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Viscosity and Fluid Flow

By: Dinesh Kumar ,PGT Physics

JNV Kurukshetra(Haryana)

Email: bihana.dinesh@gmail.com

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1–1 ■ INTRODUCTION

Fluid mechanics deals with liquids and gases in motion or at rest.

Mechanics: The oldest physical science that deals with both stationary and moving bodies under the influence of forces.

Fluid mechanics: The science that deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries.

Fluid dynamics: Fluid mechanics is also referred to as fluid dynamics by considering fluids at rest as a special case of motion with zero velocity.

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The arrangement of atoms in different phases: (a) molecules are at relatively fixed positions in a solid, (b) groups of molecules move about each other in the liquid phase, and (c) individual molecules move about at random in the gas phase.

Intermolecular bonds are strongest in solids and weakest in gases.

Solid: The molecules in a solid are arranged in a pattern that is repeated throughout.

Liquid: In liquids molecules can rotate and translate freely.

Gas: In the gas phase, the molecules are far apart from each other, and molecular ordering is nonexistent.

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Application Areas of Fluid Mechanics

Fluid dynamics is used extensively in the design of artificial hearts. Shown here is the Penn State Electric Total Artificial Heart.

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VISCOSITY

The property of a liquid by virtue of which an opposing force(internal friction) comes into play between different layers of a liquid whenever there is a relative motion between these layers of the liquid is called viscosity.

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Coefficient of Viscosity

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The coefficient of viscosity (pronounced ‘eta’) for a fluid is defined as the ratio of shearing stress(F/A) to the strain rate(v/l).

The SI unit of viscosity is poiseiulle (Pl). Its other units are N s m^-2 or Pa s. The dimensions are [ML^-1T^-1]

Effect of Temperature on Viscosity

• The Viscosity of liquids decreases with increase in temperature and increases with decrease in temperature i.e. η α

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• The viscosity of Gases increases with increase in temperature and vice versa i.e. η α

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Stoke’s Law

• According to Stoke’s Law the viscous drag(F) acting on a spherical body of radius r moving with terminal velocity v in a fluid of coefficient of viscosity η is given by , F=6ᴨηrv F

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Terminal Velocity

• When a body is dropped in a viscosity fluid, it is first accelerated and then its acceleration becomes zero and it attains a constant velocity called terminal velocity.

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where r = radius of spherical body

η= coefficient of viscosity

σ = density of fluid

ρ = density +of material of body

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Laminar versus Turbulent Flow

Laminar flow: The highly ordered fluid motion characterized by smooth layers of fluid. The flow of high-viscosity fluids such as oils at low velocities is typically laminar.

Turbulent flow: The highly disordered fluid motion that typically occurs at high velocities and is characterized by velocity fluctuations. The flow of low-viscosity fluids such as air at high velocities is typically turbulent.

Transitional flow: A flow that alternates between being laminar and turbulent.

Laminar, transitional, and turbulent flows over a flat plate.

Poiseuille’s Equation

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According to Poiseuille volume of liquid coming out the tube per second is

1. directly proportional to the Pressure difference(P)
2. directly proportional to fourth power of radius(r) of capillary tube
3. inversely proportional to coefficient of viscosity(η) of liquid
4. inversely proportional to length (l) of capillary tube

i.e.

where K= ᴨ/8

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Reynold’s Number

Where v_c= Critical velocity of liquid

ρ = density of liquid

η = coefficient of viscosity

D= Diameter of the tube

If R< 1000 , the flow of liquid is streamline or laminar

If R> 2000, The flow is turbulent

If R lies between 1000 and 2000, the flow is unstable and may change from streamline to turbulent flow

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For incompressible liquid ρ₁=ρ₂= ρ

Then A₁v₁=A₂v₂=Av

Thus Av=Constant , This is known as equation of continuity

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Steady versus Unsteady Flow

• The term steady implies no change at a point with time.
• The opposite of steady is unsteady.
• The term uniform implies no change with location over a specified region.
• The term periodic refers to the kind of unsteady flow in which the flow oscillates about a steady mean.

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Comparison of (a) instantaneous snapshot of an unsteady flow, and (b) long exposure picture of the same flow.

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Compressible versus Incompressible Flow

Incompressible flow: If the density of flowing fluid remains nearly constant throughout (e.g., liquid flow).

Compressible flow: If the density of fluid changes during flow (e.g., high-speed gas flow)

When analyzing rockets, spacecraft, and other systems that involve high-speed gas flows, the flow speed is often expressed by Mach number

Ma = 1 Sonic flow

Ma < 1 Subsonic flow

Ma > 1 Supersonic flow

Ma >> 1 Hypersonic flow

Bernauli’s Theorem

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Applications of Bernoulli's Theorem

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Limitations of Bernoulli's Theorem

• [1] The viscous drag has been neglected as we assume the flow to be non-viscous.
• [2] We also used that there is no loss of energy as the liquid moves, but some of its kinetic energy is always converted into heat due to viscous forces.
• [3] If liquid moves along a curved path then the centrifugal forces should also be considered.
• [4] We assume all the liquid particles moving with the same velocity but liquid particles near the center of tube moves faster than outer particles.

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