Fluid flow in curved geometries:Mathematical Modeling and ApplicationsFluid flow in curved geometries:Mathematical Modeling and Applications
Dr. Muhammad Sajid
Theoretical Plasma Physics DivisionPINSTECH, P.O. Nilore, PAEC, Islamabad
March 01-06, 2010Islamabad, Pakistan
Presentation Layout
Analysis of fluid behaviour
Governing equations
Manifold and Metric
∇ operations
Stretching a curved surface
Peristaltic flow in a curved channel
Summary
When a constant shear force is applied:
Solid deforms or bends
Fluid continuously deforms.
Analysis of fluid behaviour
Analysis of fluid behaviour
Analysis of any problem in fluid mechanics
necessarily includes statement of the basic laws
governing the fluid motion. The basic laws, which
applicable to any fluid, are:
Conservation of mass
Newton’s second law of motion
The principle of angular momentum
The first law of thermodynamics
The second law of thermodynamics
NOT all basic laws are required to solve any one
problem. On the other hand, in many problems it
is necessary to bring into the analysis additional
relations that describe the behavior of physical
properties of fluids under given conditions.
Many apparently simple problems in fluid
mechanics that cannot be solved analytically.
In such cases we must resort to more
complicated numerical solutions and/or
results of experimental tests.
Analysis of fluid behaviour
Governing Equations
Continuity
Equations of motion
Continuity for incompressible flow
( ) ,0=⋅+∂∂ Vρρ
∇t
∇ V 0.
( ) ,bVVV ρρ +σ∇ divt
=⎥⎦⎤
⎢⎣⎡ ⋅+∂∂
The flow of a fluid is mainly governed by the laws of conservation of mass and momentum
Manifold and metric
The mathematical model of space is a pair:
( )gM ,Differentiable Manifold
Metric
We need to review these two fundamentalconcepts
Manifold and metricWhat is a manifold?
A manifold is a geometric ‘thing’ which has open charts, subsets where a flat set of coordinates is given. In general, however, they can be built by patching together on atlas of open charts.We need to review open charts
Manifold and metric
Open charts:
The same point is contained in more than one open chart. Its description in both charts is related by a coordinate transformation
Manifold and metricManifold
Parallel transportA vector field is parallel transported along a curve, when it mantains a constant angle with the tangent vector to the curve
Manifold and metricThe metric: a rule to calculate the lenghtof curves!!
Connection and covariant derivative
TMTMTM →×∇ :A connection is a map
From the product of the tangent bundle with itself to the tangent bundle
14Fluid Mechanics Group (FMG)
In a basis...
This defines the covariant derivative of a (controvariant) vector field
Covariant derivative
Covariant derivative of a general tensorVector (first order tensor, r + s = 1)
AAA ; AAA nn
immi
m,ini
nmm
iim, Γ
ξΓ
ξ−
∂∂
=+∂∂
=
Second-order tensor (r + s = 2)
etc. TTT
T
TTT
T
TTTT
in
njm
nj
inmm
iji
m,j
innjmnj
nimm
ijm,ij
injnm
njinmm
ijijm,
•••
• −+∂∂
=
−−∂∂
=
++∂∂
=
ΓΓξ
ΓΓξ
ΓΓξ
r = 2, s = 0
r = 0, s = 2
r = 1, s = 1
Covariant derivative
For the velocity field
iji euV )(=V
( ) mmmi
miiiii VgiVgiV Γ+==⋅∇ ,;V
Gradient of the velocity field is
Where ; represent the covariant derivative
∇ operations
( )[ ]lllmlkm
mmjm
iki VgkVgggjV Γ+==∇= ,;VL
Divergence of the velocity field is
( ) ( )[ ]immjij TTT mmiijmk
jjmmimk
jjiik
kkjjii ggggkggVggg Γ+Γ+=∇⋅ ,TV
Divergence of a tensor
( )⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
Γ+
Γ+=⋅∇
immmiij
mj
mjjjmmi
mjijjjii
iiTgg
TggjTggg
,T
( ) ( )[ ]mmmk
mikkk
iii
kk VgiVgVggV Γ+=∇⋅ ,V
∇ operations
(V⋅∇) of a vector
(V⋅∇) of a tensor
Stretching a curved surface
Fig. 1 (a) Fig. 1 (b)
R
r, v
s, u
O
R
r, v
s, u
O
(a) a flat stretching sheet, (b) a curved stretching sheet.
