Properties of Fluids

Properties of Fluids

$Learning Objectives$

Define a fluid and distinguish the principal mass, weight, viscous, interfacial, and compressibility properties used in hydraulics.
Relate density, specific weight, specific volume, and specific gravity using dimensionally consistent equations.
Apply Newton's law of viscosity to Newtonian fluids and distinguish dynamic from kinematic viscosity.
Explain surface tension through the Young–Laplace pressure jump and apply it to droplets, soap bubbles, liquid jets, and capillary action.
Distinguish vapor pressure from gage pressure and identify the absolute-pressure condition associated with cavitation.
Use bulk modulus and the ideal-gas equation of state with the correct pressure, temperature, and process assumptions.

$Fluid$

A fluid is a substance that deforms continuously while subjected to any sustained shear stress. Liquids and gases therefore cannot remain in static equilibrium while a finite shear stress persists.

$Why Fluid Properties Matter$

Hydraulic analysis repeatedly converts between mass, weight, pressure, shear resistance, interfacial forces, and compressibility. The governing property must match the physical mechanism: density controls mass per volume, specific weight incorporates gravity, viscosity controls viscous shear, surface tension controls interfacial curvature effects, and bulk modulus measures resistance to volumetric compression.

Fluid-property relationships

Map of density, viscosity, surface effects, and compressibility.

$\rho$

Density is mass per unit volume at the stated thermodynamic condition.

Density

Relates fluid mass to occupied volume.

\rho = \frac{m}{V}

Variables

Symbol	Description	Unit
$\rho$	Fluid density	$kg/m^3$
$m$	Fluid mass	kg
$V$	Fluid volume	$m^3$

$\gamma$

Specific weight is gravitational weight per unit volume at the stated local gravitational acceleration.

Specific Weight

Converts density to weight per unit volume.

\gamma = \rho g

Variables

Symbol	Description	Unit
$\gamma$	Fluid specific weight	$N/m^3$
$\rho$	Fluid density	$kg/m^3$
$g$	Local gravitational acceleration	$m/s^2$

$v$

Specific volume is volume per unit mass and is the reciprocal of density for a homogeneous fluid state.

Specific Volume

Relates occupied volume to fluid mass.

v = \frac{V}{m} = \frac{1}{\rho}

Variables

Symbol	Description	Unit
$v$	Specific volume	$m^3/kg$
$V$	Fluid volume	$m^3$
$m$	Fluid mass	kg
$\rho$	Fluid density	$kg/m^3$

$SG$

Specific gravity is the dimensionless ratio of a fluid density to a stated reference-fluid density. For liquids in elementary hydraulics, the reference is commonly water at a specified reference temperature.

Specific Gravity of a Liquid

Compares liquid density or specific weight with the corresponding water reference.

SG = \frac{\rho_{\text{fluid}}}{\rho_{\text{water}}} = \frac{\gamma_{\text{fluid}}}{\gamma_{\text{water}}}

Variables

Symbol	Description	Unit
$SG$	Specific gravity	dimensionless
$\rho_{\text{fluid}}$	Fluid density	$kg/m^3$
$\rho_{\text{water}}$	Reference-water density	$kg/m^3$
$\gamma_{\text{fluid}}$	Fluid specific weight	$N/m^3$
$\gamma_{\text{water}}$	Reference-water specific weight	$N/m^3$

$Reference Values and Temperature Dependence$

For many elementary SI calculations, water is approximated as $\rho \approx 1000\ \text{kg/m}^3$ and $\gamma \approx 9.81\ \text{kN/m}^3$ . These are convenient engineering reference values, not universal constants: density and viscosity vary with temperature, pressure, and composition.

$\mu$

Dynamic viscosity is the proportionality coefficient relating shear stress to velocity gradient for a Newtonian fluid at a specified thermodynamic state.

Newton's Law of Viscosity

Relates Newtonian shear stress to the local velocity gradient.

