ENGR 5961 Fluids I · PDF file9 Introduction • In Fluid Mechanics we are considering the...

ENGR 5961 Fluids I

Dr. Y.S. Muzychka Faculty of Engineering and Applied Science

Memorial University of Newfoundland St. John’s, Newfoundland, Canada

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Course Outline •  Text:

–  Introduction to Fluid Mechanics Pritchard, Fox and McDonald, 8th ed.

•  Course Grading: –  Quizzes (2) 30% –  Labs (3-4) 10% –  Final Exam 60%

•  Lectures and Tutorials – Tuesday/Thursday (L) 2:30 – 3:45 PM – Wednesday (T) 4:00 – 5:00 PM

ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka

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Introduction •  Course Teaching Assistants

– Vandad Talimi – Mehdi Ghobadi

•  TA’s will instruct most tutorials, run the labs, and assist in grading.

•  Instructors Office Hours – TBA (see office door)

•  Students are advised to utilize the course TA’s given the large class size


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Course Material (Text) •  Introductory Concepts (Ch. 1-2) •  Fluid Statics (Ch. 3) •  Control Volume Equations (Ch. 4) •  Bernoulli Equation + Applications (Ch. 6,8) •  Similitude (Ch. 7) •  Flow in Pipes and Channels (Ch. 8) •  External Flows (Ch. 9)

– you are expected to read the text chapters, as we cannot cover all details in class.


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Additional References •  Fluid Mechanics (2008)

–  White •  Fundamentals of Fluid Mechanics (2009)

–  Munson, Young, and Okiishi •  Mechanics of Fluids (2010)

–  Potter and Wiggert •  Fluid Mechanics (2010)

–  Douglas et al. •  Fluid Mechanics (2009)

–  Cengel and Cimbala


Important Dates •  Dates to remember:

– January 5th, Lectures Begin – February 14th, Quiz #1 – February 20th – 22nd, Mid Term Break – March 20th, Quiz #2 – April 5th, Lectures End

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Frequently Asked Questions •  Is there an alternate grading scheme?

–  Not really, its up to me so long as your grade is higher than that of the official grade scheme.

•  Do we need to be able to integrate? –  Yes, occasionally, that is why you learned how to do it!

•  Do we need to know how to solve differential equations? –  Yes, occasionally we will develop some basic equations that can

easily be integrated.

•  Do we need other course concepts? –  Yes, basic physics (F = ma), statics concepts, and some calculus.

•  Will you give us old quizzes and exams? –  Many old quiz and exam problems will be covered in tutorial, that

is why you should attend them.

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Introductory Concepts and Definitions


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Introduction •  In Fluid Mechanics we are considering the

behavior of fluids in motion (dynamics) and at rest (hydrostatics): –  Fluid statics: typically considers fluid pressure on

stationary bodies and surfaces, pressure measurements, buoyancy and flotation, and fluid masses under rigid body motion, to name a few.

–  Fluid dynamics: typically considers flow fields, forces such as lift, drag, and thrust, flow in pipes, gas dynamics (compressible flows), and fluid machinery to name a few.


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Definition of a Fluid •  We all know what a fluid is. We have

experienced fluids both liquids and gases, but what is the appropriate definition of a fluid? –  Definition 1: A fluid is a substance that deforms

continuously under the application of a shear stress no matter how small that shear stress may be.

–  Definition 2: We can also define a fluid as a substance that cannot sustain a shear stress when at rest.

•  Other definitions also consider that fluids shear under the influence of their own weight.


Definition of a Fluid 11


•  Unlike the mechanics of solids, which attain a finite deformation under an applied load, fluids continue to shear under applied loads.

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Definition of a Fluid •  Fluid Mechanics principles apply to both

liquids and gases: –  Liquids: are fluids which form an interface or free

surface and have a clearly defined volume in open containers. Liquids consist of closely packed molecules. In fluid mechanics we denote free liquid surfaces using the following symbol. ____

–  Gases: are fluids which when placed in a container fill the volume completely. In some cases, we can see gases spilling out of open containers as they warm up, if the gas is initially cooler than the surroundings.

