Fluid Mechanics: Fundamental Concepts, Fluid Properties (1 of 34)
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Fluid Mechanics: Fundamental Concepts, Fluid Properties (1 of 34)

And to start off, and we talk about this is
fluid mechanics ok. The definition first of all of a fluid, the
official definition in Chapter 1, let me write that down. It’s a substance that deforms continuously
when acted upon by a shearing stress of any magnitude. How many people in here have had ME 218 strength
of materials? Ok almost everybody of course. Take a piece of copper plate, copper plate
and apply force F on it. Here’s the plate. I don’t know, an inch think, six by four
or something like that. Apply 400 pounds on it, the top. This is fixed to the base. When you apply that force to it, you know
from ME 218 for instance. There’s the plate. Now, again here’s the ground. Now I take a layer of water, I apply a force
on there. No, it doesn’t look like that when I’m done
applying force to that plate on top of water. Think what would happen if you apply a force
to that plate on top of a water layer, I apply 10 pounds, will the plate ever stop? Of course not. If I apply 10 pounds, this room is flooded
with water this high, I’m walking in the water at my knees. I put a Styrofoam plate on top of it, I push
on it 2 pounds , 2 pounds with my finger, is it going to stop me from pushing ever? No, and the key word is, oh it’s continuously. This will deform a little bit, then it’s going
to stop; but a fluid will deform continuously. So yeah, that’s fluid, air, oil, jet fuel,
you name it, these things that we live with, water, they are fluids. Now there are different kinds of fluids though. We are all use to fluids like water, or coffee. Pour it from there to there, works really
nice, works really nice. What’s in this cup went in that cup. Now fill that cup there with cold air, try
and pour cold air into that cup over there. No it won’t do it, it won’t do it, it
will spread out when it comes out, it will go all-apart in a big plume. Why? Because the molecules are not attached
to each other as strongly as they are in that liquid. Are they both fluids? sure! of course they
are. Put air in here, push on a plate in air it
will still move. Yeah they are both fluids, liquids and gases,
but not solids, not solids. Well maybe not quite, maybe not quite. Take the cap off a toothpaste, turn it upside
down, does it run out? Not really, no, unless its been in the car
for 2 hours in 100 degree temperatures. No, what makes it come out? I’ve got to press on the tube. Ok so it takes an initial stress, shearing
stress to move it, to get it moving. Once it’s moving it moves pretty good, but
it takes that initial shearing stress. It could be, you know, some very thick things
like some epoxy’s you know, you can turn them upside down, they won’t come out until
you squeeze the tube, a plunger. So yeah those things are kind of like fluid,
but we won’t discuss those in this class. We are not going to worry about you know tooth
paste epoxy and things, we will worry about water, air, oil and things like that, helium,
freon, you name it . So that’ a big different in that, and we are
also going to look at two sets of units. You know a lot of textbooks now they are almost
all in SI, which is so much easier than British system, so much easier. But unfortunately the real world doesn’t go
that way. So our textbook, I forget the percentage,
but it has quite a few, I will call it English as British scribitation. Quite few problems whether they are example
problems or homework, in English and then a lot of SI, a lot of SI. But you graduating have to be conversant in
two languages; otherwise you are not going to show off very well in the real world. You have to know two languages forward and
backwards really really well. The easy one is SI, the hard one is English
engineering. For instance, how many centimeters in a meter,
well you can guess that one ok? How many millimeters in a meter? I’ll tell
you, about a thousand. How many cubic centimeters in a liter? I’ll
tell you, it’s 1000. Do you get the point, everything in SI. Will you go out there and close that door
please? Thanks. Everything, every convergent factor. Ok let me tell you another one. 1 joule divide by 1 second, what is it? Joules
per second. Uh huh, yeah 1. Is it 1? Yeah it’s 1. Ok 1. Everything, convergent SI starts with a 1
times ten to a power, you can’t get any easier than that, now you tell me this, how many
pints in a quart, how many quarts in a gallon, how many gallons in a barrel? Ho ho ho, nothing starts with 1, nothing,
memorize that stuff? No, no, no you don’t memorize it’s right
here. You don’t memorize that stuff, but you still
have some varying instinct. Well I don’t know, but I got this thing in
my pocket, yeah I’ll tell you in a minute ok? Yeah geez, that’s ok but I mean if you divide
by gallons in a barrel no even amount will come out. No, it’s awful; it’s awful, oh power. How many foot pounds per second is a horsepower? Well you know, how about, how many watts in
a kilowatt? Well I’ll tell you, how about a thousand? 1 times 10 to the 3rd. So oh yeah, I mean don’t you wish the whole
world was SI? Of course we all do. But it’s not, and you live in a country that’s,
you live in the worst country for it, a country that is still has both unit systems, so you
have to be really really really good at both of them. That’s why, that’s why my point is, that’s
