Properties of Fluids

Fluid mechanics equations helps to predict the behavior of fluids in various flow situations. A fluid can be defined as a substance that deforms continuously under an applied shear stress.

Several properties need to be provided in order to use these equations. Some of these properties include: viscosity, pressure, density, kinematic viscosity, surface tension, specific heat, internal energy, enthalpy, and compressibility.

a) Viscosity

This is one of the most important fluid properties. Viscosity can be defined as the measure of resistance a fluid has to an externally applied shear stress.

Let us consider a fluid in between two horizontal parallel plates as shown below:

The upper plate has a contact area, A, with the fluid and is being pulled to the right by a force, F₁ at a velocity, V₁. The measurement of velocity at infinitesimally small distances along the fluid results in a velocity distribution graph as illustrated in the figure above. Since the fluid near the moving surface adheres to it, the velocity of the surface (x=Δy) is V₁. This is referred to as the nonslip condition. When x=0 (at the stationary plate) velocity is zero (again due to nonslip condition). The slope of the velocity distribution is dV₁/dy.

If a different force F₂ is applied on this system, a different slope (strain rate) of dV₂/dy is obtained. Thus, we can summarize that each applied force yields to only one shear stress and one strain rate.

Plotting τ vs dV/dy for a fluid like water results in the following graph:

As can be seen, the points lie on a straight line that passes through the origin and the slope of this line is the viscosity of the fluid as it is a measure of the fluid’s resistance to shear. i.e., viscosity indicates how a fluid will react (dV/dy) under the action of an external shear stress (τ). The above graph is characteristic of a Newtonian fluid. A Newtonian fluid is simply a fluid whose viscosity is not affected by shear rate at constant temperature. Water, oil, and air are examples of Newtonian fluids.

Since there are also non-Newtonian fluids as well, if we plot τ vs dV/dy we obtain the following rheological diagram.

Newtonian fluids obey Newton’s law of viscosity that is represented as the equation below, any fluid that doesn’t comply with this equation can be classified as a non-Newtonian fluid.

Where:

• τ = the applied shear stress (lbf=ft² or N=m²)
• μ = the absolute or dynamic viscosity of the fluid (lbf.s/ft² or N.s/m²)
• dV/dy = the strain rate (rad/s)

The dynamic viscosity is simply the resistance to movement of one fluid layer over another.

Non-Newtonian fluids are divided into three categories: time-independent, time-dependent, and viscoelastic.

    (i) Time-Independent Fluids

These fluids are divided into dilatant, pseudoplastic and Bingham plastics.

Dilatant fluids exhibit an increase in viscosity with an increase in shear stress. Examples include wet beach sand and other water solutions containing a high concentration of powder.  

A power law equation (called the Ostwald–deWaele equation) usually gives an adequate description:

where: K is called a consistency index (lbf.sⁿ/ft² or N.sⁿ/m²) and n is a flow behavior index.

Pseudoplastic fluids exhibit a decrease in viscosity with an increase in shear stress. Examples include mayonnaise, greases and starch suspensions. A power law equation also applies:

Bingham plastic fluids behave as solids until an initial yield stress τ₀ is exceeded. Beyond τ₀, Bingham plastics behave like Newtonian fluids. Examples include toothpaste, soap, paint, chocolate mixtures and paper pulp. The descriptive equation is:

   (ii) Time-Dependent Fluids

These are divided into rheopectic and thixotropic fluids.

In a rheopectic fluid, a shear stress that increases with time gives the rheopectic fluid a constant strain rate.

Thixotropic fluid behaves opposite to rheopectic fluids i.e., a shear stress that decreases with time gives a thixotropic fluid a constant strain rate.

   (iii) Viscoelastic Fluids

These are fluids that show both elastic and viscous properties, they partly recover elastically from deformations cased during flow. Examples include flour dough.

b) Pressure

Pressure is defined as a normal force per unit area existing in the fluid with units lbf/ft² or N/m². Fluids, whether at rest or moving, exhibit some type of pressure variation either with height or with horizontal distance.

In a closed-conduit flow like flow in pipes, differences in pressure throughout the conduit maintain the flow. Pressure forces are significant in this regard.

c) Density

The density of a fluid is its mass per unit volume, represented as ρ.

One quantity of importance related to density is specific weight. Whereas density is mass per unit volume, specific weight is weight per unit volume (lbf/ft³ or N/m³). Specific weight is related to density by:

SW = ρg

Another useful quantity related to density of a substance is specific gravity. The specific gravity of a substance is the ratio of its density to the density of water at 4°C:

d) Kinematic Viscosity

Kinematic viscosity, ν, with the units ft²/s or m²/s is the ratio of absolute viscosity to density:

e) Specific Heat

The specific heat of a substance is the heat required to raise a unit mass of the substance by 1°. The dimension of specific heat is energy/(mass.temperature).

The process by which the heat is added also makes a difference, especially for gases. The specific heat for a gas undergoing a process occurring at constant pressure involves a different specific heat than that for a constant volume process.

f) Internal Energy

This is the energy associated with the motion of the molecules of a substance. For example, a gas can have three types of energy:

• energy of position (potential energy),
• energy of translation (kinetic energy) and
• energy of molecular motion (internal energy).

The added heat does not increase the potential or kinetic energies but instead affects the motion of the molecules. This results in an increase in temperature. For a perfect gas with constant specific heats, internal energy is:

ΔU = c_vΔT

where ΔU is a change in internal energy per unit mass with dimensions of energy/mass. 

g) Enthalpy 

This is the total heat energy in a system equivalent to the sum of total internal energy and resulting energy due to its pressure and volume. For a perfect gas with constant specific heats, it can be shown that:

ΔH = c_pΔT

Similar to internal energy, enthalpy has the dimension of energy/mass.

h) Surface Tension

This is a measure of the energy required to reach below the surface of a liquid bulk and bring molecules to the surface to form a new area. Therefore, surface tension arises from molecular considerations and applies only for liquid–gas or liquid–vapor interfaces.

For a liquid-gas interface, attraction of molecules is due to cohesive forces. If the liquid is in contact with a solid, adhesive forces can also be considered. Remember, cohesive forces are the forces that attract molecules of the same substance whereas, adhesive forces binds molecules of different substances together.

The dimension of surface tension is energy/area (lbf/ft or N/m) and is generally denoted by the symbol, σ.

i) Bulk Modulus or Compressibility Factor

All materials, whether solids, liquids or gases, are compressible. This means that for a given volume, V of a given mass, will be reduced to V – dV when a force is exerted uniformly over its entire surface. If the force per unit area of surface increases from P to P + dP, the relationship between change of pressure and change of volume depends on the bulk modulus of the material.

Compressibility factor is expressed as follows:

The isothermal bulk modulus is the reciprocal of the compressibility factor and expressed as follows:

Since the dimensions of volume change and volume are the same, the denominator of the above equation is dimensionless. Thus the isothermal bulk modulus has the same units as pressure.

The isothermal bulk modulus can be measured experimentally, but it is more convenient to calculate it using data on the velocity of sound in the medium. The sonic velocity in a liquid or a solid is related to the isothermal bulk modulus and the density by the following equation: