8.1 Fluid statics

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8. Fluid mechanics

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Fluid statics

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This chapter covers the following:

Definitions
Manometers and barometers
Forces on submerged surfaces and the center of pressure
Buoyancy and Archimedes principle

Fluid properties: definitions

In fluid mechanics, you’ll use a few core properties again and again. Three of the most common are density, specific weight, and specific gravity. We’ll define each one and show how to use the key equations.

Density ( $ρ$ )

Definitions

Definition Density is the mass of a fluid per unit volume. It indicates how much matter is packed into a given volume.

Equation:

$ρ = \frac{m}{V}$

Where:

$ρ$ = density ( $k g / m^{3}$ or $s l ug s / f t^{3}$ )
$m$ = mass ( $k g$ or $s l ug s$ )
$V$ = volume ( $m^{3}$ or $f t^{3}$ )

Example:

The density of water at $4° C$ is:

$ρ_{w a t er} = 1000 k g / m^{3}$

Specific weight ( $γ$ )

Definitions

Specific weight is the weight of a fluid per unit volume. It is a force quantity and depends on the local gravitational acceleration.

Equation:

$γ = ρ g$

Where:

$γ$ = specific weight ( $N / m^{3}$ or $l b / f t^{3}$ )
$ρ$ = density ( $k g / m^{3}$ or $s l ug s / f t^{3}$ )
$g$ = acceleration due to gravity ( $9.81 m / s^{2}$ or $32.2 f t / s^{2}$ )

Example:

The specific weight of water at $4° C$ is:

$γ_{w a t er} = 1000 k g / m^{3} \times 9.81 m / s^{2} = 9810 N / m^{3}$

Specific gravity (SG)

Definitions

Specific gravity is the ratio of the density (or specific weight) of a fluid to the density (or specific weight) of a reference substance (usually water for liquids).

Equation (based on density):

$SG = \frac{ρ _{f l u i d}}{ρ _{w a t er}}$

Equation (based on specific weight):

$SG = \frac{γ _{f l u i d}}{γ _{w a t er}}$

Note: Since $g$ is constant, both definitions give the same value.

Example:

If a fluid has a density of $850 kg/m^{3}$ , then its specific gravity is:

$SG = \frac{850}{1000} = 0.85$

Pressure

Definitions

Pressure is defined as the normal force exerted per unit area on a surface. It is a scalar quantity and acts equally in all directions at a point in a fluid at rest.

Mathematical expression:

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

Where:

$P$ = Pressure ( $P a$ or $N / m^{2}$ )
$F$ = Normal force acting perpendicular to the surface ( $N$ )
$A$ = Area over which the force is applied ( $m^{2}$ )

Example:

Suppose a force of $100 N$ is applied on an area of $0.5 m^{2}$ . The pressure is:

$P = \frac{100}{0.5} = 200 Pa$

Viscosity

Definitions

Viscosity is the property of a fluid that resists the relative motion between adjacent layers. It is a measure of the internal friction within the fluid.

There are two types:

Dynamic (or absolute) viscosity ( $μ$ ) - measured in $P a \cdot s$ or $N \cdot s / m^{2}$ .
Kinematic viscosity ( $ν$ ) - the ratio of dynamic viscosity to density, measured in $m^{2} / s$ .

$ν = \frac{μ}{ρ}$

Where:

$ν$ = Kinematic viscosity
$μ$ = Dynamic viscosity
$ρ$ = Fluid density

Newton’s law of viscosity

Statement:

Newton’s law of viscosity states that the shear stress between adjacent fluid layers is proportional to the velocity gradient perpendicular to the direction of flow.

Mathematical expression:

$τ = μ \frac{d u}{d y}$

Where:

$τ$ = Shear stress ( $P a$ )
$μ$ = Dynamic viscosity ( $P a \cdot s$ )
$\frac{d u}{d y}$ = Velocity gradient ( $s^{- 1}$ )

This law applies to Newtonian fluids - fluids for which viscosity remains constant regardless of the applied shear rate (e.g., water, air, most common oils).

