydrostatic pressure and apply Pascal's Principle.
Understand and apply Archimedes' Principle to buoyancy problems.
State the assumptions of ideal fluid flow.
Apply the Equation of Continuity, Bernoulli's Equation, and Torricelli's theorem while identifying their assumptions and limitations.
Account for pressure head, velocity head, elevation head, and specified head loss without mixing pressure references.

Fluid mechanics is the study of fluids (liquids and gases) both at rest (statics) and in motion (dynamics). It is a foundational subject for civil engineering branches like hydraulics, water resources, and environmental engineering.

Fluid Statics

$Fluid Statics Concepts$

Fluids differ from solids because they cannot sustain shear stress while at rest. They flow and take the shape of their container.

$\rho$

Mass per unit volume. It is a fundamental property. The SI unit is $\text{kg/m}^3$ .

Density Equation

Calculates density from mass and volume.

\rho = \frac{m}{V}

Variables

Symbol	Description	Unit
$\rho$	Density	$kg/m^3$
$m$	Mass	kg
$V$	Volume	$m^3$

$Water Density$

For water at $4^\circ\text{C}$ , $\rho \approx 1000 \text{ kg/m}^3$ .

$Specific Gravity (SG)$

The ratio of the density of a substance to the density of a reference substance (usually water). It is a dimensionless number.

Specific Gravity Equation

Ratio of substance density to reference density.

SG = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}

Variables

Symbol	Description	Unit
$SG$	Specific Gravity	dimensionless
$\rho_{\text{substance}}$	Density of the substance	$kg/m^3$
$\rho_{\text{water}}$	Density of water	$kg/m^3$

Pressure ( $P$ )

$P$

Instead of dealing with forces on specific particles (which are constantly moving in a fluid), we deal with pressure.

$P$

The magnitude of the normal force exerted by a fluid per unit area of a surface. It is a scalar quantity.

Pressure Equation

Calculates pressure from force and area.

P = \frac{F}{A}

Variables

Symbol	Description	Unit
$P$	Pressure	Pa
$F$	Normal force	N
$A$	Area	$m^2$

$Pressure vs. Force$

Pressure vs. Force: Pressure acts perpendicular to any surface it contacts. While force is a vector, pressure itself has no direction.

Hydrostatic Pressure

$Hydrostatic Pressure Concepts$

The pressure at any depth in a stationary liquid depends only on the depth, the density of the liquid, and gravity.

Hydrostatic Equation

Calculates absolute pressure at a given depth.

P = P_0 + \rho g h

Variables

Symbol	Description	Unit
$P$	Absolute pressure at depth h	Pa
$P_0$	Pressure at the surface (often atmospheric)	Pa
$\rho$	Density of the fluid	$kg/m^3$
$g$	Acceleration due to gravity	$m/s^2$
$h$	Depth below the surface	m

$Pressure and Depth Relationship$

This equation shows that pressure increases linearly with depth in an incompressible fluid (like water).

$Interactive Simulation$

Use this hydrostatic model to see how depth, density, and area control pressure and resultant force.

Gauge pressure · p = ρgh

1000 kg/m³

5.0 m

2.0 m

1.5 m

Validity gate

The top edge depth is 4.00 m, so the entire gate remains below the free surface. States with h_c ≤ h/2 are prevented because the fully-submerged formulas would no longer apply.

Centroid gauge pressure

49.05 kPa

Resultant force F_R

147.15 kN

Center of pressure y_cp

5.07 m

Gate area A

3.00 m²

Top-edge pressure

39.24 kPa

Bottom-edge pressure

58.86 kPa

Fully submerged vertical gate with a linear pressure distribution and resultant below the centroid — Hydrostatic Pressure Distribution and Center of Pressure
For a fully submerged vertical plane area, pressure increases linearly with depth; the resultant is based on centroid pressure and acts below the centroid.

Pascal's Principle

$Pascal's Principle Concepts$

If you apply pressure to an enclosed, incompressible fluid, that change in pressure is transmitted undiminished to every part of the fluid and the walls of its container.

Pascal's Principle

Pressure change in an enclosed fluid.