€

∇


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Methods of Analysis •  We typically use two approaches in Fluid

Mechanics analysis: –  Integral Approach: we apply fundamental laws (e.g.

mass, energy, and momentum) to finite control volumes using integral balances and obtain equations describing the flow on a macroscopic level, i.e. the gross behaviour of fluids

–  Differential Approach: we apply fundamental laws to infinitesimal control volumes and obtain differential equations which describe the flow on a microscopic level, i.e. much detail about local fields within the fluid

–  Fluids I focuses mainly on the Integral Approach, while Fluids II focuses mainly on the Differential Approach


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Engineering Units •  Three systems:

•  SI System of Units [M,L,t,T] – force is a secondary unit •  British Gravitational System [F,L,t,T] – force is a primary unit •  English Engineering System [F,M,L,t,T] - both mass and

force are primary units


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English Engineering Units •  When is a pound not a pound?

–  In English Engineering Units of course! •  In the EEU system, the unit of force is the pound force

(lbf or lbf) and the unit of mass is the pound mass (lbm or lbm). Length is in (ft), time in (s), and temperature in (R). Both force and mass are primary. Absolutely brilliant!!!

•  A force of one pound is the force that gives a one pound mass an acceleration equal to that of the earth’s gravity g = 32.174 ft/s2. i.e. 1 lbf = 1 lbm x 32.174 ft/s2

•  Newton’s Law then becomes:

€

F =m⋅ a gc


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English Engineering Units •  The constant gc = 32.174 (lbm ft) / (lbf s2) is required in

any relationships derived from Newton’s law. •  This constant of proportionality has both units and a

value that is not equal to unity. •  Care must be taken when working in the EEU system,

to understand when a lbf or lbm are specified. The course text adopts that (lb) is (lbm). So be careful.

•  We will primarily use the SI system. Occasionally, we will do an example in the EEU system.

•  Finally, note:

€

1⋅ kg = 2.204⋅ lbm

€

1⋅ lbm = 0.45359⋅ kg


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British Gravitational Units •  In this system, force is defined in (lbf), length in (ft),

time in (s), and temperature in (R). The unit of mass the slug is secondary, such that:

•  The weight of one slug in earth’s gravity is then:

•  This leads to, using Newton’s Law from EEU, as :

€

1⋅ lbf = 1⋅ slug( ) × 1⋅ fts2⎛

⎝ ⎜

⎞

⎠ ⎟

€

32.174⋅ lbf = 1⋅ slug( ) × 32.174⋅ fts2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

1⋅ slug = 32.174⋅ lbm =14.594⋅ kg

€

1⋅ lbm =1

32.174⋅ slug = 0.45359⋅ kg


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Example - 1 •  Let’s say a person who weighs 150 lbf in earth gravity is

weighed on the moon where g = 5.348 ft/s2. What is their new weight?

•  On earth we also say they have a mass of 150 lbm! Thus:

•  In the BGU system, this person has a mass of 4.662 slug, such that:

€

F =150⋅ lbm × 5.348⋅ ft /s2

32.174⋅ lbm⋅ ftlbf ⋅ s2

= 24.93⋅ lbf

€

F = 4.662⋅ slug × 5.348⋅ ft /s2 = 24.93⋅ lbf


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Example (continued) •  Now this same person has a mass of 68.04 kg in SI units •  On earth we would find this person weighs:

•  On the moon, this person now weighs:

•  In summary, SI is simpler. The BGU system is a little more practical than the EEU, as the pound is force and not mass as well. But you should see the equality in BGU and EEU, since m = lbm/gc.