why when we select a textbook, we make sure the textbook has, if we can find it, duel
unit systems in it, because we know the more you practice the better you get. The problems is most work problems in SI where
they need to practice in English engineering. Ok well that’s neither here nor there, we
there’s problems here there will be problems on homework of both, there will be problems
on tests of both. You can use your, you can use your smart phone
for conversions, there’s conversions on the inside back cover too, SI to British gravitational. So your choice but there will be both kinds
of problems ,for homework and exams. Ok lets just briefly go over the two unit
systems. Let’s do SI first. Ok, force is in Newton’s. Our mass is in kilograms. And of course length is in meters. Ok lets do British Gravitational. Force is in pounds. Mass is in slugs. Length is in feet, and they both have seconds
for time. There is a third unit system that has force
in pounds force, mass in pounds mass, but this is the British Gravitational. You don’t put a subscript on this guy, because
you know if it’s in this system, and it says pounds it has got to be force. So you can skip putting the subscript F on
pounds. Unfortunately you don’t deal with pounds under
mass, you deal with slugs. These are not little creepy crawly slimy things
on to the floor, these are not snails they are slugs. A slug is a unit of mass, how big is it? Anybody
know how many pounds mass are in a slug? What’s the gravitation acceleration in inches,
32.2 right? There you go, 32.2 pounds mass in 1 slug. That means the slug is more massive than a
pound mass, a slug is more massive. Ok so, those are the two unit systems, when
you work on problems for homework or exams that’s what you use there. Ok now, lets take a look, if you’ve had physics
or thermo… Rho density, density mass per volume. If it’s in SI, kilograms per cubic meter. If it’s in British gravitational, mass slugs
per cubic foot. Specific weight, gamma, weight per volume. SI, weight, weight is a force. Newtons per cubic meter. British Gravitational, force, pounds per cubic
foot. Specific gravity. Ok, gamma of the fluid in question under gamma
of water at a certain temperature, like I think it’s 4 degrees, yeah 4 degrees C. R is 39 degrees F, so it’s a ratio, a ratio
is dimensionless. So it’s a dimensionless number, specific gravity. Gamma is the specific weight of the fluid
you are studying, divided by gamma of water at a certain temperature. Ideal gas, our perfect gas law. P over rho equal R times T. P and Tare absolute values. Absolute pressure, absolute temperature, we
will mention absolute pressure in just a little bit. You know absolute temperature at 273 degrees
C, degrees K. Add 460 degrees F to get degrees R, it’s in
the book. This is for gasses, so the good news is what
we are studying in this class, pretty much all the gasses behave like that. As long as you are around ambient conditions
of air, hydrogen, oxygen, nitrogen behave like that. Viscosity, viscosity is mu, sometimes called
absolute viscosity. Ok now, lets get to our definition of mu. Mu this is the big property that’s different in a thermal
class. What page is that on? Well look after if that’s ok, if I find
it. This is shearing stress, tau is
shearing stress from ME 218. Du dy, this is called a velocity gradient. So if we take a simple case again, take that
plate on water, and here is water. Maybe the velocity looks like this, this will
be capital V velocity, and then this is typical of how the velocity might look between the
plate. Let’s make it linear, make life easy right
now, we don’t want to get too exotic day 1. So this is called a linear velocity gradient
because it is a straight line. Velocity versus the height, the height is
y, the velocity is lower case u. U is a function of y, differentiate that,
that’s called a velocity gradient. You take the velocity gradient like in a bearing. Bearing, or surface that is rotating on its
stationary, stationary, moving. You got 10 weight oil, that’s mu, multiply,
where do you get mu of oil? Back of the book in the appendix . What’s the temperature? I don’t know, 120 degrees fahrenheit. Go back to the book, find mu, put it in here,
multiply it by du dy, which is v divided by the separation distance. And that’s going to give you the shearing
stress that the fluid creates. So viscosity is a property, which is useful
to determine shearing stresses. And there is another property, which is nu,
which is mu over rho; this is called kinematic viscosity. So there are two possible viscosities. One is the real thing, the real thing is this
one up here, this is really the viscosity. If someone just says viscosity to you then
you assume they mean this one. That’s why it is also called absolute and
kinematic. Just to be certain that someone knows what
you are giving them. If you say the viscosity though, the default
definition is here. Kinematic viscosity is this value divided
by the density rho. This thing comes in handy we will see later
on in some equations. It appears the ration mu divide by rho, appears
in some equations so it’s given a special name called the kinematic viscosity. Ok, the units, aw the unit system. If you don’t memorize those things, I hate
to. If you don’t memorize them, then figure
out what they are, what is this guy? Force divided by area, ok so, force divided
