FLUID MECHANICS : Fluid Properties Notes

By Deepanshu Rastogi|Updated : April 1st, 2021

Fluid Properties

Basic Concept

• A substance in the liquid / gas phase is referred to as ‘fluid’.
• The distinction between a solid & fluid is made on the basis of the substance’s ability to resist an applied shear (tangential) stress that tends to change its shape. A solid can resist an applied shear by deforming its shape whereas a fluid deforms continuously under the influence of shear stress, no matter how small is its shape. In solids, stress is proportional to strain, but in fluids, stress is proportional to ‘strain rate.’

Illustration of solid and fluid deformation

Referring to Fig., the shear modulus of solid (S ) and coefficient of viscosity (µ ) for fluid can defined in the following manner;

Here, the shear force (F ) is acting on the certain cross-sectional area ( A),

h is the height of the solid block / height between two adjacent layer of the fluid element,

∆x is the elongation of the solid block and ∆u is the velocity gradient between two adjacent layers of the fluid.

• So, a Fluid is a substance which deforms continuously, or flows, when subjected to shearing forces.
• If a fluid is at rest there are no shearing forces acting. All forces must be perpendicular to the planes which they are acting.
• Fluid can be treated as continuum and the properties at any point can be treated as bulk behavior of the fluids.

Newton’s Law of Viscosity

• The shearing force F acts on the area on the top of the element. This area is given by A = δz × δx . We can thus calculate the shear stress which is equal to force per unit area i.e.

shear stress, τ = F/A

• The deformation which this shear stress causes is measured by the size of the angle φ and is know as shear strain.
• In a solid shear strain, φ, is constant for a fixed shear stress τ.
• In a fluid φ increases for as long as τ is applied - the fluid flows.
• If the particle at point E (in the above figure) moves under the shear stress to point E’ and it takes time t to get there, it has moved the distance x. For small deformations we can write

shear strain, φ = x/y

rate of shear strain =φ/t = x/ty = x/t.1/y = u/y

where x/t = u is the velocity of the particle at E

Using the experimental result that shear stress is proportional to rate of shear strain then

τ = Constant × u/y

The term u/y is the change in velocity with y, or the velocity gradient, and may be written in the differential form du/dy . The constant of proportionality is known as the dynamic viscosity, µ , of the fluid, giving

Fluids vs. Solids

• For a solid the strain is a function of the applied stress (providing that the elastic limit has not been reached). For a fluid, the rate of strain is proportional to the applied stress.
• The strain in a solid is independent of the time over which the force is applied and (if the elastic limit is not reached) the deformation disappears when the force is removed. A fluid continues to flow for as long as the force is applied and will not recover its original form when the force is removed.

Newtonian / Non-Newtonian Fluids

• Fluids obeying Newton’s law where the value of µ is constant are known as Newtonian fluids. If µ is constant the shear stress is linearly dependent on velocity gradient. This is true for most common fluids.
• Fluids in which the value of µ is not constant are known as non-Newtonian fluids.

Other types of Fluids

• There are several categories of these, and they are outlined briefly below. These categories are based on the relationship between shear stress and the velocity gradient (rate of shear strain) in the fluid. These relationships can be seen in the graph below for several categories.

Below are brief description of the physical properties of the several categories

1. Plastic: Shear stress must reach a certain minimum before flow commences
2. Bingham plastic: As with the plastic above a minimum shear stress must be achieved. With this classification n = 1. An example is sewage sludge.
3. Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with rate of shear, e.g. colloidial substances like clay, milk, quicksand and cement.
4. Dilatant substances; Viscosity increases with rate of shear e.g. cornflour, printing inks and vinyl resin pastes.
5. Thixotropic substances: Viscosity decreases with length of time shear force is applied e.g. thixotropic jelly paints.
6. Rheopectic substances: Viscosity increases with length of time shear force is applied
7. Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic.
8. There is also one more - which is not real, it does not exist - known as the ideal fluid. This is a fluid which is assumed to have no viscosity.

