تلاایس کیناکم Fluid Mechanics

Dr. Maysam Saidi Fluid Mechanics

مکانیک سیالاتFluid Mechanics

نگاه کلی و مفاهیمOverview and concepts

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Gas Liquids Statics Dynamics

Air, He, Ar, N2, etc. Water, Oils, Alcohols, etc.

0 iF

Viscous/Inviscid

Steady/Unsteady

Compressible/

Incompressible

Course overview

0 iF

Laminar/

Turbulent

Compressibility Viscosity Vapor

Pressure

Density

Pressure Buoyancy

Stability

Concepts Fluid Statics Fluid Dynamics

Surface

Tension

Fluid Mechanics

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What is Fluid?

• Substance that continually deforms (flows) under an applied shear stress.

• Although could Includes liquids, gases, plasmas and plastic solids, usually it refers to liquid and gases.

Solid:

Deforms or breaks when a stress is applied on it.

Fluid:

Deforms continuously as long as any shear stress is applied.

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What is Mechanics?

The study of motion and the forces which cause or prevent the motion.

• Statics: The study of forces acting on the particles or bodies at rest.

• Dynamics: The study of forces acting on the particles and bodies in motion.

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Type of stresses?

Stress = Force /Area

• Normal stress:

A force acting perpendicular to the surface per unit area of the surface.

• Shear (Tangential) stress:

The force acting parallel to the surface per unit area of the surface.

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How to study fluid mechanics?

Basic laws of physics:

• Conservation of mass

• Conservation of momentum – Newton’s second law of motion

• Conservation of energy: First law of thermodynamics

• Second law of thermodynamics

+ Equation of state

Fluid properties e.g., density as a function of pressure and temperature.

+ Constitutive laws

Relationship between the stresses and the deformation of the material.

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How to study fluid mechanics?

Example:

Density of an ideal gas:

Ideal gas equation of state

Newton’s law of viscosity:

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2 3

PV=nRT,

P: pressure (N/m ),V: volume(m ),

T:temperature (K),n:number of moles.

mass nM=

V V

pM=

RT

Stress α train (deformation)

du du =

dy dy

S

: coefficient of viscosity (Dynamic viscosity)


Viscosity

It is defined as the resistance of a fluid which is being deformed by the application of shear stress.

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In everyday terms viscosity is “thickness”. Water is “thin” having a lower viscosity, while honey is “thick” having a higher viscosity.

• Common fluids, e.g., water, air, mercury obey Newton's law of viscosity and are known as Newtonian fluid.

• Other classes of fluids, e.g., paints, polymer solution, blood do not obey the typical linear relationship of stress and strain. They are known as non-Newtonian fluids.

Unit of viscosity: Ns/m2 (Pa.s)


Viscosity

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Viscosity

• Newtonian fluids: water, air.

• Pseudoplastic fluids: paint, printing ink.

• Dilatant fluids: dense slurries, wet cement.

• Bingham fluids: toothpaste, clay.

• Casson fluids: blood, yogurt.

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Newtonian (low μ)

Newtonian (high μ)

Bingham-plastic

0

c

Casson fluid

Pseudo-plastic (shear-thinning)

Dilatant (shear-thickening)

Strain rate (1/s)


Viscosity

• The Society of Automotive Engineers (SAE) has established a numerical code system for grading motor oils according to their viscosity characteristics.

• SAE viscosity gradings include the following, from low to high viscosity: 0, 5, 10, 15, 20, 25, 30, 40, 50 or 60. The numbers 0, 5, 10, 15 and 25 are suffixed with the letter W, designating they are "winter" or cold-start viscosity, at lower temperature.

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Kinematic viscosity

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• Kinematic viscosity is another way of representing viscosity

• Used in the flow equations

• The units are of L2/T or m2/s and ft2/s


Density

V

m

13

1v

The density of a fluid is defined as mass per unit volume.

• Different fluids can vary greatly in density

• Liquids densities do not vary much with pressure and temperature

• Gas densities can vary quite a bit with pressure and temperature

• Density of water at 4° C : 1000 kg/m3

• Density of Air at 4° C : 1.20 kg/m3

Alternatively, Specific Volume:

m = mass, V = volume.


Specific weight

g

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The specific weight of fluid is its weight per unit volume.

• Specific weight characterizes the weight of the fluid system

• Specific weight of water at 4° C : 9.80 kN/m3

• Specific weight of air at 4° C : 11.9 N/m3

g = local acceleration of gravity, 9.806 m/s2


Specific gravity

ref

SG

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The specific gravity of fluid is the ratio of the density of the liquid to the density of water or gas to the air.

• Gases have low specific gravities

• A liquid such as Mercury has a high specific gravity, 13.2

• The ratio is unitless.

