Viscosity

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Viscosity

The property of liquid by virtue of which it opposes the relative motion of its different layers.

It is similar to solid friction because it acts tangentially backwards and opposes relative motion and occurs due to intermolecular force. It is dissimilar to solid friction because it depends on velocity, area of contact while solid friction does not. viscosity does not depend on normal reaction but solid friction does.

Newton’s Law

viscous force is directly proportional to area of contact and velocity gradient. F =- nAdv/dx ; Where n is the proportionality constant known as coefficient of viscosity. Negative sign indicates its an opposing force.

Coefficient of Viscosity

Its SI unit is decapoise or N.s/m^2 or kg m^-1s^-1

Its CGS unit is poise or dyn.s/cm^2 or g cm^-1s^-1

n=-F when A=1 and dv/dx = 1

thus it is defined as the tangential backward viscous force acting on a unit area with unit velocity gradient.

Stoke’s Law

6πηrvt

When a solid body moves through a viscous medium a layer of liquid in contact with the solid surface is in rest and moves along with the body. The other layers oppose this movement. The viscous force increases with the increase in velocity. We assume that the viscous force F is

Fαηⁿ

Fαrⁿ

Fαvⁿ

Solve dimensionally

Fαηⁿrⁿvⁿ

F=kηⁿrⁿvⁿ

MLT²=k (ML⁻¹T⁻¹)ⁿ (L)ⁿ (LT⁻¹)ⁿ

F=6πηrvt experimentally k=6π

Terminal Velocity

A solid body falls through a viscous medium it accelerates due to g. Due to increase in velocity viscous force also increases. Soon it comes to dynamic equilibrium and goes down with constant velocity known as terminal velocity

Flow

Streamline if same path and same velocity of its preceding particle

Laminar if same velocity of the particle but may not be of different layers

Turbulent if it does not follow the path or velocity of its preceding particle

Reynold’s Number

Shows whether the flow is turbulent or streamline.

For a pipe of internal diameter D Reynold’s number

Nr= DVcρ/η

Vc is critical velocity above which the flow becomes turbulent.

If Nr

Poiseuille’s Formula

Rate of flow of liquid V/t α r⁴P/lη

Derive dimensionally again

M³L⁰T⁻¹=k (L)ⁿ (ML⁻²T⁻²)ⁿ (ML⁻¹T⁻¹)ⁿ

V/T= π/8 r⁴P/lη

Equation of continuity

a1v1=a2v2

incompressible and non-viscous liquid through a pipe of nonuniform cross section.

mass entering=mass exiting

mass=volume x density

m1= a1v1delt x rho

Bernoulli’s Theorem

a pipe of non uniform cross section

inclined

v1 velocity at a1 and v2 velocity at a2

P1 h1 pressure and height of a1 P2 h2 pressure and height of a2

word done in displacing liquid through small distance dx

W1=F1dx1=P1dV1

at a2

W2=F2dx2 = P2dV2

W=W1-W2

Change in KE

1/2mv1^2 - 1/2mV2^2

change in PE

mgh2-mgh1

total work done = change is KE + change in PE