July 2010
Chapter 2
Fundamental Concepts
2-2 Velocity Field.
2-3 Stress Field.
2-4 Viscosity.
2-5 Description & Classification of Fluid
Motions.
،‫مذكزات شزح ومتارين حملولة‬
‫امتحانات سابقة للعديد من املواد أدناه‬
ً
‫متاحة جمانا على املوقعني املذكورين‬
‫العجٍبببت أى ثعبببط الٌببببص‬
.‫ٌفعلىى هي ٌكذة علٍهن‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
timeline:
it is the line joining number of adjacent fluid particles at a given instant.
subsequent observations of such lines gives some information about the flow
field.
pathline:
it is the path of a certain particle during its moving through the flow field.
streakline:
it is the line joining all fluid particles passing through a fixed location.
streamline:
it is the tangent line to the fluid velocity direction at a given instant.
streamline 
dy


dx
u
where:
(u is the velocity x component)
(v is the velocity y component)
‫ال رزض ثأقل هي‬
.‫الزوٍش الوطلق‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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
2.12 for the velocity field V  Axy iˆ  By 2 ˆj, where A = 1 m-1sec-1,
B   12 m1 sec1 , and the coordinates are measured in meters, plot
July 2010
streamlines for positive y.
dy


dx
u
B y2
(0.5) y
y



A xy
(1) x
2x
dy
dx
2


 y  x
2 n y   n x  c1
streamline 
n y 2  n x  c1
n y 2 x  c1
x y 2  e c1
‫اثبببببببذ كبببببببل هبببببببب‬
‫رظزطٍع لكل شًء‬
.‫أًذ هظئى عٌه‬
x y2  c
x ‫ قٍن هىججخ ثن حظبة قٍن‬y ‫ هخزلفخ وئعطبء‬c ‫ ثفزض قٍن‬streamline ‫حٍث ٌوكي رطن أكثز هي‬
:ً‫الوٌبظزح كوب فً الجذو الزبل‬
c
y
0
1
2
3
4
5
10
5
x
5
1.25
0.556 0.313
0.2
0.05

y
0
1
2
3
4
5
10
10
x
10
2.5
1.11
0.625
0.4
0.1

y
0
1
2
3
4
5
10
20
x
20
5
2.22
1.25
0.0
0.2

y
0
1
2
3
4
5
10
-5

x
-5
-1.25 -0.556 -0.313
-0.2
-0.05
y
0
1
2
3
4
5
10
-10

x
-10
-2.5
-1.11 -0.625
-0.4
-0.1
y
0
1
2
3
4
5
10
-20

x
-20
-5
-2.22
-1.25
-0.0
-0.2
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010

2.21 Consider the flow field V  a x (1  b t ) iˆ  c y ˆj, where a = c = 1 sec-1,
and b = 0.2 sec-1. For the particle that passes through the point (x, y) = (1, 1) at
the instant t = 0, plot the pathline during the interval from t = 0 to t = 3 sec.
Compare with the streamline through the same point at the instant t = 0.
pathline for x-direction:
dx
dt
dx
dt
dx
x
x dx

x0 x
n x x
x
0
 Vx  a x (1  b t )
 x (1  0.2 t )
 (1  0.2 t ) dt
t
  (1  0.2 t ) dt
t0

 t  0.1t
2

t
0
n x  n x0  t  0.1 t 2
 x
n    t  0.1 t 2
 x0 
2
x
 et  0.1t
x0
x  x0 et
@ t  0,
x  et
 0.1t 2
x  1 
 0. 1 t 2
x0  1
1
‫اجعبل الزوٍبش هبى‬
.‫عالهزك الوظجلخ‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
2.21 ‫ربثع ئجبثخ طإا‬
pathline for y-direction:
Vy  c y
y

y0
dy

y
n y  y
y
t
 dt
0
 t
0
n
y
 t
y0
y  y0 e t
@ t  0,
y  et
y  1 
y0  1
2
‫ عٌذ أسهٌخ هخزلفخ خال الفززح الوطلىثخ‬y ،x ‫لحظبة قٍن‬
2 ، 1 ‫حٍث ٌوكي اطزخذام الوعبدلزٍي‬
.particle ‫ للـ‬pathline ‫حظت الجذو الزبلً وهي ثن رطن‬
t
x
y
0
1
1
0.5
1.69
1.65
1
3.00
2.72
1.5
5.61
4.48
2
11.02
7.39
2.5
22.76
12.2
3
49.4
20.1
‫وظع اللوظخ األخٍزح على عولك‬
.‫ثوثبثخ الزبج الذي ٌكلل ًجبحك‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
2.21 ‫ربثع ئجبثخ طإا‬
streamline:
dy


dx
u
dy
cy

dx
a x (1  b t )
streamline 
(1  0.2 t ) 
dy

y


y
x (1  0.2 t )
dx
x
(1  0.2 t ) n y  n x  c
n y (1
 0.2 t )
 n x  c
n y (1
 0.2 t )
 n x  c
 y1  0.2 t 
  c
n 

x


y1
 0.2 t
 x ec
x  c1 y1
@ t  0,
 0.2 t
y  1,
x  y1
y  1  c1  1
 0.2 t
:ً‫) حظت الجذو الزبل‬t = 0( ‫ عٌذ‬streamline ‫حٍث ٌوكي رطن‬
@ t  0  x  y1  y
x
y
0
0
1
1
2
2
3
3
4
4
5
5
10
10
‫الوٌبفظخ الحقٍقٍخ دائوب هب ركبىى ثبٍي‬
.‫هب قوذ ثفعله وهب أًذ قبدر على فعله‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
Stress Field
surface force:
it is the force acting on the boundries of a medium through direct contact
units =
F
F
= 2
A L
= N / m2 or lb f / ft2
body force:
it is force developed without any physical contact with other bodies such as
self weight and electromagnetic forces.
‫الببذٌي ٌظببعىى دائوببب ًحببى‬
‫الزوٍش واألفعل ٌصجحىى‬
.‫رلقبئٍب هثال لآلخزٌي‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
2.33 For the region shown in problem 2.32, determine the resultant body force
when:

2.32 A body force distribution is given as B  a xiˆ  bˆj  c zkˆ per unit
mass of the matrial acted on. The density of the material is given as
  l x 2  ry  nz . All coordinates are measured in meters. Determine the
resultant body force in the region shown when: a = 0, b = 0.1 N/kg . m, l = 2.0
kg/m5, r = 0, and n = 1.0 kg/m4


F
B 
m

F  Bm

F  Bm

 B 

2
(
5
xi

10
zk
)

(
0
.
1
x
 0.5 y ) d

342

   (0.5 x
3
 2.5 xy) i  ( x 2 z  5 yz) k dx dy dz
000
34

  (0.5 x z
3
 2.5 xyz) i

2
0
00
24

  (x
3

 ( x
y  2.5 xy ) i
2
0
3


 2.5 yz ) k
2

2
0
dx dy
 5 xy) i  (2 x 2  10 y ) k dx dy
00
2
3

 (0.5 x z
2 2

4
0

 (2 x y  15 y ) k
2
2

4
0
dx
(4 x 3  40 x) i  (8 x 2  80) k dx
0

 (1.5 x
4
 20 x ) i
2
 (261 i  312) kN

3
0

 (2.67 x
3
 80 x) k

3
0
‫اثذأ الٍىم حزى ال رٌبذم‬
‫غذا على فىاد األواى‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
2.43 two infinite parallel horizontal surfaces are separated by a film of SAE 10w
oil at T = 10 C that is d = 0.0144 in. thick. The upper surface moves with speed
V. The lower surface is stationary and contains a spring-mounted plate of area A
= 0.725 ft2. The horizontal force on the plate is F = 0.155 lbf. Calculate the speed
of the upper plate.
 

F
A
0.155 (lbf )
0725 ( ft 2 )
 0.214 psf
 0.214  (4.788  101 )  10.24 m / s
  
V
h
10.24  (2.1  101 ) 
V
(0.0177  2.54  10 2 )
V  0.018 m / s
‫األهىر الصعجخ رشجه قٍبدح الظٍبرح فً ظالم‬
‫ رظزطٍع الزؤٌخ ثقذر هب ركشفه أًبىار‬،‫اللٍل‬
.‫الظٍبرح لكٌك غبلجب طزكول الزحلخ‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
Internal and External Flows
internal flow:
it is the flow completely bounded by solid surfaces or duct.
external flow:
it is the flow over bodies immersed in an unbounded fluid.
both internal and external flows could be laminar or turbulent.
flow through a pipe:
can be categorized laminar or turbulend according to dimentionless
parameter called Reyonlds number Re.
Re   V D / 
flow is laminar if Re  2300
‫ال ًزٍجــخ ثــــذوى ألـــن‬
No gains without
pains
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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July 2010
2.79 Water at 15 C flows in a tube with an inside diameter of 50 mm. Determine
the maximum value of average velocity for which the flow would be laminar.
Laminar pipe flow is valid for Re  2300
Re   V
to get Vmax
D

 Re  2300
2300  1000  Vmax 
0.05
(1.3  10 3 )
Vmax  0.06 m / s
‫األهبًً رءوص‬
‫أهىا الوفلظٍي‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
Statics, Strength, Dynamics, Mechanical Design I, II, Vibrations, Survey, Management, Structural Analysis I, II, Fluids
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July 2010
2.80 The fluid mechanics of the human arterial system is of vital importance to
our health and safety. The specific gravity and viscosity of blood are about 1.06
and 3.3 centipoise (cp), respectively. The mean flow speed in the aorta (30 mm
i.d.) of a large human is about 0.15 m/sec. Calculate the flow Reynolds number
using these properties. Would you expect this flow to be laminar or turbulent?
γ
1.06 

 water
blood
1000
 blood  1060 kg / m3
  3.3 c
 3.3 g / m . sec
 3.3  10 3 kg / m . sec
Re 

V D

1060  0.15  0.030
3.3  10 3
 1445 (dimentionless)
 2300 laminar
‫ وال ببذ شببٍك‬،‫األهببض شببٍك رببن طببحجه‬
‫ أهبب الحبظبز‬،‫هإجل قبذ ال ٌبزن ربزفه‬
.‫فهى الظٍىلخ الىحٍذح الوزىفزح لذٌك‬
) ًً‫ أو ثبلجزٌذ االلكززو‬SMS ‫هذٌخ قٍوخ عٌـذ الزٌجٍه على كـل خطـأ ( ثزطبلخ‬
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English 123, 162, 221, TOEFL, Numerical Methods, Economy, Supply Chain
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