• Density of water: 1000 kg/m3

• Density of air: 1.2 kg/m3


Compressibility of fluids: Bulk modulus

// d

dp

vdv

dpE

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• Measure of how pressure compresses the volume/density • Units of the bulk modulus are N/m2 (Pa) and lb/in.2 (psi). • Large values of the bulk modulus indicate incompressibility • Incompressibility indicates large pressures are needed to compress the volume

slightly • It takes 3120 psi to compress water 1% at atmospheric pressure and 60° F. • Most liquids are incompressible for most practical engineering problems.

P is pressure, and is the density.


Compressibility of fluids: Speed of sound

vE

cord

dpc

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kpc

kRTc

A consequence of the compressibility of fluids is that small disturbances introduced at a point propagate at a finite velocity. Pressure disturbances in the fluid propagate as sound, and their velocity is known as the speed of sound or the acoustic velocity, c.

Isentropic Process (frictionless, no heat exchange because):

Ideal Gas and Isentropic Process:

• Speed of Sound in Air at 60 F (15 °C) 1117 ft/s or 300 m/s

• Speed of Sound in Water at 60 F 4860 ft/s or 1450 m/s

• If a fluid is truly incompressible, the speed of sound is infinite, however, all fluids compress slightly.

Ma=V/c

For Ma < 1 Subsonic Flow

For Ma > 1 Supersonic Flow


Vapor pressure: Evaporation and Boiling

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Evaporation occurs in a fluid when liquid molecules at the surface have sufficient momentum to overcome the intermolecular cohesive forces and escape to the atmosphere. Vapor Pressure is that pressure exerted on the fluid by the vapor in a closed saturated system where the number of molecules entering the liquid are the same as those escaping. Vapor pressure depends on temperature and type of fluid. Boiling occurs when the absolute pressure in the fluid reaches the vapor pressure. Boiling occurs at approximately 100 °C, but it is not only a function of temperature, but also of pressure.

Cavitation is a form of Boiling due to low pressure locally in a flow.

boiling.mov


Vapor pressure

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Surface tension

• At the interface between a liquid and a gas or two immiscible liquids,

forces develop forming an analogous “skin” or “membrane” stretched

over the fluid mass which can support weight.

• This “skin” is due to an imbalance of cohesive forces. The interior of the

fluid is in balance as molecules of the like fluid are attracting each other

while on the interface there is a net inward pulling force.

• Surface tension is the intensity of the molecular attraction per unit

length along any line in the surface.

• Surface tension is a property of the liquid type, the temperature, and the

other fluid at the interface.

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Surface tension

• The tendency of the surface of a liquid to resist a force and behave like a membrane and is a result of cohesion between liquid molecules. Adhesion: The ability of a substance to stick to an unlike substance.

• Cohesion: Various intermolecular forces that hold solids and liquids together.

• Wettability: The ability of a solid surface to reduce the surface tension of a liquid in contact with it such that it spreads over the surface and wets it.

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Surface tension: Liquid drop

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The pressure inside a drop of fluid can be calculated using a free-body diagram:

Real Fluid Drops Mathematical Model

R is the radius of the droplet, s is the surface tension, Dp is the pressure difference between the inside and outside pressure.

The force developed around the edge due to surface tension along the line:

sRFsurface 2

This force is balanced by the pressure difference Dp: 2RpFpressure D

Applied to Area

Applied to Circumference


Surface tension: Liquid drop

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Now, equating the Surface Tension Force to the Pressure Force, we can estimate Dp = pi – pe:

This indicates that the internal pressure in the droplet is greater that the external pressure since the right hand side is entirely positive.

Is the pressure inside a bubble of water greater or less than that of a droplet of water? Prove to yourself the following result:

Rp

s2D

Rp

s4D


Surface tension: Capillary action

Capillary action in small tubes which involve a liquid-gas-solid interface is caused by surface tension. The fluid is either drawn up the tube or pushed down.

h is the height, R is the radius of the tube, q is the angle of contact.

“Wetted” “Non-Wetted”

The weight of the fluid is balanced with the vertical force caused by surface tension.

Adhesion > Cohesion Cohesion > Adhesion

Adhesion

Cohesion Adhesion

Cohesion

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Rh

qs cos2

qs cos2 RFsurface

Free Body Diagram for Capillary Action for a Wetted Surface:

For clean glass in contact with water, q 0°, and thus as R decreases, h increases, giving a higher rise. For a clean glass in contact with Mercury, q 130°, and thus h is negative or there is a push down of the fluid.

hRW 2

Equating the two and solving for h:

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Surface tension: Capillary action


Laminar-Turbulent-Transition

• Fluid flow: turbulent, laminar, or transitional state

• Route to turbulence: different for different flows

• These fluid states: decides many important things

e.g, Energy dissipation, mixing etc.

Nonlinear analysis/

direct numerical simulation Linear stability analysis

‘Standard’ route to turbulence:

Laminar stable

Laminar unstable

Nonlinear instability

Turbulent flow Infinitesimal

disturbance

Roughness,

Entry effect etc.

Disturbances

grow to finite

amplitude

Transition

ReUL

“Inertial force/Viscous force’’

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تلاایس کیناکم Fluid Mechanics