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.

Density

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.

Specific Weight

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".

Viscosity

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.

Non-Newtonian Fluids:

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.

Surface tension:

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.

1. 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.
2. 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.

K=-V dp/dV

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

Vapour Pressure

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.

Thank you soo much sir

1

Thanks for the notes. Interesting

0