🔷Notes on Flow of Liquids and Viscosity🔷
(Mechanical Properties of fluids):-
Characteristic of Ideal fluid:-
(a) It is incompressible
(b) It is non-viscous
(c) Flow of ideal fluid is irrational
(d) It is capable of exhibiting steady flow
Stream line flow:- Flow of a liquid fluid is said to be streamlined if the velocity of a molecule, at any point, coincides with that of the preceding one.
Tube of flow:- A bundle of streamlines having same velocity of fluid elements, over any cross-section perpendicular to the direction of flow, is called a tube of flow.
Laminar flow:- It is a special case of streamline flow in which velocities of all the molecules on one streamline is same throughout its motion.
Turbulent flow:- Whenever the velocity of a fluid is very high or it rushes past an obstacle so that there is a sudden change in its direction of motion, the motion of fluid becomes irregular, forming eddies or whirlpools. This type of motion of fluid is called turbulent flow.
Rate of flow (Equation of continuity):-
?Equation of Continuityav= Constant (a1v1=a2v2)
Equation of continuity can be considered to be a statement of conservation of mass.
So, v ∝ 1/a
Velocity of flow of liquid varies inversely as the area of cross-section of the opening from where the liquid comes out.
Total energy of a liquid:-
(a) Kinetic energy:- It is the energy possessed by a liquid by virtue of its velocity.
K.E = ½ mv2
K.E per unit mass = ½ v2
K.E per unit volume = ½ [mv2/V] = ½ ρv2
Here, ρ is the density of liquid.
(b) Potential energy:- It is the energy possessed by a liquid by virtue of which of its position.
Potential energy = mgh
P.E per unit mass = mgh/m = gh
P.E per unit volume = mgh/V = ρgh
(c) Pressure energy:- It is the energy possessed by a liquid by virtue of its pressure.
Pressure energy = p×V = m (p/ρ)
Pressure energy per unit mass = p/ρ
Pressure energy per unit volume = p×V /V= p
Total energy:- Total energy of a liquid is the sum total of kinetic energy, potential energy and pressure energy.
E= ½ mv2 +mgh+mp/ρ
Total energy per unit mass = ½ v2 +gh+p/ρ
Total energy per unit volume = ½ ρv2 +ρgh+p
Bernoulli’s equation:- It states that the total energy of a small amount of an incompressible non-viscous liquid flowing without friction from one point to another, in a streamlined flow, remains constant throughout the displacement.
(a) ½ mv2 + mgh+ mp/ρ = Constant
(b) ½ v2 +gh+p/ρ = Constant
(c) ½ ρv2 +ρgh+p = Constant or v2/2g + h + p/ρg = Constant
The term v2/2g is called velocity head, h is called gravitational head and p/ρg is called pressure head.
Therefore Bernoulli’s theorem states that in case of an incompressible, non-viscous fluid, flowing from one point to another in a streamlined flow, the sum total of velocity head, gravitational head and the pressure head is a constant quantity.
Limitation of Bernoulli’s equation:-
(a) Force of viscosity, which comes into play in case of fluids in motion has not been accounted for.
(b) Loss of energy due to heat is not accounted for.
(c) When a fluid flows in a curved path, the energy due to centripetal force is also not accounted for.
If v is the relative velocity of top layer w.r.t. any other deeper layer (may be the lowest), then v is lesser for greater depth.
v = K/bd
or v ∝ 1/d
?It is a device used for measuring the rate of flow of liquids, generally water, through pipes.
The rate of flow of water, Q = a1a2√2hg/[a12-a22]
Torricelli’s theorem (velocity of efflux):-
It states that the velocity of efflux of a liquid (V), from an orifice, is equal to the velocity acquired by a body, falling freely (v), from the surface of liquid to the orifice.
So, V = v = √2gh
Viscosity:- Viscosity is the property of fluids by virtue of which they tend to destroy any relative motion between their layers.
Velocity gradient:- Velocity gradient is defined as the rate of change of velocity with respect to distance.
(a) Velocity gradient = dv/dr
(b) Dimension of velocity gradient = [dv/dr] = [T-1]