Geometry of the problem
Stretching a curved surface
Mathematical Formulation),( sRr +
,)(
1122
2
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−∂∂
++
∂∂
+∂∂
+−=
++
∂∂
++
∂∂
Rru
ru
Rrru
sp
RrR
Rruv
su
RrRu
ruv ν
ρ
,12
rp
Rru
∂∂
=+ ρ
( ) ⎟⎟⎠⎞
⎜⎜⎝
⎛+
= 2001Rr
gab
Curvilinear coordinates
metric
{ } ,0)( =∂∂
++∂∂
suRvRr
rcontinuity equation
components of equation of motion
Stretching a curved surfaceThe boundary conditions applicable to the present flow are
Using the similarity transformation
The problem takes the form
. as 0 ,0
,0at 0 ,
∞→→∂∂
→
===
rruu
rvasu
( ) ( ) . ,)( , , 22 raPsapfaRr
Rvfasuν
ηηρηνη ==+
−=′=
kfP+′
=∂∂
ηη
2
,)()(
22
22 ff
kkff
kkf
kk
kf
kffP
kk ′
++′′
++
+−
+′
−+′′
+′′′=+
′
ηηηηηη
, as 0 ,0,0at 1 ,0∞→→′′→′
==′=η
ηffff
Stretching a curved surface
Eliminating pressure
It is important to point out that the Crane’s problems can be recovered by taking . The numerical solution of the problem is developed by a shooting method using a Runge-Kutta algorithm.
The skin friction coefficient
.0)(
)()(
)()()(
23
2232 =′
+−′′−
+−′′′−′′′
+−
+′
++′′
−+′′′
+ ′ ffk
kfffk
kffffk
kk
fk
fk
ff iv
ηηηηηη
kf
kffC fx
1)0()0()0(Re 2/1 +′′=′
+′′=
∞→k
Stretching a curved surface
r-component of velocity, s-component of velocity
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
h
f'HhL
k = 1000 ; k = 20 ; k = 10 ; k = 5
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
1.2
h
fHhL
k = 1000 ; k = 20 ; k = 10 ; k = 5
Stretching a curved surface
Dimensionless pressure
0 1 2 3 4 5 6
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
h
PHhL k = 1000 ; k = 20 ; k = 10 ; k = 5
k −Cf Rex1/2
5 0.75763
10 0.87349
20 0.93561
30 0.95686
40 0.96759
50 0.97405
100 0.98704
200 0.99356
1000 0.99880
Skin friction coefficient
Peristaltic flow in a curved channel
Geometry of the problem
( ) ( )
( ) ( )
2, sin , Upper wall
2, sin . Lower wall
H X t a b X ct
H X t a b X ct
πλ
πλ
⎡ ⎤= + −⎢ ⎥⎣ ⎦⎡ ⎤− = − − −⎢ ⎥⎣ ⎦
Peristaltic flow in a curved channelMathematical FormulationGoverning equations in fixed frame:
( ){ }* * 0,UR R V RR X∂ ∂
+ + =∂ ∂
( )
( ) ( )
2**
* * *
2* 2 *
2 2 2* * *
1
2 ,
V V R U V U p VV R Rt R R R X R R R R R R R
R V V R UR R XX R R R R
ν⎡ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂
+ + − = − + +⎢ ⎨ ⎬∂ ∂ + ∂ + ∂ + ∂ ∂⎢ ⎩ ⎭⎣
⎤⎛ ⎞ ∂ ∂ ⎥+ − +⎜ ⎟ ⎥+ ∂⎝ ⎠ ∂ + + ⎥⎦
( )
( ) ( )
* **
* * * *
2* 2 *
2 2 2* * *
1
2 .
U U R U U UV R p UV R Rt R R R X R R R R X R R R R
R U U R VR R XX R R R R
ν⎡ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂
+ + − = − + +⎢ ⎨ ⎬∂ ∂ + ∂ + + ∂ + ∂ ∂⎢ ⎩ ⎭⎣
⎤⎛ ⎞ ∂ ∂ ⎥+ − +⎜ ⎟ ⎥+ ∂⎝ ⎠ ∂ + + ⎥⎦
Peristaltic flow in a curved channelGoverning equations in wave frame:
( ){ }* * 0,ur R v Rr x∂ ∂
+ + =∂ ∂
( ) ( ) ( )
( ) ( )
2**
* * *
2* 2 *
2 2 2* * *
1
2 ,
R u c u cv v v p vc v R rx r R r x R r r R r r r
R v v R uR r xx R r R r
ν+ + ⎡ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂
− + + − = − + +⎢ ⎨ ⎬∂ ∂ + ∂ + ∂ + ∂ ∂⎢ ⎩ ⎭⎣
⎤⎛ ⎞ ∂ ∂ ⎥+ − +⎜ ⎟ ⎥+ ∂⎝ ⎠ ∂ + + ⎥⎦
( ) ( ) ( )
( ) ( )
* **
* * * *
2* 2 *
2 2 2* * *
1
2 .
