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IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2024.06.01 Introducing a competency framework for languag learning materials ...
boundary layer good 2.ppt
1. 1
SOLUTION FOR THE BOUNDARY LAYER ON A FLAT PLATE
Consider the following scenario.
1. A steady potential flow has constant velocity U in the x direction.
2. An infinitely thin flat plate is placed into this flow so that the plate is
parallel to the potential flow (0 angle of incidence).
Viscosity should retard the flow, thus creating a boundary layer on either
side of the plate. Here only the boundary layer on one side of the
plate is considered. The flow is assumed to be laminar.
Boundary layer theory allows us to calculate the drag on the plate!
x
y d
U
U
u
plate
2. 2
A STEADY RECTILINEAR POTENTIAL FLOW HAS ZERO
PRESSURE GRADIENT EVERYWHERE
x
y d
U
U
u
plate
A steady, rectilinear potential flow in the x direction is described by the
relations
According to Bernoulli’s equation for potential flows, the dynamic
pressure of the potential flow ppd is related to the velocity field as
Between the above two equations, then, for this flow
0
y
v
,
U
x
u
,
Ux
const
)
v
u
(
2
1
p 2
2
pd
0
y
p
x
p pd
pd
3. 3
BOUNDARY LAYER EQUATIONS FOR A FLAT PLATE
x
y d
U
U
u
plate
For the case of a steady, laminar boundary layer on a flat plate at 0
angle of incidence, with vanishing imposed pressure gradient, the
boundary layer equations and boundary conditions become (see Slide
15 of BoundaryLayerApprox.ppt with dppds/dx = 0)
0
y
v
x
u
y
u
y
u
v
x
u
u 2
2
U
u
,
0
v
,
0
u y
0
y
0
y
Tangential and normal velocities
vanish at boundary: tangential
velocity = free stream velocity far
from plate
0
y
v
x
u
y
u
dx
dp
1
y
u
v
x
u
u 2
2
pds
4. 4
NOMINAL BOUNDARY LAYER THICKNESS
x
y d
U
U
u
plate
Until now we have not given a precise definition for boundary layer
thickness. Here we use d to denote nominal boundary thickness, which
is defined to be the value of y at which u = 0.99 U, i.e.
U
99
.
0
)
y
,
x
(
u y
d
x
y
u
U
u = 0.99 U
d
The choice 0.99 is arbitrary; we could have chosen 0.98 or 0.995 or
whatever we find reasonable.
5. 5
STREAMWISE VARIATION OF BOUNDARY LAYER
THICKNESS
Consider a plate of length L. Based on the estimate of Slide 11 of
BoundaryLayerApprox.ppt, we can estimate d as
or thus
where C is a constant. By the same arguments, the nominal boundary
thickness up to any point x L on the plate should be given as
d UL
,
)
(
~
L
2
/
1
Re
Re
2
/
1
2
/
1
U
L
C
or
U
L
~
d
d
2
/
1
2
/
1
U
x
C
or
U
x
~
d
d
x
y d
U
U
u
plate
L
6. 6
SIMILARITY
One triangle is similar to another triangle if it can be mapped onto the
other triangle by means of a uniform stretching.
The red triangles are similar
to the blue triangle.
The red triangles are not
similar to the blue triangle.
Perhaps the same idea can be applied to the solution of our problem:
0
y
v
x
u
,
y
u
y
u
v
x
u
u 2
2
U
u
,
0
v
,
0
u y
0
y
0
y
7. 7
SIMILARITY SOLUTION
Suppose the solution has the property that when u/U is plotted against y/d
(where d(x) is the previously-defined nominal boundary layer thickness) a
universal function is obtained, with no further dependence on x. Such a
solution is called a similarity solution. To see why, consider the sketches
below. Note that by definition u/U = 0 at y/d = 0 and u/U = 0.99 at y/d = 1,
no matter what the value of x. Similarity is satisfied if a plot of u/U versus
y/d defines exactly the same function regardless of the value of x.
plate
x
y d
U
U
u
u
U
x1 x2
0
0
1
1
u/U
y/d
profiles at
x1 and x2
0
0
1
1
u/U
y/d
profile at x1
profile at x2
Similarity
satisfied
Similarity not
satisfied
8. 8
SIMILARITY SOLUTION contd.
So for a solution obeying similarity in the velocity profile we must have
where g1 is a universal function, independent of x (position along the
plate). Since we have reason to believe that
where C is a constant (Slide 5), we can rewrite any such similarity form as
Note that is a dimensionless variable.
If you are wondering about the constant C, note the following. If y is a
function of x alone, e.g. y = f1(x) = x2 + ex, then y is a function of p = 3x
alone, i.e. y = f(p) = (p/3)2 + e(p/3).
d
)
x
(
y
f
U
u
1
2
/
1
2
/
1
U
x
C
or
U
x
~
d
d
x
U
y
U
x
y
,
)
(
f
U
u
2
/
1
9. 9
BUT DOES THE PROBLEM ADMIT A SIMILARITY
SOLUTION?