by area. The area of this is length squared. What’s this guy? Velocity, length over time. What’s this guy? Length. Ok, cancel, cancel. So figured out the mu is F, this guy is in
the denominator, this guy is in the denominator, put him up here, FT over L squared. This nu kinematic viscosity, divide that guy
by mass divided by length cubed, density. Mass divided by length cubed. We go back here. Viscosity, English units, what are they? Pounds
second per foot squared. Look at it. Pound over there, pound second per foot squared,
yeah. I don’t memorize that stuff, I derive one
if I have to, if I have to. So there is no reason to memorize it because
you know you can derive it, where does it come from? The basic governing equation right
there. Ok, now let’s go on to our last two properties. Next one, by the way before we forget, lets
just go ahead and put this graph up here. This is shearing stress Tau right there, this
is du dy velocity gradient, sometimes called the rate of shearing strain, but the velocity
gradient is sufficient. This is air, this is water, and this is oil. Oh yeah we know intuitively oil is viscous. It will give you very high shear stress for
certain value of du dy, given the value of du dy, air is not very viscous, water is in
the middle of the road, oils are very viscous. Notice those are linear too, which is very
important, those are linear. Those fluids are called Newtonian. There many which are not linear, which we
won’t will study in ME 311, Examples, tooth paste, no its not Newtonian. Latex paint, no it’s not. When you put that paint on a brush, put that
brush in water, pull it out of there, oh the water just runs off. You would be running for the wall, to get
up to the wall before it ran off your paint brush. Now you put that in there, in latex paint,
take it out there, and turn the brush, it stays 124 00:22:27,989 –>00:22:32,259 on the brush fibers until you get to the wall, then you just roll it on or paint it on. Yeah it’s different, it’s not linear, it’s
not linear, the graphs are in the book that way. quick sad, not linear, no no. The more you try and get out, the harder it
is for you. You got to go really really slowly, dont try
and pull your foot out, you go in deeper. It’s all in this graph, tells the whole story. I’ll just show you a couple, some go like
this, some go like that, here is quick sand, here is latex paint, there is tooth paste,
you know, maybe stuff like that. That’s just for your own sideline right now,
what your worried about, are the 3 things us ME’s and CE’s look at most in life. Aero’s & ME’s, Aero’s & CE’s, umm mostly
ME’s, engines and things like that you know. So we are not going to worry about those other
exotic ones, we are going to worry about things like these 3 right here. They are all going to be Newtonian Fluids
in ME311. Ok let’s go to 1 more, ok lets see here we
go; surface tension. Surface tension, the symbol for that is sigma,
there’s sigma, and sigma is the force per unit length, so sigma has dimensions of force
per length. Here is a reservoir; I put this tube in there,
inverted. The liquid will rise up in that reservoir
something like this, you might have seen a mercury barometer on the wall, and the mercury
rises up that glass tube and you read off what the mercury level is. The guy night on the news says, “Ok the
weather forecast, cold front is coming on from Canada, and the barometer is dropping,
its right now 30.05 inches of mercury.” We say, “What do you mean there is mercury
dropping out of the sky on people’s heads?” No, no, now ask any of your friends at night,
say to them, “What does he mean when he says the pressure is 30 inches of mercury?” Most folks don’t have the slightest of course,
they don’t, ask them their tire pressure, oh they will tell you 30-32 pounds in my tires. Do they really mean 32 pounds of air is in
your tire? No, no , they taking shortcuts, they mean
to say 32 psi in my tire. Well, if you put mercury in here, fill the
tube with mercury, invert it, thumb off there, the mercury rises up. At this point right here, mercury does this. Around the glass perimeter, where the glass
and mercury are in contact, the mercury is being pulled down, that’s surface tension,
that’s surface tension. Take water, here is a tube of water. Do the same thing, it doesn’t work quite as
well. But it looks like that, the water is being
pulled up by the glass it seems like, so it’s a property, it’s a property of fluid and temperature. So that’s why it’s force per unit length,
what length is this length? It’s the perimeter of the glass tube, the
perimeter pi times d. So that’s a phenomenon that occurs quite often
in fluids that you should be aware of. What’s this thing called? Surface tension. Try and put, you can do it if you do it really
carefully, a dime, a dime on water if you are really really really careful, you can
float it on water, it won’t sink, it won’t sink. Something is holding, pulling it up around
the perimeter, something is pulling it up, and that’s the water doing that, and that’s
the surface tension in the water. So those are all part of the concept of surface
tension. Ok I think that pretty much covers all of
our important properties in there, so let’s take a look then at pressure. All right, now let’s go back over here to
this thing on here, let’s talk about pressure. Its one of the big concepts in our first fluids
course What do you call pressure in ME218 strengths?