Example:

A fluid has a viscosity $μ = 0.02 P a \cdot s$ . If the velocity gradient $\frac{d u}{d y} = 100 s^{- 1}$ , then the shear stress is:

$τ = 0.02 \times 100 = 2 Pa$

Surface tension

Definitions

Surface tension is the property of a liquid that allows it to resist an external force, due to the cohesive nature of its molecules. Molecules at the surface experience a net inward force because they are not surrounded by similar molecules on all sides.

Mathematical expression:

Surface tension $σ$ is defined as the force per unit length acting along the surface of a liquid at rest:

$σ = \frac{F}{L}$

Where:

$F$ = Force acting along the surface ( $N$ )
$L$ = Length over which the force acts ( $m$ )
$σ$ = Surface tension ( $N / m$ )

Example:

Water droplets tend to form spherical shapes because surface tension minimizes the surface area.
A small insect like a water strider can walk on water due to the surface tension forming a ‘film’ at the surface.

Capillarity (capillary action)

Definitions

Capillarity is the rise or fall of a liquid in a small diameter tube (capillary tube) caused by the interaction of adhesive and cohesive forces.

Adhesion: Attraction between liquid and tube wall
Cohesion: Attraction between liquid molecules

Mathematical expression:

The height of capillary rise (or fall) $h$ is given by:

$h = \frac{2 σ cos θ}{ρ g r}$

Where:

$h$ = Capillary rise or fall (m)
$σ$ = Surface tension of the liquid (N/m)
$θ$ = Contact angle between liquid and solid surface
$ρ$ = Density of the liquid (kg/m^3) - $r$ = Radius of the capillary tube (m)

Example:

Water rises in a narrow glass tube due to strong adhesion (water to glass) and moderate cohesion (water to water).
Mercury falls in a glass tube due to strong cohesion (mercury to mercury) and weak adhesion (mercury to glass), forming a convex meniscus.

The pressure field in a static fluid

In a static fluid, pressure increases with depth. The pressure difference between two points is given by $P_{2} - P_{1} = - γh$ , where point $2$ is located a vertical distance $h$ above point $1$ , and $γ$ is the specific weight of the fluid.

Absolute pressure = atmospheric pressure + gauge pressure reading

Absolute pressure = atmospheric pressure - vacuum gauge pressure reading

Manometers and barometers

Manometer problem-solving procedure

The following is a general procedure for solving all manometer problems:

Start at one end (or any meniscus if the circuit is continuous) and write the pressure there in an appropriate unit or in an appropriate symbol if it is unknown.
Add the pressure change, in the same unit, from one meniscus to the next:
- Add if the next meniscus is lower.
- Subtract if the next meniscus is higher.
Continue until the other end of the gage (or the starting meniscus) is reached and equate the expression to the pressure at that point, whether known or unknown.

The expression will contain one unknown for a simple manometer or will give a difference in pressure for a differential manometer.

Example:

A U-tube manometer containing Hg (specific gravity = 13.6) has its right limb open to the atmosphere. The left limb is full of water and connected to a pipe containing water under pressure. Task: Find the pressure of water in the pipe above atmospheric pressure, given the manometer readings as shown in the figure.

A U-tube manometer setup showing the height difference of fluid columns used to measure pressure at point i based on the liquid level differences. — U-tube manometer

Given:

Height difference of mercury column: $h_{H g} = 10 c m$
Specific gravity of mercury: $S G_{H g} = 13.6$

Solution

Equating the pressure at points 2 and 3, we have:

$h_{1} \times 1 + 0.05 \times 1 = 0.15 \times 13.6$

or,

$h_{1} = 0.15 \times 13.6 - 0.05 \times 1 = 1.99 m of water$

Therefore,

$p_{1} = γ h_{1} = 9.81 \times 1.99 = 19.52 kN/m^{2}$

Another device that works on the same principle as the manometer is the simple barometer.

