Fluid Mechanics/Fluid Properties

In addition to the properties like mass, velocity, and pressure usually considered in physical problems, the following are the basic properties of a fluid

Density

The density of a fluid, is generally designated by the Greek symbol $\rho$ (rho) is defined as the mass of the fluid over an infinitesimal volume. Density is expressed in the British Gravitational (BG) system as slugs/ft³, and in the SI system kg/m³.

$\rho =\lim _{\Delta V\rightarrow 0}{\frac {\Delta m}{\Delta V}}\,={\frac {dm}{dV}}$

If the fluid is assumed to be uniformly dense the formula may be simplified as:

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

Specific Weight (Weight Density)

The specific weight of a fluid is designated by the Greek symbol $\gamma$ (gamma), and is generally defined as the weight per unit volume. The units for gamma are lb/ft³ and N/m³ in the imperial and SI systems, respectively.

$\gamma =\rho$ * g

g = local acceleration of gravity and $\rho$ = density

Note: It is customary to use:

g = 32.174 ft/s² = 9.81 m/s²

$\rho$= 1000 kg/m³

Relative Density (Specific Gravity)

The relative density of any fluid is defined as the ratio of the density of that fluid to the density of the standard fluid. For liquids we take water as a standard fluid with density ρ=1000 kg/m³. For gases we take air or O₂ as a standard fluid with density, ρ=1.293 kg/m³.

Dynamic Viscosity

Viscosity is the property of fluid which defines the interaction between the moving particles of the fluid. It is the measure of resistance to the flow of fluids. The viscous force is due to the intermolecular forces acting in the fluid. The flow or rate of deformation of fluids under shear stress is different for different fluids due to the difference in viscosity. Fluids with high viscosity deform slowly.

Viscosity (represented by μ, Greek letter mu) is a material property, unique to fluids, that measures the fluid's resistance to flow. Though a property of the fluid, its effect is understood only when the fluid is in motion. When different elements move with different velocities, each element tries to drag its neighbouring elements along with it. Thus, shear stress occurs between fluid elements of different velocities.

Velocity gradient in laminar shear flow

The relationship between the shear stress and the velocity field was studied by Isaac Newton and he proposed that the shear stresses are directly proportional to the velocity gradient or we can say shear strain rate. $\tau =\mu {\frac {\partial u}{\partial y}}$ The constant of proportionality is called the coefficient of dynamic viscosity.

Another coefficient, known as the kinematic viscosity ($\nu$, Greek nu) is defined as the ratio of dynamic viscosity and density. I.e., $\nu =\mu /\rho$ It is the property of a fluid that quantifies resistance to flow of the fluid.

Dimensionless parameters

Dimensionless parameters are used to simplify analysis, and describe the physical situation without referring to units. A dimensionless quantity has no physical unit associated with it. They arose from dimensional analysis techniques. These numbers have many applications in fluid mechanics as well as in related subjects like aerodynamics and convective heat transfer.

Reynolds Number

Reynolds number (after Osborne Reynolds, 1842–1912) is used in the study of fluid flows. It compares the relative strength of inertial and viscous effects.

The value of the Reynolds number is defined as: $Re={\frac {\rho VL}{\mu }}={\frac {VL}{\nu }}$

where ρ(rho) is the density, μ(mu) is the absolute viscosity, V is the characteristic velocity of the flow, and L is the characteristic length for the flow.

Example 0.1: Reynold's number for flat plate flow

Air at 293K temperature, and 1.225 kg m−3 density is flowing past a flat plate at 1 m s−1. What's the Reynold's Number 1 m downstream from the leading the edge of the plate?

Absolute viscosity for air is 1.8 × 10-5 N s m−2. $Re={\frac {\rho VL}{\mu }}={\frac {1.225(1)(1)}{1.8E10^{-5}}}=68,055$

Additionally, we define a parameter ν(nu) as the kinematic viscosity.

Low Re indicates creeping flow, medium Re is laminar flow, and high Re indicates turbulent flow.

Reynolds number can also be transformed to take account of different flow conditions. For example the Reynolds number for flow within a pipe is expressed as

$Re={\frac {\rho ud}{\mu }}$

where u is the average fluid velocity within the pipe and d is the inside diameter of the pipe.

Application of dynamic forces (and the Reynolds number) to the real world: sky-diving, where friction forces equal the falling body's weight. (jjam)