Reynolds Number

• Reynolds number is a dimensionless number, whose value gives an idea whether the flow would be turbulent or laminar.
• Types of flow are classified as 2 types:laminar flow and turbulent flow.
• Reynolds number helps us to determine whether the flow is laminar or turbulent.
• It is denoted by R_e. where ‘_e’ shows Reynolds.
• Expression: R_e=ρvd/ η;
  • where ρ = density of the fluid,
  • v=velocity of the fluid,
  • d=diameter of the pipe through which the fluid flows
  • η=viscosity of the fluid.

How does Reynolds number (R_e) distinguish laminar flow from tubular?

• If the value of Reynold’s number (R_e) reaches 1000 then the flow is laminar.
• When the value of Reynold’s number(R_e)is greater than 2000 then the flow is turbulent.
• If the value of (R_e)is between 1000 and 2000 then the flow is unstable.The flow is in intermediate stage.
  • At this state it has some characteristics of laminar flow and some of turbulent flow.

Alternative expression of R_e: Inertial force/force of viscosity

• By using R_e= ρvd/ η
• Multiplying both numerator and den by v:- R_e= ρv²d/ ηv
• By rearranging, ρv²/( ηv/d)
• Multiplying both numerator and den by A:- R_e= ρv²A/(ηv/d)A
  • Where
    • ρv²A = Inertial force
    • (ηv/d)A = Force of viscosity

(a)Calculating inertial force

• Inertial force = ma
  • =ρV xv/t = (ρVxAxdisplacement)/t
  • =ρvAv =ρv²A

(b)Calculating Force of viscosity:-

• Coefficient of viscosity η =stress/shearing strain
  • F/A/(x/lt)
  • F/A/v/l =Fl/Av
  • η=Fl/Av
  • F = ηAv/l
  • =(ηv/l)A (Expression is same as   )

Turbulence:boon or bane

• Turbulence has both advantages and disadvantages.
• Advantages:-
  • Promotes mixing and increases the rates of transfer of mass,momentum and energy. For example: – Mixer and Grinder or a juice mixer.

Grinding of flour

• Disadvantages:-
  • Dissipates Kinetic energy in the form of heat.

Problem:- The flow rate of water froma tap of diameter 1.25 cm is 0.48 L/min.The coefficient of viscosity of water is10^-3 Pa s. After sometime the flow rate isincreased to 3 L/min. characterise the flowfor both the flow rates.

Answer:- Let the speed of the flow be v and thediameter of the tap be d = 1.25 cm. Thevolume of the water flowing out per second isQ = v × π d² / 4

v = 4 Q / d²

We then estimate the Reynolds number to be

R_e = 4 ρ Q / π d η

= 4 ×103 kg m^–3 × Q/(3.14 ×1.25 ×10^-2 m ×10^-3 Pa s)

= 1.019 × 108 m^–3 s Q

Since initially

Q = 0.48 L / min = 8 cm³ / s = 8 × 10^-6 m³ s^-1,

We obtain,

R_e = 815

Since this is below 1000, the flow is steady.After some time when

Q = 3 L / min = 50 cm³ / s = 5 × 10^-5 m3 s^-1,

We obtain,

R_e = 5095

Liquid Surfaces

• Certain properties of free surfaces:-
• Whenever liquids are poured in any container they take the shape of that container in which they are poured and they acquire a free surface.
  • Consider a case if we pour water inside the glass it takes the shape of the glass with a free surface at the top.
  • Top surface of the glass is a free surface. Water is not in contact with anything else,it is in contact with the air only.
  • This is known as free surfaces.
• Liquids have free surfaces. As liquids don’t have fixed shape they have only fixed volume.
• Free surfaces have additional energy as compared to inner surfaces of the liquid.

Surface Energy

• Surface energy is the excess energy exhibited by the liquid molecules on the surface compared to those inside the liquid.
• This means liquid molecules at the surface have greater energy as compared to molecules inside it.
  • Suppose there is a tumbler and when we pour water in the tumbler,it takes the shape of the tumbler.
  • It acquires free surface.
• Case 1: When molecules are inside the liquid:-
  • Suppose there is a molecule inside the water,there will be several other molecules that will attract that molecule in all the directions.
  • As a result this attraction will bind all the molecules together.
  • This results in negative potential energy of the molecule as it binds the molecule.
  • To separate this molecule huge amount of energy is required to overcome potential energy.
  • Some external energy is required to move this molecule and it should be greater than the potential energy.
  • Therefore in order to separate this molecule a huge amount of energy is required.
• Therefore a large amount of energy is required by the molecules which are inside the liquid.
• Case2: When the molecules are at the surface:-
  • When the molecule is at the surface, half of it will be inside and half of it is exposed to the atmosphere.
  • For the lower half of the molecule it will be attracted by the other molecules inside the liquid.
  • But the upper half is free. The negative potential energy is only because of lower half.
  • But the magnitude is half as compared to the potential energy of the molecule which is fully inside the liquid.
  • So the molecule has some excess energy, because of this additional energy which the molecules have at the surface different phenomenon happen like surface energy, surface tension.
• Liquids always tend to have least surface are when left to itself.
• As more surface area will require more energy as a result liquids tend to have least surface area.

Surface energy for two fluids in contact

• Whenever there are two fluids,in contact, surface energy depends on materials of the surfaces in contact.
• Surface energy decreases if the molecules of the two fluids attract.
• Surface energy increases if molecules of the two fluids repel.