GATE Mechanical involves a very important subject of Fluid mechanics, which can be a scoring subject provided students study it well and thoroughly. In this set of notes, we will look at the basic properties of fluids, which is the foundation to learn the subject of fluid mechanics for GATE Mechanical and UPSC ESE mechanical engineering.
Fluid mechanics is the study of fluids either in motion or at rest, and the subsequent effects of the fluids upon the boundary conditions. Both gases and liquids are classified as fluids.
The study of fluid properties involves the basic understanding of flow or static condition of fluids. The important properties are:
- Density (ρ)
- Viscosity ()
- Surface tension (σ)
- Bulk modulus (K)
- Vapour pressure (Pv)
From the point of view of fluid mechanics, all matter consists of only two states, i.e. fluid and solid. The main distinction between a solid and a fluid lies in their respective reaction towards applied shear and tangential stress.
Points to remember:
- A solid can resist shearing by static deformation, but a fluid cannot do so. Even on applying a very small amount of shear on the fluid, the result will be fluid motion, and this motion and deformation will continue till the time the shearing is applied.
- A fluid at rest is in a state of zero shear stress. This condition is termed as hydrostatic stress in structural analysis. • In hydrostatic stress condition, the Mohr circle of stress reduces to a point and there is no shear stress on any plane of the element.
A fluid further can be divided as liquid and gas. The main distinction between a liquid and a gas is the effect of cohesive forces on them.
A liquid, with relatively low value of molecular spacing (molecules are closely packed as compared to gas) exhibit strong cohesive forces, and tends to retain its volume. On the other hand, gases have large molecular spacing and exhibit low cohesive forces and have a tendency to freely expand until it is confined.
Point to remember:
- As gases have no definite volume and tend to expand freely when left to itself, they create an atmosphere which is essentially hydrostatic.
It is defined as the mass per unit volume.
It is denoted by Greek letter ‘Rho’ symbolised as ρ. It is highly variable in gases and it increases proportionately with the gas pressure. Density of liquids is almost constant, i.e. the density of water is 1000 kg/m3.
Points to remember:
- In general, at atmospheric pressure, liquids are almost three times denser than the gases.
- The densest common liquid is mercury (Hg) with a density value of 13,600 kg/m3 and the least dense gas is hydrogen with a density of 0.0838 kg/m3.
If the density of a fluid varies significantly with moderate changes in pressure or temperature, then the fluid is termed as compressible fluid. Generally gases and vapours can be regarded as compressible fluids.
If the density variation of a fluid is small due to changes in temperature or pressure, then the fluid is termed as incompressible fluid. All liquids are classified under this category.
It is defined as the weight per unit volume (N/m3), and is denoted by the letter߱ . Mathematically, it can be written as, .
Where g = acceleration due to gravity = 9.8 m/s2 The specific weight of water is 9800 N/m3 at 200oC and atmospheric pressure.
Specific Gravity (S.G.)
It is defined as the ratio of fluid density to the density of a standard reference fluid.
Specific gravity of mercury is 13600/1000 = 13.6. It is also referred to as Relative density "R.D".
When a fluid is subjected to shearing, it results in the movement of the fluid creating a strain rate which is inversely proportional to a property called the coefficient of viscosity (μ) . The resulting shear stress (τ) is expressed as, .
The figure shown below, illustrates the velocity profile and development of shearing in an elemental fluid element. It shows that shear stress is maximum at the wall. Further, the velocity ሺuሻ is zero at the wall, otherwise known as no‐slip condition.
The units of μ are N‐s/m2, and the term is known as the coefficient of dynamic viscosity or simply viscosity. More popular unit for expressing value of viscosity is Poise (P).
1 Poise = 0.1 N-s/m2
Equation ሺ1ሻ shown above is known as Newton’s Law.
The fluids which follow equation (1) are known as Newtonian fluids.
The value of viscosity of Newtonian fluids varies with temperature and pressure. The viscosity of a fluid varies very weakly with pressure. Temperature, on the other hand has a strong effect on the viscosity. For gases, viscosity increases with an increase in temperature, and decreases with an increase in temperature.
Point to remember:
- For common engineering applications, we tend to neglect the pressure effects.
The ratio of dynamic viscosity to the fluid density is defined as kinematic viscosity (ν). Mathematically, ν = μ/ρ.
The units of kinematic viscosity are, Stokes.
1 stoke = 1 cm2/sec = 10-4 m2/s
Point to remember:
- The term ‘kinematic’ comes into picture, as the mass units are cancelled.
Flow between plates:
In this problem, we have a fixed lower plate and a moving upper plate, moving with a constant velocity V. The condition is illustrated in the figure below,
The clearance between the plates is ‘h’, fluid is a Newtonian fluid with no slip condition at the bottom plate. It is assumed that there is zero acceleration and no pressure variation in the direction of flow. We can conclude, that the shear stress is constant throughout the fluid, and,
Upon integration of the above equation, we get,
The above expression validates that the velocity distribution is linear. The terms ‘a’ and ‘b’ are constant values and can be evaluated from the no‐slip condition at the upper and lower walls as follows,
The boundary conditions give us,
Therefore the velocity profile between the plates is given by the relation,
From the above matheatical expression, we can observe that the viscous stresses are negligible in magnitude, although viscosity, as a property, has profound effects on the fluid motion.
Fluids which do not obey Newton’s laws are called Non‐Newtonian fluids.
