Concept of Viscosity:

Viscosity is a frictional force in fluids. Usually, viscosity in the liquid is the characteristic to effect of cohesive forces of molecules comparatively close together in the liquid. When liquids flow by pipes they have frictional force between layers. Same thing occurs when gases move or when objects move by air. However while molecules of the liquid are extremely close to each other, those of gases are very far apart for cohesion to be effective. Viscosity of fluids influences volume of fluid flowing per second as fluid flows in the pipe. Viscosity also influences lubrication of moving parts in machines as in car engines. It is not simple to realize that when the liquid flows along the pipe, there are layers of fluid moving with various velocities. Molecules appear to be moving fastest at centre along the axis of the pipe. While velocity reduces as one move towards wall of the pipe, velocity tends to zero.

Velocity Gradient in Moving Fluids:

When the fluid is moving from pipe or solid object is moving through a fluid, layer of fluid in make contact with sides of the pipe or the surface of the object tends to be in the similar state of motion as the object with which it is in contact; i.e., layer of fluid along the side of the pipe is at rest, while that in contact with moving object is carried along at the similar velocity as object. If difference in velocity between fluid at the sides of the pipe and that at the center, or between moving object and fluid through which it is moving, is not very great, then fluid flows in continuous, smooth layers; i.e., the flow is laminar.

Difference in velocity between adjacent layers of fluid is called as velocity gradient and is provided by v/x, where v is velocity difference and x is distance between layers. To keep one layer of fluid moving at the greater velocity than adjacent layer, a force F is essential, resulting in the shearing stress F/A, where A is area of surface in contact with layer being moved.

Velocity Gradient = velocity/distance.

Dimensional Formula of Velocity Gradient = M⁰L⁰T^-1. From dimensional formula SI unit of Velocity Gradient is s^-1

Coefficient of Viscosity η:

When fluids flow from pipes, outermost layer touching sides of the pipe, have minimum velocity. As one approaches centre of the tube, the velocity of innermost (or centre) layer is maximum. Hence, the velocity gradient develops inside the pipe.

Newton defined that under condition of steady flow, frictional force (F) between two layers moving fluid is proportional to:

• velocity gradient G and
• area (A) parallel to fluid flow of the liquid under consideration

Hence, F ∝ AG

Therefore F = ηAG

This direction of force (F) is opposite to direction of velocity. If A = 1, G = 1, then η = F. Here η is the constant called as coefficient of viscosity of fluid. η is stated as frictional force needed to maintain the unit velocity gradient between two layers each of unit area.

N(Newton) and Unit of area (A) is m² (meter square). Unit of velocity gradient (G) is s^-1. Then,

Therefore unit of η is N/m²s^-1 = Nm^-2s

The dimensions of force (F) = ML/T² for N is unit of force

The dimensions of area (A) = L² for m² is unit of area

The dimensions of velocity gradient (G) = T¹ for s^-1 is unit of velocity gradient.

Therefore the dimensions of η = ML/T² x 1/L² ÷ 1/T = ML^-1T^-1

It can be noted that η is also stated in terms of stress.

η = Tangential stress/Velocity Gradient

Force per unit Area/Velocity Gradient

= (F/A)/(Δv/Δr)

Poise (pronounced as puaz) is adopted for unit of η after name of scientist Poiseiulle (pronounced as puasoy0. S.I. unit of η is Pa s (Pascal-series or decapoise. 1 decapoise = 1Nsm^-2 = 10 poise.

Variation η of with Temperature:

Viscosity of motor oil is 2 poise at 20^oC. Viscosity of water is 1 x 10^-3 poise at 10^oC. Viscosity reduces with increase in temperature; though, values of most of motor oils are such that they are independent of temperature.

Poiseuille's Formula:

Poiseuille's formula assists us to find out coefficient of viscosity of the liquid by observing rate of low of liquid. It is the expression which associates volume of liquid flowing through pipe per second (V/t), coefficient of viscosity (η poises), radius of the pipe (r) and Pressure gradient (P/L)

Formula was derived by dimensional analysis

V/t = (Π/8)Pr⁴/ ηL

Value Π/8 calculated experimentally.

Where, p = difference in pressure between ends of pipe

r = radius of pipe

L = length of pipe

η = coefficient of viscosity

Stokes' Law and Terminal Velocity:

When any object rises or falls through the fluid, a viscous drag will occur, whether it is a parachutist or spacecraft falling from air, a stone falling through water or the bubble rising through fizzy lemonade. Mathematics of viscous drag on uneven shapes is hard. Formula was first recommended by Stokes and is thus called as Stokes' law.

Let the sphere falling through the viscous fluid. As sphere falls so velocity increases until it reaches the velocity called as terminal velocity. At this velocity frictional drag because of viscous forces is just balanced by gravitational force and velocity is constant. At this speed:

Viscous drag = 6Πηrv = Weight = mg

The given formula can be proved.

Frictional force (F) = 6Πηrv (Stokes' law)

Terminal velocity is highest velocity achievable by the object as it falls through air. It happens once the sum of drag force (F_d) and buoyancy equals downward force of gravity (F_G) applying on the object. As the net force on object is zero, the object has zero acceleration.

Measuring η by Falling Sphere:

This is the good method utilized to find out coefficient of viscosity of the liquid through determination of terminal velocity.

Liquid is put in tube which is set vertically. We utilize a sphere of 2mm in diameter of tube is 2cm. Error is negligible even when in theory diameter of tube is expected to be large compared with diameter of the sphere. Therefore correction factor may be ignored. Density of sphere ρ is determined by using Archimedes' principle. Density of liquid (ρ') is also found using specific gravity bottle i.e. by using relationship

ρ' = ρ'rρw

Where ρ' is density of liquid, ρ'r is relative density of liquid, and ρ′w is density of water But,

 = ρ'r = ρ'/ρw

 = Mass of liquid/Mass of equal volume of water.

The terminal velocity V_T is determined by noting the times taken by the sphere to move through AB and BC at t₁ and t₂ respectively. If AB is made to be equal to BC, therefore at terminal velocity t₁ should be equal to t₂ when observed. If not then the experiment has failed because terminal velocity has been attained.

If t₁ = t₂ then terminal velocity has been attained

Therefore V_t = (AB+BC)(t₁ + t₂)

Micrometer screw gauge is utilized to find out radius of the sphere. Value η of can then is determined by using relation.

[(η2r²g(ρ-ρ'))/9V]poise

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