€

F = 68.04⋅ kg × 9.81⋅ m /s2 = 667.46⋅ N

€

F = 68.04⋅ kg ×1.638⋅ m /s2 =110.9⋅ N


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Engineering Units •  Note that in the EEU system of units, the consequence

of defining a pound mass as being equal to the mass that gives a pound force in earth gravity, is that the force unit and mass unit are not equal. Hence the gc conversion factor required in Newton’s Law, F = ma/gc

•  In the BGU system, this is not a problem, as the slug is essentially the combined unit of mass, i.e. slug = lbm /gc . It has been converted for you, so Newton’s law is still F = ma.

•  In SI, there are no consistency issues, as F = ma as well.

•  Your Fluids text uses the SI and BGU systems, unlike your Thermodynamics text which used the SI and EEU systems.


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Engineering Units •  A final word of caution on units. Some engineering texts

use the EEU system and SI, others use the BGU system and SI, if they are dual unit.

•  Most texts prior to 1976 use the EEU system when tabulating properties.

•  Always keep a handy reference of conversion factors nearby. Your book has a nice one on the inside cover.

•  Don’t panic when reading your book. In fluid mechanics, many problems are easily solved the in EEU system, since the properties are given in terms of lbm! But there are instances where a simple conversion is required.


Properties of Fluids •  Fluid as a Continuum

– The concept of a continuum as applied to fluids refers to the state whereby the fluid properties vary smoothly such that we may apply differential calculus methods in our analysis.

–  In most engineering systems the dimensions of a system are usually substantially larger than the smallest critical volume used to define the continuum hypothesis

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€

ρ = limδV→δV *

δmδV⎛

⎝ ⎜

⎞

⎠ ⎟


Properties of Fluids 23


Properties of Fluids •  In Fluid Mechanics we frequently make

use of the following physical properties: – Density, ρ – Viscosity, µ – Surface Tension, σ – Speed of Sound, a – Vapor Pressure, pv

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Properties of Fluids •  Density: the density of a fluid is defined

in the same manner as it is for solids, simply put:

•  In fluid mechanics we frequently make use of other related properties such as the specific weight:

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€

ρ =massvolume

=kgm3

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

γ = ρg =Nm3

⎡

⎣ ⎢ ⎤

⎦ ⎥


Properties of Fluids •  We frequently make use of relative

properties in the context of liquids and gases, by defining ratios of a fluid’s density to that of water or air, i.e. specific gravity:

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€

SGgas =ρgas

ρair,@20C=

ρgas1.204 kg /m3[ ]

€

SGliquid =ρliquid

ρwater,@4C=

ρliquid1000 kg /m3[ ]

* Densities are usually specified at 20 C


Properties of Fluids •  Viscosity: the viscosity of a fluid is a measure of the

fluidity of a substance. For example we know that water flows more easily than honey.

•  Formally, the property we define as viscosity relates the shear stress in a fluid to the shear rate. For most common fluids this can be easily defined as:

•  µ is the viscosity coefficient, and the above equation is often called Newton’s Law of Viscosity

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€

τ = µ∂u∂y~ µ

uh


Properties of Fluids •  A fluid which follows Newton’s law of viscosity is

called Newtonian. •  Many fluids are Newtonian in character, e.g. water,

oil, air, honey, to name a few. •  Many other fluids are non-Newtonian. The

behaviour of these fluids does not follow Newton’s law of viscosity, e.g. ketchup, starch mixtures, gels, toothpaste, mayonaise, and silly putty*, to name a few.

•  Other fluids have time or shear dependent viscosity.

