♦️Revision Notes on Flow of Liquids and Viscosity♦️

♦️Revision Notes on Flow of Liquids and Viscosity♦️(2/3)

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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]

(c) Direction of velocity gradient is perpendicular to the direction of flow, directed in the direction of increasing velocity.

(d) Average velocity gradient:- Average velocity gradient is the difference between velocities of two layers separated a unit distance apart.

Average velocity gradient = Δv/Δr

Newton’s law of viscosity:-

In accordance to Newton’s law of viscosity, the viscous drag force depends upon the nature of fluid along with following factors:-

(a) F∝A (common area of two layers)

(b) F∝dv/dr (velocity gradient)

(c) So, F =ηA (dv/dr)

Here η is called coefficient of viscosity of fluid.

Coefficient of viscosity of fluid (ηv) or fugitive elasticity:-

ηv = shear stress/velocity gradient = (F/A)/(dv/dr)

Modulus of rigidity(ηr):-

ηr = shear stress/shear strain = (F/A)/(θ) = (F/A)/(dx/dr)

Here, θ = dx/dr = displacement gradient

Coefficient of viscosity (Absolute viscosity or Dynamic viscosity):-

F= ηA (dv/dr) if A = 1, dv = 1, dr =1, F = η

Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain (or resist) a unit relative velocity between two layers a unit distance apart.

Or

Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain a unit velocity gradient between its layers.

Unit of η:-

S.I:- η = 1 deca poise = 1 N sec/m2

Co-efficient of viscosity of a fluid is said to be one deca-poise if a tangential force of 1 N per meter square is required to maintain a relative velocity of 1 ms-1 between its layer 1 m apart.

C.G.S:- η = 1 poise = 1 dyn sec/cm2

Coefficient of viscosity of a fluid is said to be one poise if a tangential force of 1 dyn per square cm is required to maintain a relative velocity of 1 cms-1 between its layers 1 cm apart.

Relation between deca-poise and poise:-

1 deca-poise = 10 poise

Dimension formula for η:-

η = Fdr/Adv = [M1L-1T-1]

Fluidity:- Reciprocal of coefficient of viscosity of a fluid is called its fluidity.

Fluidity = 1/η

Unit of fluidity: poise-1

Dimension of fluidity: [M-1L1T1]

Kinematic viscosity:- Kinematic viscosity of a fluid is defined as the ration between its coefficient of viscosity to the density of fluid.

Kinematic viscosity = η/ρ

Units of kinematic viscosity:- C.G.S – 1 stoke = cm2 s-1

Kinetic viscosity of a fluid having its dynamic viscosity one poise and density one g cm-3 is said to be 1 stoke.

Dimensional formula of kinematic viscosity = η/ρ = [M0L2T-1]

Critical velocity (Reynold’s Number):- Critical velocity (vc) is the maximum velocity of the flow of liquid flowing in a streamlined flow.

vc = NR η/ρD

Here η is the coefficient of viscosity of liquid, ρ is the density of liquid and D is the diameter of the tube.

Reynold’s Number, NR = ρvcD/ η

Stokes law:- In accordance to Stoke’s law, force of viscosity F depend upon,

(a) Co-efficient of viscosity of fluid η

(b) Radius of the moving body r

(c) Velocity of body v

So, force of viscosity, F = 6π η r v

Terminal velocity:- v = 2/9 [r2 (ρ-σ)/η]

η = 2/9 [r2 (ρ-σ)g/v]

Variation of viscosity with a change in temperature and pressure:-

(a) Effect of temperature:-

η= A /(1+Bt)c

Here A, B and C are constants.

Again, ηv1/2 = Aec/vt

Here, A and C are constants and v is the relative velocity.

(b) Effect of pressure:- Co-efficient of viscosity of liquids increases due to an increase in pressure but there is no relation, so far, to explain the effect.

Change in viscosity of gases:-

(a) Effect of temperature:- Co-efficient of viscosity of a gas at a given temperature is given by,

η= η0AT1/2

Here T is the absolute temperature of gas.

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