Properties of Fluid

• Any characteristic of a system is called property. It may either be intensive (mass independent) or extensive (that depends on size of system). The state of a system is described by its properties. Most common properties of the fluid are:
1. Pressure ( p): It is the normal force exerted by a fluid per unit area. More details will be available in the subsequent section (Lecture 02). In SI system the unit and dimension of pressure can be written as, N/m2 and ML-1 T-2 , respectively.
2. Density: The density of a substance is the quantity of matter contained in unit volume of the substance.

It is expressed in three different ways; mass density (ρ = mass/volume),

specific weight(ρg ) and relative density/specific gravity water SG = ρ/ρwater

The units and dimensions are given as, For mass density; Dimension: M L-3 Unit: kg/m3

For specific weight; Dimension: ML-2 T-2 Unit: N/m3

The standard values for the density of water and air are given as 1000kg/m3 and 1.2 kg/m3 , respectively. Many a times the reciprocal of mass density is called as specific volume ( v ).

3. Temperature (T ): It is the measure of hotness and coldness of a system. In thermodynamic sense, it is the measure of internal energy of a system. Many a times, the temperature is expressed in centigrade scale (°C) where the freezing and boiling point of water is taken as 0°C and 100°C, respectively. In SI system, the temperature is expressed in terms of absolute value in Kelvin scale (K = °C+ 273).

4. Viscosity: Viscosity is a measure of a fluid’s resistance to flow. It determines the fluid strain rate that is generated by a given applied shear stress.

Velocity profile and shear stress

• A Newtonian fluid has a linear relationship between shear stress and velocity gradient:

• The shear stress is proportional to the slope of the velocity profile and is greatest at the wall.
• The no‐slip condition: at the wall velocity is zero relative to the wall. This is a characteristic of all viscous fluid.
• The linearity coefficient in the equation is the coefficient of viscosity, µ(Ns/m2), We can also use the kinematic viscosity ν(m2/s) =µ/ρ
• Temperature has a strong and pressure has a moderate effect on viscosity. The viscosity of gases and most liquids increases slowly with pressure.
• Gas viscosity increases with temperature. Two common approximations are the power law and the Sutherland law
• Liquid viscosity decreases with temperature and is roughly exponential.

5. Thermal Conductivity(k ): It relates the rate of heat flow per unit area (q)to the temperature gradient dT/dx and is governed by Fourier Law of heat conduction i.e.

q = -k.dT/dx

In SI system the unit and dimension of pressure can be written as, W/m.K and MLT-3 θ-1 , respectively

6. Surface Tension:

When a liquid and gas or two immiscible liquids are in contact, an unbalanced force is developed at the interface stretched over the entire fluid mass. The intensity of molecular attraction per unit length along any line in the surface is called as surface tension. For example, in a spherical liquid droplet of radius (r), the pressure difference (∆p) between the inside and outside surface of the droplet is given by,

∆p = 2 σ/ r

Reason: - Cohesive force b/w molecules.

Definition: - Force required to maintain unit length of the film in equilibrium, means force per unit length

Unit:- (N/m)

→ Due to surface tension

Increasing internal pressure of droplet.

The tendency of liquid droplet to attain minimum surface area at a given volume, only for this reason, shape of droplet is “Sphere”.

NOTE:-

Minimum surface area at a given volume = surface area of sphere.

Dependency of surface tension:-

Temperature:-

If temperature increases, cohesive force decreases and this will results in decrease in surface tension

If continuous decreasing in temperature takes place than surface tension becomes zero at “critical point of temperature”.

Surfactants:-

→ Reduce the surface tension

Ex. Organic solute

Some salt [NaCl] increase the surface tension

Curved surface indicate pressure difference (mean pressure jump)
Pressure higher on concave side (in given figure)

Pressure difference between pi and p0 for

Soap bubble:

Water Droplet:

Where σ is surface tension and R is the radius of curvature for bubble or droplets.