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Properties of Fluids •  Pseudoplastic Fluids: are shear thinning, that is their

fluidity increases with the shear rate, and their apparent viscosity decreases. e.g. ketchup

•  Dilatent Fluids: are shear thickening, that is their fluidity decreases with the shear rate, and their apparent viscosity increases. e.g. polymer solutions, starch mixtures

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€

τ = K ∂u∂y⎛

⎝ ⎜

⎞

⎠ ⎟

n

= K ∂u∂y

n−1

apparent−viscos ity

∂u∂y

=η∂u∂y

€

n <1 shear − thinningn >1 shear − thickening


Properties of Fluids •  Bingham Plastic Fluids: behave as a solid until

an initial yield stress is applied. e.g. toothpaste, thick suspensions

•  Other fluids also have apparent viscosity which depends on the shear history, i.e. they are time dependent. –  Thixotropic: apparent viscosity decreases with time

under constant shear stress –  Rheopectic: apparent viscosity increases with time

under constant shear stress

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€

τ = τ0 + µp∂u∂y


Properties of Fluids 31


Properties of Fluids •  Viscoelastic Fluids: are substances that possess

characteristics of both solids and fluids. That is after an applied stress is removed they partially return to their original shape.

•  Some examples are silly putty and asphalt. These substances at rest appear as solids, flow slowly under the influence of gravity, and also possess elasticity.

•  Understanding how fluids shear is a subject in itself called Rheology.

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Properties of Fluids •  Measurement of Viscosity: viscosity of

fluids is easily measured using a number of devices based on simple principles.

•  Two common types of viscometers are rotational devices and flow devices.

•  Rotational devices are based on simple Couette Flow (shear driven).

•  These utilize integrated forms of Newton’s law of viscosity.

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Example - 2 •  A simple viscometer called a disk on plate viscometer,

is based on measuring the tourque T required to spin a finite disk of radius R at constant angular speed ω in contact with a thin film of fluid of thickness h. Develop an expression for the viscosity.

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Example - 3 •  A simple viscometer called a cone on plate

viscometer, is based on measuring the tourque T required to spin a finite conical disk of radius R at constant angular speed ω in contact with a thin film of fluid of variable thickness h(θ). Develop an expression for the viscosity.

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Properties of Fluids •  Additional Configurations (text problems)

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Properties of Fluids •  Surface Tension – is a measure of the molecular

attraction along a liquid surface formed between a liquid and gas or two immiscible fluids. These forces cause the surface to behave as if it were a thin membrane stretched over the surface. We use the symbol σ and its units are: [N/m].

•  It is related to the capillary rise or dip of a fluid.

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Properties of Fluids •  The force balance between gravity and surface

tension leads to (assuming h > r), see sketch in class:

•  The contact angle θ is a measure of wettability of a fluid with a surface.

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€

2πrσ cosθ ≈ πr2ρgh

€

h ≈ 2σcosθρgr


Example - 4 39

•  Consider a very long small diameter capillary tube of diameter 1 mm. Determine the capillary rise of water assuming the contact angle is nearly zero. What happens when the diameter is reduced to 100 µm?


Properties of Fluids •  Speed of Sound – the speed of sound in a fluid is a

thermodynamic property. It is quite important in the analysis of compressible flows. In general it is defined as:

•  In compressible flows we make use of the Mach number defined as:

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€

a =dpdρ

General

a = kRT Ideal −Gas

€

a =1481m /s Watera = 343m /s Air

€

Ma =Va

*If Ma > 0.3, compressibility effects are strong


Properties of Fluids •  Vapor Pressure – is the pressure exerted by a saturated

vapor on a liquid surface at a given temperature. It is the same as the saturation pressure used in thermodynamics. When the fluid pressure is lower than the vapor pressure the fluid boils. This can be achieved two ways: –  pv = 2.34 kPa at 20 C –  pv = 101 kPa at 100 C

•  When the pressure in a liquid drops below the vapor pressure, small bubbles form, called cavitation bubbles. These can have destructive effects on propellers, create vibrations, or lead to pressure waves.

Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John’s, Newfoundland, Canada

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Pressure •  What is pressure?

–  pressure is the normal or compressive force per unit area exerted by a fluid at a point.

–  consider a small infinitesimal area A in medium of fluid at rest. At some point on this surface, a normal force is exerted by the fluid on the top and bottom of this area. The pressure is defined in the limit as this area becomes smaller, until it is the smallest area that can be considered measureable:

–  Pressure is an intensive property.