- Dliatant fluid or a shear thicknening fluid increases flow resistance with an increasing applied stress
- Pseudoplastic fluid or a shear thinning fluid decreases flow resistance with an increasing applied stress. If the thinning effect is very strong, then the flow is termed as Plastic
- Bingham plastics show that the flow is linear after the yielding, but in actual the flow may be non linear too.
The transient effect in Non-Newtonian flow can be illustrated by the rheopectic and thixotropic fluids.
- Rheopectic fluids show an increase in the shear stress with respect to time.
- Thixotropic fluids show a decrease in the shear stress with respect to time.
A liquid being unable to expand freely, will form an interface with a second liquid or gas. This causes a phenomenon to occur, surface tension.
Surface tension is defined as the apparent tensile stress that acts whenever a liquid has a density interface, such as when the liquid comes in contact with a gas, vapour or another liquid. The effect of surface tension is that, the liquid surface at an interface appears to act as a stretched elastic membrane. The symbol to denote surface tension is ‘σ’ and its units are N/m.
There are two types of forces that act between molecules and matter.
- The first kind of force is directly proportional to the product of the masses of two molecules and inversely proportional to the square of the distance between their centres of masses. This force is referred to as mass attraction.
- The second kind of force is an electro-chemical attractive force between molecules, which results in cohesion and adhesion
- Adhesion is the attractive force between molecules of solid and liquid or between molecules of two different liquids which do not mix
- Cohesive forces are the resistive forces that exist between molecules of the same substance
- The relative magnitude of cohesive and adhesive forces will decide whether the liquid will or will not wet the given solid surface
- If the adhesive forces between a liquid and a solid surface are stronger, then liquid will wet the solid surface
- If the cohesive forces among the liquid itself are stronger, then the liquid molecules will resist such adhesion, and will not wet the surface causing the liquid to form spherical beads
It is due to the phenomena of surface tension, that the liquid drops appear to be spherical.
- The cohesive force between gas molecules or between liquid molecules and gas molecules is negligible, because of greater intermolecular distances
- Presence of gas in case of liquid-gas interface has negligible effect on curvature of the surface
The pressure inside a spherical bubble is given as, or .
Where, p = pressure inside the droplet, in excess of the atmospheric pressure, d = diameter of the droplet and σ = surface tension of the liquid.
The above relation shows that, the pressure increases with the decrease in the size of the drop.
The pressure inside a liquid jet is given as,
, where p = pressure inside the jet, in excess of the atmospheric pressure, d = diameter of the jet, σ = surface tension of the liquid and L = length of the jet.
Another important surface effect is the contact angle (θ) which appears when a liquid interface intersects with a solid surface, as illustrated in the figure below,
- If the value of θ<90o then the liquid is said to wet the solid
- If the value of θ>90o then the liquid is non-wetting
Point to remember:
- Water is wetting to a clean glass surface as θ≈0o
Capillary Rise or Capillary Depression
This is one of the most important applications of cohesion and adhesion, concerning small diameter tubes and in interstices of porous materials.
If the adhesive forces predominate, the liquid will wet the glass surface and liquid will rise in a vertical capillary tube dipped in the liquid.
If the surface tension predominates over adhesion, then the liquid does not wet the surface and will tend to depress the point of contact causing capillary depression.
The value of capillary rise or fall (Δh) can be given as, .
- Smaller the tube radius, greater will be the capillary rise
- The curved surface of liquid in the tube is known as meniscus
- For tubes of diameter 6 mm or more, the capillary rise is negligible for water
Bulk Modulus (K)
Fluids can be compressed by applying external pressure, and they expand back to their original volume once this external pressure is removed. This indicates the elastic property of a fluid. Due to the compressible nature, the fluid density changes with pressure. But this change in density due to pressure variations is very small in case of liquids, and due to this fact liquids are considered to be incompressible.
But the mass density of gases changes appreciably with variation in pressure, which concludes that the gases are compressible.
The compressible fluids are characterized by the property of Bulk Modulus of Elasticity (K), which is defined as the ratio of volume change under a pressure increase of Δp.
Negative sign indicates a decrease in volume upon increase in pressure.
The value of bulk modulus of elasticity is not constant for a fluid but necessarily depends upon pressure.
- As for gases, there is a very fine relation between pressure and temperature, therefore, the K for a gas is dependent upon temperature also.
- The units for bulk modulus are N/m2
- In case of liquids, the effect of compressibility can be neglected, however in some cases like sudden closure of valves, it must be taken into account
- In cases like air flowing in a ventilating system the gas may be treated as incompressible because the pressure variation is so small and change in air density is negligible
All liquids show a tendency to evaporate when exposed to the atmosphere. The rate of evaporation depends upon:
- Nature of the liquid
- Temperature of the liquid
- Condition of atmosphere just above the liquid surface
Consider a closed container partially filled with a liquid and is maintained at a constant temperature. As the liquid evaporates the number of molecules in the region above the liquid start to increase. As this is happening, simultaneously a small number of molecules re-enter the liquid.
With the passage of time, the concentration of vapour molecules above the liquid surface increase, such that an equilibrium conditions exists with the help of which air above the liquid surface is saturated with vapour molecules.
The pressure exerted by the vapour molecules (or saturated vapour molecules) on the liquid surface is called vapour pressure.
- A liquid with high vapour pressure will evaporate more readily as compared to the liquid with a low value of vapour pressure
- When the vapour pressure of a liquid is slightly greater than the pressure exerted on the surface, the liquid will boil
- If the pressure at any point in the liquid approaches the vapour pressure, the liquid starts vapourising and creates bubbles or pockets of gases and vapours. This phenomenon is called cavitation.