7. Capillary action: Capillary action (sometimes capillarity, capillary motion, or wicking) is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. It occurs because of intermolecular forces between the liquid and surrounding solid surfaces. If the diameter of the tube is sufficiently small, then the combination of surface tension (which is caused by cohesion within the liquid) and adhesive forces between the liquid and container wall act to propel the liquid.

Capillary action of water compared to mercury, in each case with respect to a polar surface such as glass

Capillary Effect:

Reason:- Cohesive force or surface tension and Adhesive forces. (Both force responsible for Capillary effect)

• Curved free surface inside the capillaries is called meniscus.
• Rise or fall of liquid inside the tube is due to contact angle b/w liquid surface and capillary tube.

NOTE:
if  then

• Level of liquid inside the tube is rise
• Liquid is known as Wetting liquid
• In this case: cohesive force >adhesive force

If  then

• Level of liquid fall inside the tube
• liquid is known as Non-wetting liquid
• In this case:-

Angle b/w tangent to the liquid surface and solid surface at the contact point.

Height of a meniscus

The height h of a liquid column is given by

where γ is the liquid-air surface tension (force/unit length), θ is the contact angle, ρ is the density of liquid (mass/volume), g is the local acceleration due to gravity (length/square of time), and r is the radius of tube.

Thus the thinner the space in which the water can travel, the further up it goes.

Observations

• For water –glass interface
So  this results in

• Height of capillary rise is a function of

• If diameter of tube > 1 cm than Capillary effect negligible

8. Bulk Modulus of Elasticity:

• Compressibility of liquid is measured by bulk modulus of elasticity.
• Bulk modulus is represented the compressive stress per unit volumetric strain.
• Bulk modules  k
• K → always positive or is a positive quantity having unit of pressure .
• Truly incompressible substance: means
So, K (bulk modulus) = ∞

Note:

K increase means Resistance to further compression increases.

• For liquid K increases with decreases in temperature: with decrease in temperature cohesive force between molecules increases, which results in higher resistance to further compression.
• For gases K increases with increases in temperature: With increase in temperature, collision between gas particle increases and results in higher internal pressure so the resistance to further compression increases.

9. Vapour Pressure and cavitation:

• Saturation Temperature:
For a given pressure, the temperature at which a pure substance changes phase is known as saturation temperature
• Saturation Pressure:
At a given temperature, the pressure at which a pure substance changes phase.

Example: at 1 atm pressure (const. pressure) saturation temperature is 100 c and at constant temp. 100 c saturation pressure for water is 1 atm.

Vapour Pressure:

• For liquid, pressure exerted by its vapour, in phase equilibrium with its liquid at a given temperature
• Vapour pressure increases [with temperature with increases and rate molecules escaping liquid surface increasing
• When vapour pressure equal to pressure on the liquid – boiling occur.

Cavitation:

• Cavitation is a phenomenon which occurs in a liquid flow system.
• If liquid undergo pressure below vapour pressure during flow, than sudden vaporization takes place
• Vapour bubbles collapse as they are swept from the low pressure region, generating highly destructive pressure waves.
• Cavitation can also occur if a liquid contains dissolved air or other gases, {Reason-Solubilites Decrease with decreasing pressure}
• Risk of cavitation is greater at higher temperature.

Example: Given a flow system (water) and Temperature is 36 c. Find the minimum pressure to avoid cavitation?

Solution: Minimum pressure to avoid cavitation is equal to vapour pressure of that liquid at given temperature for water

Note:

1. Partial pressure is the pressure exerted by a component in a mixture of gases.
2. for pure substance vapour pressure and saturation pressure both are equal.
3. If external pressure is equal to or less than the vapour pressure, boiling of liquid will start no matter how much temperature.

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