Pressure? Stress, what’s stress? Force per unit area,
what’s pressure? Force per unit area. Right right, different but just different,
those guys are solids, these guys are liquids. Ok, let’s start off by absolute pressure. We will put them all down, Gage pressure. Ok, absolute, referenced to 0 pressure. Referenced to local atmospheric pressure. We’ll use our standard pressure, 14.7 psi. And I think for our, well just use 101 kpa. It’s like 101.3 something something something,
101 kpa. So these are the standard pressure values,
we are not going to use the value in Pomona which is I don’t know 800 feet above sea
level. These guys are at sea level, and I don’t
know 34 degrees data too I don’t know, something like that. They are standard values, we will use those. Ok gage pressure, reference to local atmospheric
pressure. The world local means where you are when you
make the measurement, local means where you are when you make the measurement. If you are in Pomona at Cal Poly Pomona, it’s
the pressure here, that’s’ the local atmospheric pressure. If you are in Denver Colorado, it’s the pressure
in Denver Colorado, at that time of the day, that day of the year, so that’s reference
to where you are. Ok now let’s talk about these guys, gage pressure
can be positive or negative. These guys are always positive because their
references to zero, unless there is a negative absolute pressure. Negative gage pressures are sometimes called
vacuum pressures. So hear the word vacuum pressure, and some
gages will show you positive pressures and vacuum pressures, that’s what they mean by
that. So let’s draw a little picture here to show
us these pressures. The pressures, I will put a little board with
the values of some conversion factors too for you. Let’s start off with zero pressure down here. Zero absolute, and let’s say that this is
the local atmospheric pressure. And let’s call that 101 kpa. And lets call the pressure up here PA. If I measure PA from the local atmospheric
pressure, reference to local atmosphere pressure, that’s this line right here. And PA is going to be, this is given, 200
kpa, gage. This is 101 kpa, absolute. This pressure is 101 plus 200, its not to
scale, 301. If the pressure is down here, and let’s say
that it’s minus 50 kpa gage vacuum. Then the absolute pressure, always measured
from the bottom down up to here, 101 minus 50, 51. 52 kpa and that’s absolute. In the textbook I think he gives us a big
hint, lets see what pressures are her, this picture is in the book too by the way, yeah
so on page 50. Ok here’s what he says in italics. In this textbook, pressures will be assumed
to be gage pressures unless there is specifically designated as absolute. In this book, or homework, in an exam, or
a problem, the pressures are assumed to be gage unless it specifically says absolute. So If I give you a problem on an exam, and
it says the pressure is 50 kpa, that means you assume that’s gage. Unless I put that, then it’s absolute, so
that’s default rule. The default rule is, pressures that just are
in kpa, assume that they are gage, unless they are absolute, there will be a parentheses
with abs after. Or the problem will say the absolute pressure
is 50 kpa. Ok, in English engineering or British Gravitational,
if someone tells you pressure of the air in your tire, make it easy, is 30 psi, how did
they measure? Oh probably a tire pressure gauge I would
suspect, tire pressure gauge. How does a tire pressure gauge work? Oh a typical example, there is a spring in
this thing and the stem pops up and you read it, or there’s a circular gauge you put
on your tire and read it. Take the spring loaded one, what’s the pressure
in your tire doing to that thing? Oh it’s compressing the spring as it pushes it up. What’s it working against? What’s outside
that pencil like device? Atmospheric pressure. So what’s it measuring the pressure with respect
to? Uh huh, 30 psi means that 30 psi gage, and
we don’t say 30 psi gage, we say my tire pressure is 30 psig, that’s standard practice, psig
means gage. If somebody says well then what’s the absolute
pressure of the air in my tires, there it goes, there it is. You’re up here now at 30, 30 above local
atmosphere, 30, local atmosphere, I think I erased it, yup there it is, 14.7. 30 plus 14.7, 44.7 psia, means absolute, that’s
just the way that most engineers work, they use psig for gage pressure, psia for absolute
pressure. But in the world of SI, in the world of SI