\tau = \mu \frac{du}{dy}

Variables

Symbol	Description	Unit
$\tau$	Shear stress	Pa
$\mu$	Dynamic viscosity	Pa·s
$du/dy$	Velocity gradient normal to the flow	1/s

Newtonian shear between layers

Velocity gradient and tangential shear in a fluid layer.

$Scope of Newton's Viscosity Law$

The linear relation $\tau=\mu\,du/dy$ applies directly to Newtonian fluids. For non-Newtonian fluids, apparent viscosity may depend on shear rate, time, or deformation history, so one constant $\mu$ does not describe the full constitutive behavior.

$\nu$

Kinematic viscosity is dynamic viscosity divided by density. It has dimensions of momentum diffusivity and does not itself represent a ratio of viscous force to inertial force.

Kinematic Viscosity

Normalizes dynamic viscosity by fluid density.

\nu = \frac{\mu}{\rho}

Variables

Symbol	Description	Unit
$\nu$	Kinematic viscosity	$m^2/s$
$\mu$	Dynamic viscosity	Pa·s
$\rho$	Fluid density	$kg/m^3$

$Common Viscosity Units$

$1\ \text{P}=0.1\ \text{Pa·s}$ , $1\ \text{cP}=10^{-3}\ \text{Pa·s}$ , $1\ \text{St}=10^{-4}\ \text{m}^2/\text{s}$ , and $1\ \text{cSt}=10^{-6}\ \text{m}^2/\text{s}$ . For liquids such as water, viscosity generally decreases as temperature rises over ordinary engineering ranges.

$Interactive Fluid-Property Exploration$

Use the simulation to compare how changes in fluid type and properties affect viscous and interfacial behavior. Treat displayed property values as model inputs and keep temperature dependence in mind when applying tabulated data.

Fluid Properties Explorer

Learning objective: Connect representative fluid properties to viscous shear, capillary response, and model-validity limits without treating illustrative material values as universal constants.

Representative properties near 20°C. Actual values vary with temperature, composition, and surface condition. Contact angle is an explicit modeling assumption for the glass-liquid pair.

Newtonian reference fluid

NIST-consistent reference values near 20 °C

Capillary diameter (

d

)2.0 mm

Moving-plate speed (

U

)0.50 m/s

Idealized equilibrium capillary rise

h=4\sigma\cos\theta/(\gamma d)

14.9 mm

Assumed contact angle, $\theta$: 0°
Capillary length, $\ell_c$: 2.73 mm
Bond number, $Bo=(r/\ell_c)^2$: 0.13

Jurin's law here assumes equilibrium in a circular vertical tube and neglects contamination, dynamic contact-angle hysteresis, evaporation, detailed meniscus-volume corrections, and finite-reservoir effects.

Newtonian Couette flow

Plate gap, $H$: 5.0 mm
Shear rate, $du/dy=U/H$: 100 s⁻¹
Shear stress, $\tau=\mu U/H$: 0.100 Pa
Plane-Couette $Re_C=\rho(U/2)(H/2)/\mu$: 623

Transition caution: $Re_C$ is at or above the experimentally observed sustained-turbulence onset of about 360 ± 10 for plane Couette flow. The displayed straight-line profile and $\tau=\mu U/H$ remain the laminar solution, but a sufficiently disturbed physical flow may not remain laminar.

The relation assumes steady, laminar, isothermal flow of a Newtonian fluid, no slip at both plates, negligible pressure gradient, and a uniform 5 mm gap. The Reynolds-number convention uses half the wall-speed difference and half the plate gap.

Density

998 kg/m³

Specific weight

9.79 kN/m³

Dynamic viscosity

0.001002 Pa·s

Kinematic viscosity

1.00e-6 m²/s

Surface tension

0.0728 N/m

Contact angle

0° assumed

$\sigma$

Surface tension is the tangential interfacial force per unit length associated with molecular cohesion at a fluid interface. It is equivalently surface energy per unit area for a reversible change in interface area.

Young–Laplace Pressure Jump

Relates pressure difference across a curved interface to its principal radii of curvature.