\Delta P = \text{constant everywhere}

Variables

Symbol	Description	Unit
$\Delta P$	Change in pressure	Pa

$Hydraulic Lifts$

This is the principle behind hydraulic lifts. A small force $F_1$ applied over a small area $A_1$ creates a pressure $\Delta P = F_1/A_1$ . This exact same pressure appears on a larger area $A_2$ , producing a much larger force $F_2$ .

Hydraulic Lift Force Equation

Calculates the output force of a hydraulic lift given the input force and area ratio.

F_2 = F_1 \left( \frac{A_2}{A_1} \right)

Variables

Symbol	Description	Unit
$F_2$	Output force on the larger piston	N
$F_1$	Input force on the smaller piston	N
$A_2$	Area of the larger piston	$m^2$
$A_1$	Area of the smaller piston	$m^2$

Archimedes' Principle (Buoyancy)

$Archimedes' Principle (Buoyancy) Concepts$

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

Buoyant Force ( $F_B$ )

Calculates the upward buoyant force on an object in a fluid.

F_B = \rho_{\text{fluid}} V_{\text{submerged}} g

Variables

Symbol	Description	Unit
$F_B$	Buoyant force	N
$\rho_{\text{fluid}}$	Density of the fluid	$kg/m^3$
$V_{\text{submerged}}$	Submerged volume of the object	$m^3$
$g$	Acceleration due to gravity	$m/s^2$

$Density and Flotation$

Notice that the buoyant force depends on the density of the fluid, not the object, and the volume of the object that is actually submerged ( $V_{\text{submerged}}$ ).

If an object is denser than the fluid ( $\rho_{obj} > \rho_{fluid}$ ), it will sink (its weight $W_{obj}$ is greater than the maximum $F_B$ ).
If it is less dense ( $\rho_{obj} < \rho_{fluid}$ ), it will float. In equilibrium floating, it displaces a volume of fluid whose weight exactly equals its own total weight ( $F_B = W_{obj}$ ).

$Interactive Simulation$

Use this buoyancy model to compare object weight with displaced fluid weight.

Floats

600 kg/m³

1000 kg/m³

1.0 m³

Governing Relations

Weight

W = \rho_{obj} V g

Buoyant Force

F_B = \rho_{fluid} V_{disp} g

Weight

5,886 N

Buoyant Force

5,886 N

Net Vertical Force

0 N

Displaced Volume

0.60 m³

Fraction Submerged

60.0%

Hydraulic force multiplication and buoyancy force balance shown in separate panels — Pascal and Archimedes Force Balances
Hydraulic systems transmit pressure changes through an enclosed fluid, while buoyancy equals the weight of displaced fluid.

Fluid Dynamics

$Fluid Dynamics Concepts$

When fluids move, things get complicated quickly. In introductory physics and engineering, we usually start with an idealized model of fluid flow: ideal fluid flow.

$Assumptions of the Introductory Ideal-Flow Model$

Steady flow: the velocity field at a fixed point does not change with time.
Incompressible flow: density remains constant along the modeled motion.
Inviscid flow: viscous dissipation is neglected.
Along a streamline: the basic Bernoulli relation is applied between points on the same streamline unless stronger conditions justify a broader application.

The Equation of Continuity

$The Equation of Continuity Concepts$

For an incompressible fluid flowing steadily through a pipe of varying cross-sectional area, the volume flow rate ( $Q$ ) must remain constant everywhere. What goes in must come out.

$Q$

The volume of fluid passing a given cross-section per unit time. The SI unit is $\text{m}^3/\text{s}$ .

Volume Flow Rate Equation

Calculates volume flow rate from area and fluid speed.

Q = \frac{\Delta V}{\Delta t} = A v

Variables

Symbol	Description	Unit
$Q$	Volume flow rate	$m^3/s$
$\Delta V$	Change in volume	$m^3$
$\Delta t$	Time interval	s
$A$	Cross-sectional area	$m^2$
$v$	Fluid speed	m/s

Equation of Continuity

Relates area and velocity at two points in steady flow.

A_1 v_1 = A_2 v_2

Variables

Symbol	Description	Unit
$A_1$	Cross-sectional area at point 1	$m^2$
$v_1$	Fluid speed at point 1	m/s
$A_2$	Cross-sectional area at point 2	$m^2$
$v_2$	Fluid speed at point 2	m/s

$Flow Speed Example$

This explains why a river speeds up when it passes through a narrow gorge ( $A_2 < A_1 \implies v_2 > v_1$ ).

