Samenvatting Formuleblad Fluid Mechanics

Hydrostatics



e
.
2 .
3
Hydrodynamics
Pressure 4 = -p Navier Stokes :



[Pa N/m2]=


!
-


toot =

Fvolumeforces "Fourface forces
horizontal
altijd gelijk


·
de
Op -



#(x ax)dF F'(x + dx) F 2x)
F(x
+ = +
-g
- -
dF = -

F(x + dx) + F(x)
p(x + dx)S + P(x)S

dric
dF
-




↓
=




x + dx
P(x)]S
[P(x
Sir Gey

si
dF + dx) -




ex
= -




+


When the is the same both sides
pressure on


↑ (x + dx) P(x) and dF 0

#-
iviscastane
=
=




-
F JP(z)
-rw) +zw
Bdz e termen !
Schrijf P in
=
>
-

va n z
7

e
= E
+ +
Spdz


Cart gradient
=




Cross dot
product
thevelocita
and

exex =
1 x 1 x Sin(o2 = water a
because decreases because
xsin()
>
-




- -
& x1
=
er




1
of friction


·
+
=
2x .

COS(0) 1

er
I X
=
- X
ex

x cost Euler Assume and constant
:

yo p
=
x =
dX dz
dV
dy
= C 6
E




· -
Torque i


.
of fluid
* # x rexf Newton on cube at rest
we er = =

>
-




· erx
jp
p
-

M ~xFx 0 +
>
= -




-
=


~

the
7
i and introduce
by
↳infid atrest
xFx divide
Hoal ut de
gradient head p
=




= p
,
,

w
Patm ,
use
piezometric .




Denys000kg/m)
clay
volume
0 a
g(h)
=
=
total volume = -




Pascal :




& usea
/saltwater Patm
Ptop barrel +
pgH For barrel [Path-Pbarrel]S
=

=



(1-0) x pwater + ↓ Xpclay
Pfluid Pin barrel Patm PGH
=

+

PgHS
=
=




eparticles
+ water 1
Poutside barrel Patr
=


:




Integral nemen ove r z
vessel
Viscosi viscosity) ↑ closed :


honey (high viscosity
in



acceleration
advective
Reynolds
,


(20) P(z) number Re
:
-Viscous
- =




#nematic
20
=
2 -




"inertial
force"
2
[v/t] [v2/7= =
(1 +2]

-4
h 20
=
20 + P(z) -




-
2n
-

=




Low Re number >
- laminar flow viscosity v = Torricelli :

-
Stress and :
pressure h Patm temporal advective viscous
forces
Stress T =
=


4
=


y t P9 (unsteady [yv/((p(4)] =
(2/t7]




i
shear strain
v
= =
tan (
A
air Re1 /
- +
v -g-steada =




(
=
shear rate j =



It Re +
v -g- inviscid
=




# therefore
dux
jdy
=




I jaux 93jdy (vx*x Vyy v)
breedte in
= ① de =
papier = + + +



local
O
total advective

jy #(2) pg[ex-eitan derivative derivative
/1117
=
derivative




-
Van Euler naar Bernoulli :


I
& in a 1)
along a streamline c) steady flow
(peconst
wjPgatan
Fz


- friction ( 0) incompressible
-




4)
=




3) no
=




Ex =
T
-z-
PgV

+
-




·
area
Archimedes :
pgz) ( pgz)
P + =
+ p +




No
-

Exercise :

= + h) = h
(*) ydy y ye
+


==
-

or




Fi
pwaterV Ftot
(pcube-pwater) V 2 )
-fly) frictio
Q
= -
=
=
2
use
fly) =
:



Forchimedes Patn]a Pat]
=



-Sugd + +

Grgdta +
Mass conservation :


m(t dt) m(t) drin(t) -



Amout(t)
Rheometer :
#archimedes
+ -
=




The
torque is pugae =+




↑
↑= reix Fe proportional
Shear Stress T
to the
la =
Usub
rF(r) sin) ei =) rF(r)
M(r) =
=


and measured



Notes
device
↑ =
= =

ni
=
h by the


exert an
wal a
a water change jet can o a


↑
=
=

time
of momentum over
Vena contracta :
Rin F
= veugh :S
Q-VSuc Vous
*
=




7
Types of slopes Opdrachten tank -Fer
rotating
.



: v =
rw a=


* river with more
friction -
equilibrium depth
de
-desdo
Mi *
Rotating Cylinder
ww

dar
exa --

Belanger equationisvalidforsteadyou-dimensio
a se
M2 ↳
dexdad
S
Ihethbth
-
My
loss
. -


friction
-,
--
--
experience
&

-
BC
--
* Pitot Lube (Bemoullil


Vort
D




↓ In
&
-
O


Si
App-
In
-


> > de VBz 0
dar
=


constant so point C is
hent -next

how
H =
+ hbend +
VaX 0
-
=




Vcz 0
a
stagnation point he his ha
Bo of fluid in
=




velocity
⑭
dd >
de Vaz =
o Pc =
PB +
pV - => P



·
de



+e
estimated from
z

VDz = 0 tube can be

H-h
dc > d ed
height difference

"
S3
+
longer accelerating
+


, terminal velocity Gur
velocity when
-- is no
-
BC
=

[F = -



and
*
largest errors come from energy losses
thaxit
end
effect corrections (when liquid enters & Whent
exits the
capidary
hEnt + holdw) + hexit

he' Up In