\Delta p = \sigma\left(\frac{1}{R_1}+\frac{1}{R_2}\right)

Variables

Symbol	Description	Unit
$\Delta p$	Pressure on the concave side minus pressure on the convex side	Pa
$\sigma$	Surface tension	N/m
$R_1$	First principal radius of curvature	m
$R_2$	Second principal radius of curvature	m

$Droplets, Soap Bubbles, and Cylindrical Jets$

For a spherical liquid droplet, $R_1=R_2=R$ , so a single interface gives $\Delta p=2\sigma/R$ . A soap bubble has two liquid-gas interfaces, so the total pressure jump is $4\sigma/R$ when both interfaces have nearly the same radius. For a cylindrical liquid jet, one principal radius is $R$ and the other is effectively infinite, giving $\Delta p=\sigma/R$ .

Common Surface-Tension Pressure Jumps

Special cases of the Young–Laplace relation used in elementary hydraulics.

\Delta p_{\text{droplet}}=\frac{2\sigma}{R}, \qquad \Delta p_{\text{soap bubble}}=\frac{4\sigma}{R}, \qquad \Delta p_{\text{jet}}=\frac{\sigma}{R}

Variables

Symbol	Description	Unit
$R$	Droplet, bubble, or jet radius	m
$\sigma$	Surface tension	N/m
$\Delta p$	Excess pressure associated with curvature	Pa

$h$

Capillary rise or depression is the change in liquid level inside a small tube caused by surface tension and the contact angle between the liquid interface and the solid wall.

Capillary Rise or Depression in a Circular Tube

Balances the vertical surface-tension force with the weight of the displaced liquid column.

h = \frac{4\sigma\cos\theta}{\gamma d}

Variables

Symbol	Description	Unit
$h$	Signed capillary height relative to the outside free surface	m
$\sigma$	Surface tension	N/m
$\theta$	Contact angle measured through the liquid	degrees or rad
$\gamma$	Liquid specific weight	$N/m^3$
$d$	Inside tube diameter	m

Capillary rise and meniscus

Surface tension, contact angle, and capillary height.

$Capillary Sign Convention$

With the contact angle measured through the liquid, $\theta<90^\circ$ gives $\cos\theta>0$ and a rise, whereas $\theta>90^\circ$ gives $\cos\theta<0$ and a depression. Use the actual liquid-solid contact angle when it is provided rather than assuming perfect wetting.

$p_v$

Vapor pressure is the saturation pressure of a substance at a specified temperature when its liquid and vapor phases are in equilibrium.

$Cavitation Uses Absolute Pressure$

Cavitation can begin when the local absolute liquid pressure falls to or below the saturation vapor pressure at the local temperature. A negative gage pressure does not by itself establish cavitation because gage pressure is referenced to the local atmosphere, not to a vacuum.

$K$

Bulk modulus is a positive measure of resistance to volumetric compression. It relates an increase in pressure to the corresponding fractional decrease in volume, or equivalently to the fractional increase in density.

Differential Bulk Modulus

Defines local volumetric stiffness for a compressible fluid.

K=-V\frac{dp}{dV}=\rho\frac{dp}{d\rho}

Variables

Symbol	Description	Unit
$K$	Bulk modulus	Pa
$p$	Pressure	Pa
$V$	Fluid volume	$m^3$
$\rho$	Fluid density	$kg/m^3$

Small Finite Compression Approximation

Estimates a small volume change when bulk modulus is approximately constant over the pressure interval.

K \approx -\frac{\Delta p}{\Delta V/V_1}

Variables

Symbol	Description	Unit
$\Delta p$	Pressure increase over the interval	Pa
$\Delta V$	Final volume minus initial volume	$m^3$
$V_1$	Reference initial volume	$m^3$

$Liquid Compressibility in Engineering Models$

Water near ordinary ambient conditions has a bulk modulus of roughly $2.2\ \text{GPa}$ , although the value varies with temperature and pressure. Treating a liquid as incompressible is often appropriate for steady-flow calculations, but compressibility becomes important in transients such as water hammer.

Fluid compression and bulk modulus

Pressure increase associated with a small volume reduction.