A simple mercury barometer showing how atmospheric pressure balances the weight of a liquid column to a height. — A simple barometer

$p_{a t m} = p_{A} = p_{v} + γh = p_{B} + γh$

where, $P_{v} = vapor pressure of the barometer fluid$

Forces on submerged surfaces and the center of pressure

A diagram illustrating hydrostatic forces on an inclined submerged surface and the location of the resulting center of pressure relative to the centroid. — Forces on a submerged surfaces and the center of pressure

The pressure at a point a vertical distance $h$ below the surface is:

$P = P_{atm} + γh$

where:

$P$ = pressure
$P_{a t m}$ = atmospheric pressure
$P_{C}$ = pressure at the centroid of area
$P_{CP}$ = pressure at the center of pressure
$y_{C}$ = slant distance from liquid surface to the centroid of area
$y_{C} = \frac{h _{C}}{s i n θ}$
$h_{C}$ = vertical distance from liquid surface to centroid of area
$y_{CP}$ = slant distance from liquid surface to center of pressure
$h_{CP}$ = vertical distance from liquid surface to center of pressure
$θ$ = angle between liquid surface and edge of submerged surface
$I_{x}$ = moment of inertia about the centroidal x-axis

If atmospheric pressure acts above the liquid surface and on the nonwetted side of the submerged surface:

$y_{CP} = y_{C} + \frac{I _{x}}{y _{C} A}$

or,

$y_{CP} = y_{C} + γ sin θ \cdot \frac{I _{x}}{P _{C} A}$

Wetted side force:

$F_{R} = (P_{atm} + γ y_{C} sin θ) A$

If $P_{a t m}$ acts on both sides:

$F_{n e t} = (γ y_{C} sin θ) A$

Buoyancy and Archimedes principle

The buoyant force exerted on a submerged or floating body is equal to the weight of the fluid displaced by the body.
A floating body displaces a weight of fluid equal to its own weight; i.e., a floating body is in equilibrium.
The center of buoyancy is located at the centroid of the displaced fluid volume.

$F_{B} = γ V_{f}$

where:

$F_{B}$ = buoyancy force ( $l b f$ )
$γ$ = specific weight of fluid ( $l b f / f t^{3}$ )
$V_{f}$ = volume of dissipated fluid ( $f t^{3}$ )

Fluid properties: definitions

Density ( $ρ$ ): mass per unit volume, $ρ = \frac{m}{V}$
Specific weight ( $γ$ ): weight per unit volume, $γ = ρ g$
Specific gravity (SG): ratio of fluid density to water density, $SG = \frac{ρ _{f l u i d}}{ρ _{w a t er}}$
Pressure ( $P$ ): normal force per unit area, $P = \frac{F}{A}$
Viscosity:
- Dynamic viscosity ( $μ$ ): internal friction, units $P a \cdot s$
- Kinematic viscosity ( $ν$ ): $ν = \frac{μ}{ρ}$
Newton’s law of viscosity: $τ = μ \frac{d u}{d y}$ (shear stress proportional to velocity gradient)
Surface tension ( $σ$ ): force per unit length at liquid surface, $σ = \frac{F}{L}$
Capillarity: rise/fall in tube, $h = \frac{2 σ c o s θ}{ρ g r}$
Pressure in static fluid: increases with depth, $P_{2} - P_{1} = - γh$
- Absolute pressure = atmospheric pressure + gauge pressure

Manometers and barometers

Manometer problem-solving:
- Start at one end, assign pressure
- Add pressure if moving down, subtract if up
- Continue until circuit closes, solve for unknown
U-tube manometer: measures pressure difference using fluid column heights and densities
Barometer: measures atmospheric pressure using height of liquid column, $p_{a t m} = p_{v} + γh$

Forces on submerged surfaces and the center of pressure

Pressure at depth: $P = P_{a t m} + γh$
Center of pressure (vertical distance): $y_{CP} = y_{C} + \frac{I _{x}}{y _{C} A}$
- $y_{C}$ : centroid distance; $I_{x}$ : moment of inertia; $A$ : area
Hydrostatic force on surface: $F_{R} = (P_{a t m} + γ y_{C} sin θ) A$
Net force (if $P_{a t m}$ acts both sides): $F_{n e t} = (γ y_{C} sin θ) A$

Buoyancy and Archimedes principle

Buoyant force equals weight of displaced fluid, $F_{B} = γ V_{f}$
Floating body: displaces fluid weight equal to its own weight (equilibrium)
Center of buoyancy: centroid of displaced fluid volume

Next | 8.2 Mass and energy conservation

Fluid statics

This chapter covers the following:

Definitions
Manometers and barometers
Forces on submerged surfaces and the center of pressure
Buoyancy and Archimedes principle