€

p = limA→A '

FA⎛

⎝ ⎜

⎞

⎠ ⎟ normal


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Pressure •  For a fluid at rest, pressure is the same in all directions

at this point. But can vary from point to point, e.g. hydrostatic pressure.

•  For a fluid in motion additional forces arise due to shearing action and we refer to the normal force as a normal stress. The state of stresses in a fluid in motion is dealt with in this course as well.

•  In the context of thermodynamics, we think of pressure as absolute, with respect to pressure of a complete vacuum (space) which is zero.

•  In Fluid Mechanics we often use gage pressure and vacuum pressure.


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Pressure •  Absolute Pressure

–  Force per unit area exerted by a fluid

•  Gage Pressure –  Pressure above

atmospheric –  Pgag=Pabs - Patm

•  Vacuum Pressure –  Pressure below

atmospheric –  Pvac=Patm - Pabs


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Pressure •  Common Pressure Units are:

– Pa (Pascal), mmHg (mm of Mercury), atm (atmosphere), psi (lbf per square inch)

•  1 Pa = 1 N/m2 (S.I. Unit) – 1 kPa =103 Pa – 1 bar = 105 Pa (note the bar is not an SI unit) – 1 MPa = 106 Pa

•  1 atm = 760 mmHg = 101,325 Pa = 14.696 psi


Velocity Fields •  We frequently work with velocity fields:

46


Velocity Fields •  In general, the velocity at any point in

space and time is defined as a vector:

•  The velocity vector can be resolved into components:

•  This defines the velocity of a particle of fluid at point (x,y,z) at some time t.

47

€

V = V (x,y,z,t)

€

V = uˆ i + vˆ j + w ˆ k


Velocity Fields •  We define several types of flow lines:

–  Pathline: is the path of a single moving particle of fluid. We create it, by connecting many points in time for a particle.

–  Streakline: is the path created by joining many particles that passed through the same point in space.

–  Streamlines: are drawn in a flow field, so that at any instant they are tangent to the direction of flow at every point in the flow field. Streamlines are commonly used to visualize the flow.

–  Timelines: if a number of adjacent particles are marked at some point, they form a timeline. At some point later in time the same particles are displaced and provide some information about the flow field.

–  In a steady flow field, streamlines, pathlines, and streaklines are coincident.

48


Velocity Fields

Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John’s, Newfoundland, Canada

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Stress Fields •  In a fluid we can resolve forces into

components and define stresses in an orthogonal system of coordinates, much the same as we do in solid mechanics:

50


Stress Fields •  Notation for stress components: the first

subscript denotes the axis which the stress plane is normal to, the second the direction it acts in:

51


Stress Fields •  3D state of stress on a fluid element:

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Classification of Fluid Mechanics 53


Classification of Fluid Mechanics •  Viscous Flow: viscosity dominates the

flow, and friction forces are dominant •  Inviscid Flow: is a frictionless flow (µ =

0), inertial and pressure forces are dominant

54


Classification of Fluid Mechanics •  Laminar Flow: is a flow in which the fluid

particles move in smooth layers or laminas. •  Turbulent Flow: is one in which fluid

particles rapidly mix due to random three dimensional motion.

55


Classification of Fluid Mechanics •  Internal Flow: flows are bounded by a

wall of a tube or channel.

•  External Flow: a body is completely immersed in the fluid.

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Classification of Fluid Mechanics •  Incompressible Flows: are those in which

changes in fluid density are negligible. •  Compressible Flows: are flows in which there

are appreciable changes in density, usually with gases. In liquid flows, compressibility effects result in cavitation and water hammer.

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Summary •  Material covers Chapters 1 and 2 from

the text book. •  Please read (carefully) pages 1 – 45. •  There is more detail in the book. •  We have covered all of the topics at

varying levels of detail. •  Additional problems will be assigned and

others covered in tutorial sessions.

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ENGR 5961 Fluids I · PDF file9 Introduction • In Fluid Mechanics we are considering the...

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