Maybe, maybe not, you never know until you try. The problem is:
This problem can be reduced with the streamfunction (u = /y, v = -
/x) to:
Note that the stream function satisfies continuity identically. We are not
using a potential function here because boundary layer flows are not
potential flows.
0
y
v
x
u
,
y
u
y
u
v
x
u
u 2
2
U
u
,
0
v
,
0
u y
0
y
0
y
3
3
2
y
y
x
y
x
y
U
y
,
0
x
,
0
y y
0
y
0
y
10. 10
SOLUTION BY THE METHOD OF GUESSING
We want our streamfunction to give us a velocity u = /y satisfying the
similarity form
So we could start off by guessing
where f is another similarity function.
But this does not work. Using the prime to denote ordinary differentiation
with respect to , if = f() then
But
x
U
y
,
)
(
f
U
u
)
(
F
y
)
(
F
y
u
x
U
y
so that )
(
F
x
U
1
U
u
11. 11
SOLUTION BY THE METHOD OF GUESSING contd.
So if we assume
then we obtain
This form does not satisfy the condition that u/U should be a function of
alone. If F is a function of alone then its first derivative F’() is also a
function of alone, but note the extra (and unwanted) functionality in x via
the term (Ux)-1/2!
So our first try failed because of the term (Ux)-1/2.
Let’s not give up! Instead, let’s learn from our mistakes!
)
(
F
)
(
F
x
U
1
U
u
not OK OK
12. 12
ANOTHER TRY
This time we assume
Now remembering that x and y are independent of each other and
recalling the evaluation of /y of Slide 10,
or thus
Thus we have found a form of that satisfies similarity in velocity!
But this does not mean that we are done. We have to solve for the
function F().
)
(
F
x
U
)
(
F
U
x
U
)
(
F
x
U
y
)
(
F
x
U
y
u
)
(
F
)
(
f
,
)
(
f
U
u
13. 13
REDUCTION FROM PARTIAL TO ORDINARY
DIFFERENTIAL EQUATION
Our goal is to reduce the partial differential equation for and boundary
conditions on , i.e.
to an ordinary differential equation for and boundary conditions on f(),
where
To do this we will need the following forms:
3
3
2
y
y
x
y
x
y
U
y
,
0
x
,
0
y y
0
y
0
y
x
U
y
,
)
(
F
x
U
x
U
y
x
2
1
x
U
y
2
1
x
2
/
3
14. 14
The next steps involve a lot of hard number crunching. To evaluate the
terms in the equation below,
we need to know /y, 2/y2, 3/y3, /x and 2/yx, where
Now we have already worked out /y; from Slide 12:
Thus
REDUCTION contd.
3
3
2
y
y
x
y
x
y
x
U
y
,
)
(
F
x
U
x
2
1
x
,
x
U
y
)
(
F
x
U
U
y
)
(
F
U
y2
2
)
(
F
U
y
)
(
F
x
U
U
y
)
(
F
x
U
U
y3
3
15. 15
Again using
we now work out the two remaining derivatives:
REDUCTION contd.
x
U
y
,
)
(
F
x
U
x
2
1
x
,
x
U
y
)
(
F
)
(
F
x
U
2
1
x
)
(
F
x
U
)
(
F
x
U
2
1
)
(
F
x
U
x
x
)
y
(
F
x
U
2
1
y
)
(
F
)
(
F
)
(
F
x
U
2
1
)
(
F
)
(
F
x
U
2
1
y
x
y
y
x
2
17. 17
Now substituting
into
yields
or thus
Similarity works! It has cleaned up the mess into a simple (albeit
nonlinear) ordinary differential equation!
REDUCTION contd.
)
(
F
)
(
F
x
U
2
1
x
)
y
(
F
x
U
2
1
y
x
2
)
(
F
U
y
)
(
F
x
U
U
y2
2
)
(
F
x
U
U
y3
3
3
3
2
y
y
x
y
x
y
F
x
U
F
)
F
F
(
x
U
2
1
F
F
x
U
2
1 2
2
2
0
F
F
F
2
18. 18
From Slide 9, the boundary conditions are
But we already showed that
Now noting that = 0 when y = 0, the boundary conditions reduce to
Thus we have three boundary conditions for the 3rd-order differential
equation
BOUNDARY CONDITIONS
)
(
F
)
(
F
x
U
2
1
x
)
(
F
U
y
0
F
F
F
2
U
y
,
0
x
,
0
y y
0
y
0
y
x
U
y
1
)
(
F
,
0
)
0
(
F
,
0
)
0
(
F
19. 19
There are a number of ways in which the problem
can be solved. It is beyond the scope of this course to illustrate numerical
methods for doing this. A plot of the solution is given below.
SOLUTION
0
F
F
F
2
1
)
(
F
,
0
)
0
(
F
,
0
)
0
(
F
Blasius Solution, Laminar Boundary Layer
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
f, f', f''
f()
f'()
f''() F
F
F
F
,
F
,
F
20. 20
To access the numbers, double-click on the Excel spreadsheet below.