you don’t say, you don’t say this, there is no such unit kpag, it’s no, there is
no kpag in the world. So what you do? You have a rule; textbook
rule says if I don’t put anything after kpa, you assume that it’s a gage. If it’s absolute, I’ll put abs after it. Just so on a exam you don’t say professor
Biddle what do you mean by that kpa, I’ll say go back to default what did it say? You only get confused on an exam at the time
of the exam, it makes life miserable. Ok so, that’s the pressure now, and the guy
that says what do you want your tires at 35 pounds? Say no I want 35 psi, because they take a
short cut, they just say pounds, they really mean psi of course, but they say pounds. Ok now let’s take a look at some problems,
similar to homework. Lets see which one I looked at here; I will
take this one first because this is a new one. 177, 177 you have 178 and 180 they are all
the same kind of problem, they’re viscosity problems, viscosity problems. So we don’t need this. Ok I will read it for you, it’s a snow sled. The sled shown in the picture slides along
a thin horizontal layer of water, between the ice and the runners. So here’s the sled, problem 177, here’s
ice, I’m going to expand it up, here’s water, here ‘s the runner on the sled, a
blade, that’s a skate blade. The horizontal force that the water puts on
the runners is equal to 1.2 pounds. Here’s the, ok ill put this for you guys. The sleds’ speed is 50 feet per second. The sled, the runner, the blade runs on water
not ice, water, the total area of both the runners in contacts with the water is 0.08
square feet. The viscosity of water if they didn’t give
it to you, it’s in the back of the book, but they gave it to you, viscosity of water 3.5. So this is mu, they didn’t say kinematic
they just said viscosity assume its absolute. 3.5 times 10 to the minus 5 pound seconds
per foot squared. Determine the thickness of the water layer
under the runners. D, find d. Keep reading; assume a linear velocity distribution
in the water layer, assume a linear velocity distribution in the water layer. So draw this. V is the velocity at the top of the water
layer, it goes down to 0 at the ice, the velocity at a solid surface is assumed to be at rest
for the water, it’s called the no slip condition. Right here, this is called no slip. It means the water doesn’t slip along the
ice, the velocity of the water at the surface of the ice is assumed to be zero. We will make that assumption pretty much throughout
this whole course, the no slip condition. Ok, so I erased it, yeah I erased our equation,
so we have our tau equal mu du/dy,. Our shear stress we know is force over area,
mu du/dy, we are measuring y from here up like this. Ok force is 50 pounds force, 50 pounds, aww 1.2lbs,
velocity is 50. Ok 1.2 divided by
area 0.08, equal mu, ok there is mu, right
there 3.5 times 10 to the minus five, pound second per foot squared. Multiply by du/dy, don’t forget it’s a linear
profile, so you know lucky you, du/dy if it’s linear, equal delta u over delta y. Delta u at the top, v at the bottom, zero. Delta y at the top, d, at the bottom, zero. there it is, du/dy, is delta u over delta
y, equal v over D. V, 50, d is going to be in feet, check it
out, it better come out right here. Ok seconds divide by seconds. Feet, feet squared, feet squared here, feet,
feet gone, the right hand side, pounds per foot squared, left hand side, pounds per foot
squared, we are good to go. Solve for d, it will be in feet, 11.7 10 to
the minus 4. 1 ten thousandths of an inch, 1 ten thousandths
of an inch, that’s the amount of liquid water between the blade surface and the ice on the
Kings hockey people. Not a lot, it doesn’t take a lot, it doesn’t
take a lot, but there is a layer of water under there. I want to show you how you set up the homework
problems, and I will go over it again when you pass your first homework set in, but do
that , the first thing you do, is you draw a sketch is it is appropriate. I mean were engineers, we love sketches. If you are a mathematician, you love equations
and you hate sketches; you want the theoretical solution. If you are an engineer you love pictures,
and you will do the math because you have to do the math, you don’t love the math maybe
but you will do it, because you have to do, it’s in your toolbox. So most engineers think more clearly if they
have a picture, on exam I guarantee, that if you try and sketch something on exam, whether
