Fluid Mechanics

Fluid Mechanics

Learning Objectives

Define density and pressure in the context of 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 and Bernoulli's Equation to fluid dynamics.

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.

Density ( $\rho$ )

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	$kg/m^3$

P

Pressure ( $P$ ) Concepts

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

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.

Interactive Physics Simulation

Hydrostatic Pressure Field & Gate Simulator

Submerge a rectangular vertical gate in a fluid field. Watch the hydrostatic pressure prism grow linearly with depth, and observe how the resultant force anchors precisely at the Center of Pressure (y_cp).

Fresh Water

Fluid Preset

Centroid Depth (hc)5.0 m

Fluid Density (ρ)1000 kg/m³

Gate Vertical Height (h)2.0 m

Gate Horizontal Width (b)1.5 m

Governing Formulas

Resultant Hydrostatic Force

F_R = P_c \cdot A = (\rho g h_c) \cdot (b \cdot h)

Vertical Center of Pressure

y_{cp} = h_c + e = h_c + \frac{h^2}{12 h_c}

Pressure at Centroid (Pc)

49.05 kPa

Resultant Force (Fr)

147.15 kN

Centroid Depth (hc)

5.00 m

Center of Pressure (y_cp)

5.07 m

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.

Interactive Physics Simulation

Archimedes' Principle & Buoyancy

Adjust the density of the block and the fluid. Archimedes' principle states that the buoyant force equals the weight of the fluid displaced by the object.

Floats

Block Density600 kg/m³

Styrofoam (100)Wood (600)Brick (2000)

Fluid Density1000 kg/m³

Oil (800)Water (1000)Honey (1400)

Block Volume1.0 m³

Governing Formulas

Weight of Block

W = m \cdot g = \rho_{obj} \cdot V \cdot g

Buoyant Force

F_b = \rho_{fluid} \cdot V_{sub} \cdot g

Weight of Block

5,886 N

Buoyant Force

5,886 N

Fraction Submerged

60.0%

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 Ideal Fluid Flow

STEP-BY-STEP

Steady Flow: The velocity of the fluid at any given point is constant over time.
Incompressible: The density of the fluid is constant (a very good assumption for liquids).
Nonviscous (Inviscid): Internal friction within the fluid is zero.
Irrotational: Fluid particles do not rotate about their own centers of mass.

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.

Volume Flow Rate ( $Q$ )

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
$P_1$	Absolute pressure at point 1	Pa
$v_1$	Fluid speed at point 1	m/s
$y_1$	Elevation at point 1	m
$P_2$	Absolute pressure at point 2	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

The Bernoulli Effect: If a fluid flows horizontally ( $y_1 = y_2$ ), the equation simplifies to $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$ . This shows that where the speed of the fluid is high, the pressure is low, and vice-versa. This principle is key to understanding airplane lift, carburetors, and the Venturi tube.

Interactive Simulation

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

Interactive Physics Simulation

Continuity & Bernoulli Venturi Simulator

Adjust the upstream and throat cross-sectional areas. Observe how fluid streamlines compress and accelerate inside the narrow constriction, inducing a corresponding pressure drop.

Upstream Area (A1)0.12 m²

Constricted Throat Area (A2)0.05 m²

Volumetric Flow Rate (Q)0.24 m³/s

Total Head Loss (h_L)0.4 m

Governing Formulas

Continuity Equation

Q = A_1 \cdot v_1 = A_2 \cdot v_2

Bernoulli's Energy Balance

\Delta h = \frac{v_2^2 - v_1^2}{2g} + h_L

Upstream Velocity (v1)

2.00 m/s

Constriction Velocity (v2)

4.80 m/s

Total Head Drop (Δh)

1.37 m

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.

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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

Hydrostatic Pressure Field & Gate Simulator

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

Archimedes' Principle & Buoyancy

Fluid Dynamics

Fluid Dynamics Concepts

Assumptions of Ideal Fluid Flow

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

Interactive Simulation

Continuity & Bernoulli Venturi Simulator

Torricelli's Theorem

Torricelli's Theorem Concepts

Torricelli's Theorem

Free Fall Analogy

Variation of Pressure with Depth

The Hydrostatic Paradox

Density ( $\rho$ )

Pressure ( $P$ )

Pressure ( $P$ ) Concepts

Pressure ( $P$ )

Buoyant Force ( $F_B$ )

Volume Flow Rate ( $Q$ )