Bernoulli's Equation

$Bernoulli's Equation Concepts$

Bernoulli's equation is essentially a statement of the conservation of mechanical energy applied to ideal fluid flow. It relates pressure, flow speed, and elevation along a streamline.

The work done on a fluid element as it moves through a pipe changes its kinetic and potential energy.

Bernoulli's Equation

Conservation of energy principle applied to ideal fluid flow.

P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2

Variables

Symbol	Description	Unit
$P_1$	Pressure at point 1, using a consistent absolute or gauge reference	Pa
$v_1$	Fluid speed at point 1	m/s
$y_1$	Elevation at point 1	m
$P_2$	Pressure at point 2, using the same reference as point 1	Pa
$v_2$	Fluid speed at point 2	m/s
$y_2$	Elevation at point 2	m
$\rho$	Fluid density	$kg/m^3$
$g$	Acceleration due to gravity	$m/s^2$
$\frac{1}{2}\rho v^2$	Dynamic pressure (kinetic energy per unit volume)	Pa
$\rho g y$	Static pressure due to elevation (potential energy per volume)	Pa

$The Bernoulli Effect$

For steady, incompressible, inviscid flow along a streamline at the same elevation, Bernoulli's equation reduces to $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$ . Within those assumptions, a section with greater speed has lower static pressure. This relation is directly useful in Venturi-type flows; by itself it is not a complete theory of aerodynamic lift.

$Pressure Reference in Bernoulli Calculations$

Bernoulli's equation may be written using absolute pressure or gauge pressure when the same reference is used at every point. Do not mix one absolute pressure with another gauge pressure in a single balance.

Mechanical-Energy Equation with Head Loss

Head-form energy balance for incompressible flow when irreversible losses are represented by h_L.

\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+z_1 = \frac{P_2}{\rho g}+\frac{v_2^2}{2g}+z_2+h_L

Variables

Symbol	Description	Unit
$P/\rho g$	Pressure head	m
$v^2/2g$	Velocity head	m
$z$	Elevation head	m
$h_L$	Head loss between sections	m

$Bernoulli Assumptions and Limitations$

The basic Bernoulli equation requires the stated ideal-flow assumptions along the chosen streamline. Real pipe systems can require head-loss terms, pump/turbine work terms, and compressibility or unsteady-flow models. A correct numerical result begins by selecting the equation that matches the physical system.

$Interactive Simulation$

Use this pipe-flow model to connect continuity with the Bernoulli pressure-head tradeoff.

Gauge pressure head · horizontal datum

0.12 m²

0.05 m²

0.18 m³/s

8.0 m

0.4 m

Head accounting

Q = A₁v₁ = A₂v₂.

p₁/ρg + v₁²/2g = p₂/ρg + v₂²/2g + h_L.

Both displayed pressures are gauge pressures. Adding the same atmospheric pressure to both sides would give an equivalent absolute-pressure balance.

Velocity v₁

1.50 m/s

Velocity v₂

3.60 m/s

Gauge pressure p₁

78.48 kPa

Gauge pressure p₂

69.20 kPa

Velocity head v₁²/2g

0.11 m

Velocity head v₂²/2g

0.66 m

Total head H₁

8.11 m

Total head H₂

7.71 m

Horizontal varying-area pipe with continuity, pressure-head columns, and a decreasing energy grade line — Continuity, Pressure Head, and Energy Loss
Area controls velocity through continuity; Bernoulli head accounting separates pressure head, velocity head, elevation head, and head loss.

Torricelli's Theorem

$Torricelli's Theorem Concepts$

A direct application of Bernoulli's equation is finding the speed of fluid exiting a small hole at the bottom of an open tank. If the hole is a distance $h$ below the surface, the surface area is much larger than the hole ( $v_{surface} \approx 0$ ), and both are at atmospheric pressure, Bernoulli's equation simplifies to Torricelli's Theorem.

