💧Fluid Mechanics Unit 2 Review

Fluid properties like density, viscosity, and specific gravity are the foundation of everything in fluid mechanics. You need these to predict how fluids behave under different conditions, whether you're sizing a pipe, analyzing drag on a body, or calculating pressure distributions. This section covers the core physical properties, the distinction between ideal and real fluids, and how temperature and pressure change things.

Density and Specific Properties

Density ( $\rho$ ) is mass per unit volume. It tells you how much "stuff" is packed into a given space of fluid.

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

Units: $\frac{kg}{m^3}$ (SI), $\frac{slug}{ft^3}$ (English)
Water at 4°C has a density of 1000 $\frac{kg}{m^3}$ . Air at standard conditions is only 1.225 $\frac{kg}{m^3}$ , roughly 800 times less dense than water.

Specific weight ( $\gamma$ ) is weight per unit volume. It connects density to gravitational force:

$\gamma = \rho g$

Units: $\frac{N}{m^3}$ (SI), $\frac{lb}{ft^3}$ (English)
Water at 4°C has a specific weight of 9810 $\frac{N}{m^3}$ . Mercury is much heavier at 133,100 $\frac{N}{m^3}$ , which is why it's used in manometers.

Specific gravity (SG) is the ratio of a substance's density to the density of a reference substance, usually water at 4°C:

$SG = \frac{\rho_{substance}}{\rho_{reference}}$

Dimensionless (no units).
SG < 1 means the substance is lighter than water and will float. SG > 1 means it's heavier and will sink.
Typical values: oil is around 0.8, glycerin is about 1.26, mercury is 13.6.

Density and specific properties, Density and its uses

Ideal vs. Real Fluids

The distinction between ideal and real fluids is about simplifying assumptions. Ideal fluid models strip away complexity so you can focus on the core physics. Real fluid behavior is what you actually encounter in practice.

Ideal fluids are assumed to be:

Incompressible: constant density regardless of pressure changes
Inviscid: zero viscosity, meaning no resistance to shear stress
Non-conducting: no thermal conductivity, so heat transfer within the fluid is ignored

These assumptions are useful for problems like potential flow around an airfoil or inviscid flow through a nozzle, where viscous effects are small compared to pressure and inertia forces.

Real fluids have:

Compressibility: density changes with pressure, especially important for gases at high speeds
Viscosity: resistance to flow that causes energy dissipation (think of stirring honey vs. water)
Thermal conductivity: heat can transfer within the fluid

Every engineering fluid is a real fluid. Air flowing over an aircraft wing, water moving through a pipe, oil lubricating a bearing: all of these require accounting for viscosity and, in some cases, compressibility.

Density and specific properties, Specific Heat | Boundless Physics

Viscosity in Fluid Mechanics

Viscosity ( $\mu$ ) quantifies a fluid's resistance to deformation under shear stress. A fluid with high viscosity (like honey) resists flowing; a fluid with low viscosity (like water) flows easily.

Dynamic (absolute) viscosity relates shear stress to the velocity gradient in a fluid. This relationship is Newton's law of viscosity:

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

where $\tau$ is the shear stress, $\mu$ is the dynamic viscosity, and $\frac{du}{dy}$ is the velocity gradient (how quickly the fluid velocity changes as you move perpendicular to the flow direction).

Units: $Pa \cdot s$ (SI), $\frac{lb \cdot s}{ft^2}$ (English)
Water at 20°C has a dynamic viscosity of about 1.002 mPa·s. Honey at room temperature is roughly 10,000 mPa·s, about 10,000 times more viscous.

Kinematic viscosity ( $\nu$ ) is dynamic viscosity divided by density:

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

Units: $\frac{m^2}{s}$ (SI), $\frac{ft^2}{s}$ (English)
Air at standard conditions has a kinematic viscosity of about $1.46 \times 10^{-5}$ $\frac{m^2}{s}$

Kinematic viscosity shows up frequently because it combines the effects of viscosity and density into a single quantity. It appears directly in the Reynolds number ( $Re = \frac{VL}{\nu}$ ), which determines whether flow is laminar or turbulent.

Why viscosity matters across fluid mechanics:

It drives boundary layer formation near solid surfaces, where the fluid velocity transitions from zero at the wall to the freestream value
It determines the flow regime (laminar vs. turbulent) through the Reynolds number
It controls pressure drop in pipes and ducts (described by the Hagen-Poiseuille equation for laminar flow)
It influences heat transfer in fluids through the Prandtl number ( $Pr = \frac{\nu}{\alpha}$ )

Temperature and Pressure Effects

Temperature and pressure both change fluid properties, but their effects differ significantly between liquids and gases.

Temperature effects on density:

Liquids: Density decreases slightly with increasing temperature due to thermal expansion. The change is relatively small for most engineering problems.
Gases: Density decreases with increasing temperature according to the ideal gas law: $\rho = \frac{P}{RT}$ . At constant pressure, doubling the absolute temperature cuts the density in half.

Temperature effects on viscosity (this is a common exam topic because liquids and gases behave oppositely):

Liquids: Viscosity decreases with increasing temperature. As temperature rises, molecular cohesion weakens and the fluid flows more easily. Think of how honey pours faster when warmed.
Gases: Viscosity increases with increasing temperature. Higher temperatures mean more molecular agitation and more momentum transfer between gas layers.

Pressure effects on density:

Liquids: Only a very slight increase in density with increasing pressure. Liquids are nearly incompressible, so for most problems you can treat liquid density as constant with respect to pressure.
Gases: Density increases directly with increasing pressure, again following $\rho = \frac{P}{RT}$ . At constant temperature, doubling the pressure doubles the density.

Pressure effects on viscosity:

Liquids: Negligible for most engineering applications. Liquid molecules are already closely packed, so moderate pressure changes don't significantly alter viscous behavior.
Gases: Viscosity is essentially independent of pressure for most engineering conditions. This holds true up to about 10 atmospheres; beyond that, pressure effects can become relevant.

💧Fluid Mechanics Unit 2 Review