Recall that
SOLUTION contd.
F F' F''
0 0 0 0.33206
0.1 0.00166 0.033206 0.332051
0.2 0.006641 0.066408 0.331987
0.3 0.014942 0.099599 0.331812
0.4 0.02656 0.132765 0.331473
0.5 0.041493 0.165887 0.330914
0.6 0.059735 0.198939 0.330082
0.7 0.081278 0.231892 0.328925
0.8 0.106109 0.264711 0.327392
0.9 0.134214 0.297356 0.325435
1 0.165573 0.329783 0.32301
1.1 0.200162 0.361941 0.320074
1.2 0.237951 0.393779 0.316592
1.3 0.278905 0.42524 0.312531
1.4 0.322984 0.456265 0.307868
1.5 0.370142 0.486793 0.302583
1.6 0.420324 0.516761 0.296666
1.7 0.473473 0.546105 0.290114
1.8 0.529522 0.574763 0.282933
1.9 0.5884 0.602671 0.275138
2 0.65003 0.62977 0.266753
2.1 0.714326 0.656003 0.257811
By interpolating on the table, it is seen that u/U = F’ = 0.99 when
= 4.91.
)
(
F
U
u
21. 21
Recall that the nominal boundary thickness d is defined such that u = 0.99
U when y = d. Since u = 0.99 U when = 4.91 and = y[U/(x)]1/2, it
follows that the relation for nominal boundary layer thickness is
Or
In this way the constant C of Slide 5 is evaluated.
NOMINAL BOUNDARY LAYER THICKNESS
91
.
4
x
U
d
91
.
4
C
,
U
x
C
2
/
1
d
22. 22
Let the flat plate have length L and width b out of the page:
The shear stress o (drag force per unit area) acting on one side of the
plate is given as
Since the flow is assumed to be uniform out of the page, the total drag
force FD acting on (one side of) the plate is given as
The term u/y = 2/y2 is given from (the top of) Slide 17 as
DRAG FORCE ON THE FLAT PLATE
L
b
0
y
0
y
o
y
u
y
u
L
0
o
o
D dy
b
dA
F
)
(
F
x
U
U
y
y
u
2
2
23. 23
The shear stress o(x) on the flat plate is then given as
But from the table of Slide 20, f’’(0) = 0.332, so that boundary shear stress
is given as
Thus the boundary shear stress varies as x-1/2. A sample case is
illustrated on the next slide for the case U = 10 m/s, = 1x10-6 m2/s, L = 10
m and = 1000 kg/m3 (water).
DRAG FORCE ON THE FLAT PLATE contd.
)
0
(
F
x
U
U
)
0
(
F
x
U
U
o
Ux
,
)
(
332
.
0
U
x
2
/
1
x
2
o
Re
Re
24. 24
Boundary Shear Stress
0
0.0001
0.0002
0.0003
0 0.02 0.04 0.06 0.08 0.1
x (m)
o
(Pa)
Sample distribution of shear stress o(x) on a flat plate:
DRAG FORCE ON THE FLAT PLATE contd.
x
U
U
332
.
0
o
Note that o = at x = 0.
Does this mean that the drag force FD is
also infinite?
U = 0.04 m/s
L = 0.1 m
= 1.5x10-5 m2/s
= 1.2 kg/m3
(air)
25. 25
No it does not: the drag force converges to a finite value!
And here is our drag law for a flat plate!
We can express this same relation in dimensionless terms. Defining a
diimensionless drag coefficient cD as
it follows that
For the values of U, L, and of the last slide, and the value b = 0.05 m, it
is found that ReL = 267, cD = 0.0407 and FD = 3.90x10-7 Pa.
DRAG FORCE ON THE FLAT PLATE contd.
2
/
1
D
2
/
1
L
0
2
/
1
L
0
2
/
1
L
0
o
D
bL
U
U
664
.
0
F
L
2
dx
x
dx
x
U
bU
332
.
0
dx
b
F
bL
U
F
c 2
D
D
UL
,
)
(
664
.
0
c 2
/
1
D Re
Re
26. 26
The relation
is plotted below. Notice that the plot is carried only over the range 30
ReL 300. Within this range 1/ReL is sufficiently small to justify the
boundary layer approximations. For ReL > about 300, however, the
boundary layer is no longer laminar, and the effect of turbulence must be
included.
DRAG FORCE ON THE FLAT PLATE contd.
UL
,
)
(
664
.
0
c 2
/
1
D Re
Re
Blasius Drag Law for Laminar Flow over Flat Plate
0.01
0.1
1
10 100 1000
ReL
c
D
27. 27
The solution presented here is the Blasius-Prandtl solution for a boundary
layer on a flat plate. More details can be found in:
Schlichting, H., 1968, Boundary Layer Theory, McGraw Hill, New York, 748
p.
REFERENCE