Torricelli's Theorem

Fluid exit speed from an open tank.

v_{exit} = \sqrt{2gh}

Variables

Symbol	Description	Unit
$v_{exit}$	Exit velocity of fluid	m/s
$g$	Acceleration due to gravity	$m/s^2$
$h$	Depth of hole below fluid surface	m

$Free Fall Analogy$

This shows the fluid exits with the same speed an object would have if dropped in free fall from a height $h$ .

Variation of Pressure with Depth

$The Hydrostatic Paradox$

The equation $P = P_0 + \rho gh$ implies that the pressure at a given depth in a static fluid is independent of the shape of the container or the total volume of fluid present. This leads to the "Hydrostatic Paradox":

If you have three containers of wildly different shapes (one wide, one narrow, one conical) but all filled to the exact same height $h$ with the same fluid, the pressure at the bottom of all three containers is identical. Consequently, if the bottom area $A$ is the same, the total force on the bottom of each container is exactly the same, regardless of how much total fluid is inside!

Key Takeaways

Fluid Statics is governed by pressure increasing with depth ( $P = P_0 + \rho gh$ ) and Pascal's Principle (pressure applied is transmitted undiminished).
Archimedes' Principle states that buoyancy is an upward force equal to the weight of displaced fluid ( $F_B = \rho_f V_{sub} g$ ).
Fluid Dynamics for ideal fluids relies on conservation principles.
The Equation of Continuity ( $A_1v_1 = A_2v_2$ ) is the conservation of mass/volume flow.
Bernoulli's Equation ( $P + \frac{1}{2}\rho v^2 + \rho gy = \text{const}$ ) is the conservation of energy, showing that pressure drops when speed increases horizontally.

$References$

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Learning Objectives

Fluid Statics

Fluid Statics Concepts

Density (ρ\rhoρ)

Density Equation

Water Density

Specific Gravity (SG)

Specific Gravity Equation

Pressure (PPP)

Pressure (PPP) Concepts

Pressure (PPP)

Pressure Equation

Pressure vs. Force

Hydrostatic Pressure

Hydrostatic Pressure Concepts

Hydrostatic Equation

Pressure and Depth Relationship

Interactive Simulation

Pascal's Principle

Pascal's Principle Concepts

Pascal's Principle

Hydraulic Lifts

Hydraulic Lift Force Equation

Archimedes' Principle (Buoyancy)

Archimedes' Principle (Buoyancy) Concepts

Buoyant Force (FBF_BFB​)

Density and Flotation

Interactive Simulation

Fluid Dynamics

Fluid Dynamics Concepts

Assumptions of the Introductory Ideal-Flow Model

The Equation of Continuity

The Equation of Continuity Concepts

Volume Flow Rate (QQQ)

Volume Flow Rate Equation

Equation of Continuity

Flow Speed Example

Bernoulli's Equation

Bernoulli's Equation Concepts

Bernoulli's Equation

The Bernoulli Effect

Pressure Reference in Bernoulli Calculations

Mechanical-Energy Equation with Head Loss

Bernoulli Assumptions and Limitations

Interactive Simulation

Torricelli's Theorem

Torricelli's Theorem Concepts

Torricelli's Theorem

Free Fall Analogy

Variation of Pressure with Depth

The Hydrostatic Paradox

References

$Learning Objectives$

$Fluid Statics Concepts$

$\rho$

$Water Density$

$Specific Gravity (SG)$

Pressure ( $P$ )

$P$

$P$

$Pressure vs. Force$

$Hydrostatic Pressure Concepts$

$Pressure and Depth Relationship$

$Interactive Simulation$

$Pascal's Principle Concepts$

$Hydraulic Lifts$

$Archimedes' Principle (Buoyancy) Concepts$

Buoyant Force ( $F_B$ )

$Density and Flotation$

$Interactive Simulation$

$Fluid Dynamics Concepts$

$Assumptions of the Introductory Ideal-Flow Model$

$The Equation of Continuity Concepts$

$Q$

$Flow Speed Example$

$Bernoulli's Equation Concepts$

$The Bernoulli Effect$

$Pressure Reference in Bernoulli Calculations$

$Bernoulli Assumptions and Limitations$

$Interactive Simulation$

$Torricelli's Theorem Concepts$

$Free Fall Analogy$

$The Hydrostatic Paradox$

$References$