it is a free body diagram which is essential, ME 214. What it does to you, is it builds a time delay
in your brain, you don’t start writing down and putting numbers or something. And you read the problem, when you sketch
it your mind is working in the background, believe it or not, and it makes life a lot
easier if you do it that way, I guarantee it. Because the way you get flustered & frustrated
is trying start putting numbers on that handheld calculator right away, oh that just destroys
you. If your unsure of yourself that puts you down,
way down there, what you do is you build about 60 second time delay in there so your mind
starts to think about that problem, and the best way to do it, is to draw a picture. When you draw that picture, your mind starts
to think, it’s amazing, the engineering thought process. So anyway, if it’s appropriate, it’s not
all the time, but if it’s appropriate draw a sketch. Once you draw the sketch, I don’t care the
problem has it listed there in paragraph 1, if you put these guys down and look at the
picture you start to see things. When you write the equation down you start
to things, what do I know, what don’t I know, I know it, I know it, I know it, I know it,
I don’t know it, your mind starts to think like that. So much of engineering is having an approach
to solve a problem. We are not backyard you know in your garage
making something, you know that’s you know, where not doing that in your garage, we engineers
are trained to think correctly, it’s pretty fascinating, pretty fascinating. Ok next thing you do, put the equation down,
in symbolic terms. I don’t want to see any numbers in there,
until you write the equation down so I know you are using the right equation, write the
equation. Then you put the numbers in them, then you
put the units by each number, I am repeating things I know I’ve been beating your heads
for 3,4, or 5 years, but I am going to do it one more time. Put the units in, cancel them out like this,
and end up with what you want. All those steps are important steps, as you
do them in the right order, you will be amazed of how things kind of fall out. There might be three or four possible equations,
you might write down three equations. You look at the equations, you look at this
over here, and you say uh no that’s not going to work, one equation two unknowns. Uh no that’s not going to work, one equation
three unknowns, um yeah, he is going to work. So you don’t always know, you don’t always
know which equation might work at the start, sometimes you do, sometimes you don’t. But if you don’t, write down the equations
you think might work, and stare at it, stare at them a while, maybe something will jump
out at you, hopefully, hopefully. Whether it is homework or an exam, now let
me tell you something else, you don’t get ready for an exam by copying the solution
manual. No, no, no, I mean you, tell Kershaw did you
read how to throw the curve ball before the game? He said, are you kidding me? No, I don’t
do stuff like that. I throw it myself 50 times on the sidelines. I developed a new pitch, a slider. How? No, no, I practiced it; I throw a thousands
times on the sideline. You don’t get good until you suffer through
some homework, and say gosh I just don’t get this. I went through that, I get so mad sometimes,
I take my fist and I go, I don’t know why I’m not getting this, at home, in my bedroom
with some nice music on and a Pepsi in my hand. But I still got mad; I got so frustrated and
flustered, but I worked my way through it, and boy you are better for it, you are better
for the pain. The way you prepare for your exams is you
are not surprised. And how are you not surprised? You look at
all the problems I worked in class, you read the example problems in the textbook, and
you go over homework. Don’t read their solutions, because you won’t
be ready for the exam then. Oh sure you’ve seen the solutions, but on
an exam, you won’t be ready, you have got to practice it yourself, and put yourself
in a tough spot, in a real tough spot. Ok we have finished chapter 1, here are your
assignments the next 2 class meetings, oh and by the way my office hours I can make
other times Monday, Tuesday, and Wednesday I’m here all 3 days. If those office hours don’t match with yours
let me know, we can meet some other time of day that’s fine, all right we will see you
then on Wednesday.


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