$Ideal-Gas Equation of State$

For an ideal gas, absolute pressure, density, specific gas constant, and absolute temperature are related by the equation of state $p=\rho RT$ .

Ideal-Gas Equation of State

Relates ideal-gas pressure, density, and absolute temperature.

p = \rho R T

Variables

Symbol	Description	Unit
$p$	Absolute gas pressure	Pa
$\rho$	Gas density	$kg/m^3$
$R$	Specific gas constant	J/(kg·K)
$T$	Absolute temperature	K

$Absolute Quantities in Gas Calculations$

Use absolute pressure and absolute temperature in $p=\rho RT$ . For dry air, $R\approx287\ \text{J/(kg·K)}$ . A Celsius temperature or gage pressure cannot be substituted directly into the ideal-gas equation.

$Bulk Modulus of an Ideal Gas$

For an ideal gas, the effective bulk modulus depends on the compression process. Isothermal compression gives $K=p$ , while a reversible adiabatic (isentropic) process gives $K=kp$ , where $k=c_p/c_v$ is the specific-heat ratio. The pressure in these expressions is absolute.

Key Takeaways

Density is mass per unit volume; specific weight is density multiplied by local gravitational acceleration; specific volume is the reciprocal of density.
Specific gravity is a dimensionless comparison with a stated reference fluid and reference condition.
Dynamic viscosity controls Newtonian shear stress, while kinematic viscosity is $\mu/\rho$ and has dimensions of momentum diffusivity.
Surface-tension pressure jumps follow interface curvature; droplets, soap bubbles, and cylindrical jets are special cases of the Young–Laplace relation.
Capillary rise or depression depends on surface tension, tube diameter, specific weight, and the contact angle measured through the liquid.
Vapor-pressure and cavitation checks require absolute pressure at the relevant temperature.
Bulk modulus measures volumetric stiffness; finite-difference formulas are approximations over a stated pressure interval.
Ideal-gas calculations require absolute pressure and absolute temperature, and gas bulk modulus depends on the thermodynamic process.

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Learning Objectives

Fluid

Why Fluid Properties Matter

Density, ρ\rhoρ

Density

Specific Weight, γ\gammaγ

Specific Weight

Specific Volume, vvv

Specific Volume

Specific Gravity, SGSGSG

Specific Gravity of a Liquid

Reference Values and Temperature Dependence

Dynamic Viscosity, μ\muμ

Newton's Law of Viscosity

Scope of Newton's Viscosity Law

Kinematic Viscosity, ν\nuν

Kinematic Viscosity

Common Viscosity Units

Interactive Fluid-Property Exploration

Fluid Properties Explorer

Idealized equilibrium capillary rise

Newtonian Couette flow

Surface Tension, σ\sigmaσ

Young–Laplace Pressure Jump

Droplets, Soap Bubbles, and Cylindrical Jets

Common Surface-Tension Pressure Jumps

Capillary Rise or Depression, hhh

Capillary Rise or Depression in a Circular Tube

Capillary Sign Convention

Vapor Pressure, pvp_vpv​

Cavitation Uses Absolute Pressure

Bulk Modulus, KKK

Differential Bulk Modulus

Small Finite Compression Approximation

Liquid Compressibility in Engineering Models

Ideal-Gas Equation of State

Ideal-Gas Equation of State

Absolute Quantities in Gas Calculations

Bulk Modulus of an Ideal Gas

$Learning Objectives$

$Fluid$

$Why Fluid Properties Matter$

$\rho$

$\gamma$

$v$

$SG$

$Reference Values and Temperature Dependence$

$\mu$

$Scope of Newton's Viscosity Law$

$\nu$

$Common Viscosity Units$

$Interactive Fluid-Property Exploration$

$\sigma$

$Droplets, Soap Bubbles, and Cylindrical Jets$

$h$

$Capillary Sign Convention$

$p_v$

$Cavitation Uses Absolute Pressure$

$K$

$Liquid Compressibility in Engineering Models$

$Ideal-Gas Equation of State$

$Absolute Quantities in Gas Calculations$

$Bulk Modulus of an Ideal Gas$