R u c u c vu u u R p uc v r Rx R R R x R R R R x r R r r
R u u R vv R xx r R r R
ν+ + ⎡ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂
− + + − = − + +⎢ ⎨ ⎬∂ ∂ + ∂ + + ∂ + ∂ ∂⎢ ⎩ ⎭⎣
⎤⎛ ⎞ ∂ ∂ ⎥+ − +⎜ ⎟ ⎥+ ∂⎝ ⎠ ∂ + + ⎥⎦
Peristaltic flow in a curved channelNon-dimensional variables:
Stream function: 2 .aπδλ
=where
2 *
2 , , , , Re ,
2 , , ,
x r u v cax u va c c
a H RP p h kc a a
π ρηλ µ
πλµ
= = = = =
= = =
, ku vk x
ψ ψδη η
∂ ∂= − =
∂ + ∂
Under the long wavelength approximation we have
0,Pη∂
=∂
( ) ( )2
2
1 1 1 0.P kx k k k
ψ ψηη η η η⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂
− − + − − =⎨ ⎬ ⎜ ⎟∂ ∂ ∂ + ∂⎝ ⎠⎩ ⎭
Peristaltic flow in a curved channelEliminating pressure
( ) ( )2 2
2 2
1 1 1 0.kk k
ψ ψηη η η η η
⎧ ⎫⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂⎪ ⎪+ + − =⎨ ⎬ ⎨ ⎬⎜ ⎟∂ ∂ ∂ + ∂⎝ ⎠⎪ ⎪⎩ ⎭ ⎩ ⎭
, 1 at 1 sin ,2
, 1 at 1 sin ,2
q h x
q h x
ψψ η φη
ψψ η φη
∂= − = = = +
∂∂
= = = − = − −∂
Solution of the problem
( ) ( )( ) ( ) ( ) ( )2 2
12 3 4ln 1 ln
2 2 4k kCk k C C k C
η ηψ η η η
⎧ ⎫+ +⎪ ⎪= + + + − + + + +⎨ ⎬⎪ ⎪⎩ ⎭
( ) ( ){ } 3121 ln 1 .
2 2 2CC k ku k k C
kη ηη η
η⎡ ⎤+ +
= + + + + − + +⎢ ⎥+⎣ ⎦
( )( ) ( ) ( )( ) ( ) ( ) ( )
1 2 22 22 2 2 2 2 2
8 2
4 ln ln 2 ln ln
hk h qC
h k h k k h k h h k k h k h
− +=− + − − + + − − − +
Peristaltic flow in a curved channel
The axial pressure gradient turned out to be( )
( ) ( ) ( )( ) ( ) ( ) ( )2 22 22 2 2 2 2 2
8 2
4 ln ln 2 ln ln
h h qdpdx h k h k k h k h h k k h k h
+=− + − − + + − − − +
The dimensionless pressure rise over one wavelength is defined by 2
0
.dpP dxdx
π
λ∆ = ∫
( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )
2 2
2 2 22 22 2 2 2 2 2
2 2 2 ln ln
4 ln ln 2 ln ln
h q hk k h k h k h k hC
h k h k k h k h h k k h k h
− + + − − − + +=− + − − + + − − − +
( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )
22 2
3 2 22 22 2 2 2 2 2
2 ln
4 ln ln 2 ln ln
k hh k h qk hC
h k h k k h k h h k k h k h
−⎛ ⎞− + ⎜ ⎟+⎝ ⎠=− + − − + + − − − +
( ) ( )( )( ) ( ) ( ) ( )( )
( ) ( ) ( )( ) ( ) ( ) ( )( )
23 2 2 2 2 2
4 2 22 22 2 2 2 2 2
2 2 2 ln
2 2 ln 2 2 ln
2 4 ln ln 2 ln ln
k hhk h hk h q k q h kk h
h q h k q k h h k q k hC
h k h k k h k h h k k h k h
−⎛ ⎞− + + + − ×⎜ ⎟+⎝ ⎠
− − + + + − + − + += −
− + − − + + − − − +
Summary
The ∇ operations for curvilinear
coordinates have been discussed.
Flow of a viscous fluid due to
Stretching a curved surface is
analyzed.
Peristaltic flow in a curved channel is
investigated.