Momentum Heat Mass TransferMHMT4
Newtonian fluids and Navier Stokes equations. Steady and transient flow between parallel plates, flow in pipe, annular gap, hydraulic diameter. Non-Newtonian fluids. RMW equation. Thixotropic fluids.
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
Navier-Stokes equations Non-Newtonian fluids
sourceDt
D
Unknowns / Equations
There are 10 unknowns (assuming isothermal flow):
u,v,w, (3 velocities), p, xx, xy,…(6 components of symmetric stress tensor)
And the same number of equations
Continuity equation
3 Cauchy’s equations
6 Constitutive equations
( ) 0ut
Dup g
Dt
2 ( )( )3
uII
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MHMT4
Substituting the constitutive equation for viscous stresses (Generalised Newtonian Fluid)
into the divergency term of the Cauchy’s equation
Dup g
Dt
2 ( )( )3
uII
Navier Stokes equations
gives, see the next slide…
Navier Stokes equations
22 ( ( )( )) ( ( )( ( ) )) ( ( )( ) )
3 3Tu
II II u u II u
2 2
2( ) ( ) ( ( ) )
3
2( ) ( ) ( ( ))
3
2( ) ( )
3
( )3
ij j i kij
i i i i j i k
j i i
i i i j j i
j i i i i
i i i j j i j i j i
j i
i i i j
u u u
x x x x x x x
u u u
x x x x x x
u u u u u
x x x x x x x x x x
u u
x x x x
2 2
3i i
j i j i
u u
x x x x
( ) 2( ( ) ) ( ) ( ) ( ) ( ) ( )
3 3T II
II u II u u u II
These terms are ZERO for incompressible fluids
These terms are small and will be replaced by a parameter sm
Divergence of viscous stresses
This is the same, but written in the index notation (you cannot make mistakes when calculating derivatives)
MHMT4
Navier Stokes equations
( ( ) ) m
Dup II u s g
Dt
General form of Navier Stokes equations valid for compressible/incompressible Non-Newtonian (with the exception of viscoelastic or thixotropic) fluids
Special case – Newtonian fluids with constant viscosity (compressible)
2 21( ) ( )
3i i i k
k ik i k k i k
u u u upu g
t x x x x x x
MHMT4
2 1( ( ))
3
Dup u u g
Dt
This term is zero for incompressible liquids
Navier Stokes equationsSpecial cases
2D flow (liquids) in Cartesian coordinate system2 2
2 2
2 2
2 2
( ) ( )
( ) ( )
x x x x xx y x
y y y y yx y y
u u u u upu u g
t x y x x y
u u u u upu u g
t x y y x y
MHMT4
2D flow in cylindrical coordinate system
2
2
2
2
1( ) ( ( ) )
1( ) ( ( ) )
r r r r rr z r
z z z z zr z z
u u u ru upu u g
t r z r r r r z
u u u u upu u r g
t r z z r r r z
Navier Stokes equationsSpecial cases
2D compressible flow formulated in terms of stream function and vorticity reduces number of equations (continuity equation is automatically satisfied) and eliminates pressure.
MHMT4
2 2
2 2
2 2
2 2
( ( )) ( ) ( )y yx xx y
u gu gu u
t x y x y x y x y
x y
This term is zero for
incompressible liquids
y xu u
x y
, x yu u
y x
These equations follow from the Navier Stokes equations using vorticity and stream function according to the previously introduced definitions
Navier Stokes solutionsMHMT4
Modigliani
The convective acceleration term makes Navier Stokes nonlinear and therefore analytical solutions can be found only when this term disappears (flow in straight pipes) or is very small comparing with the viscous term (Re<1, creeping flow).
uu
Drag flowLaminar flow between parallel plates
2
2( )x x x
x
u u upu
t x x y
MHMT4
x
ux(y)
y U
H
Steady drag flow (no pressure gradient)2
20 xu
y
21 cycux x
Uu y
H
Shear stress (constant in the whole gap)yx
U
H
Transient drag flow (U-unit step of velocity of plate)2
2x xu u
t y
x
ux(t,y)
yU
H
(t)Momentum of -layer
1( )
2U t
Stress in -layer( )
U
t
Stress is a flux of momentum 1( )
( ) 2
Udt Ud t
t
21( )
4dt d t
Penetration depthMHMT4
x
ux(t,y)
yU
H
(t)
2
0 0
1( )
4
t
dt d t
( ) 4 4t t t
Integration
yields expression for thickness of the accelarated fluid layer (penetration depth)
This solution is only an approximation, because the linear velocity profile with a turning point at is not an exact solution of the Navier Stokes equation. Exact solution exists in form of an infinite series for a finite thickness of gap H and for the case that H is defined by error function, giving more accurate prediction of the penetration depth
( )t t
Extensional flowMHMT4
In the simple shear flow between parallel plates velocities are defined in terms of the rate of shear
In the simple extensional flow (uniform stretching of incompressible fluid in the x-direction) velocities are defined in terms of rate of elongation
0 zyx uuyu [1/s]
1 1
2 2x y zu x u y u z
[1/s]
x
z
y
Constitutive equation for Newtonian liquid gives
3 timeselongationalshear viscosityviscosity(Trouton's ratio)
2
3
xx yy zz
xx yye
Flow in a circular pipeMHMT4
D=2Rz
uz(r)dr
dz
r
Steady fully developed laminar flow – only one non-zero velocity component uz(r)
Balance of forces for control volume (ring dr x dz)
0
))()((2)(2
z
p
rrz
rdzdrrdzr
z
p
zrdrdz
rzrzzz
rzrzrz
zz
Constitutive equation (Newtonian liquid)
0 ( )z z r zzz rz
u u u u
z r z r
Navier Stokes in z-direction
z
p
r
ur
rr
z
p
r
u
r
u
r
z
zz
))((1
)1
(2
2
21
2
ln4
)( CrCdz
dprruz
2
2( ) (1 ( ) )4z
R dp ru r
dz R
General solution
Boundary conditions r=0,R
Flow in a circular pipeMHMT4
12 4 42 * *2 *
0 0 0
2 ( ) 2 (1 ( ) ) (1 )4 2 8
R R
z
R dp r R dp R dpV ru r dr r dr r r dr
dz R dz dz
Volumetric flowrate
Hagen Poisseuille law 4
8
R dpV
dz
Darcy Weisbach equation for calculation of pressure drop in channels
DuuL
pD
D
Lup ff
642
2
12
2
2
32
D dpu
dz
Re
64f Re
uD
Reynolds number is an indicator of laminar/turbulent flow regime. Above the value Re=2300 (in US) or 2100 (in EU ) the flow is mostly turbulent.
The complete profile including the turbulent flow regime is presented in the Moody diagram (see next page) or described by using correlation Churchill S.W.: Friction factor equation spans all fluid-flow regimes. Chemical Engineering, 1977, 84 pp.91-92.
Flow in a circular pipeMHMT4
21
2 f
Lp u
D
Frinction factor f depends upon Re
and relative roughness
Pressure drop in a circular pipe Darcy Weisbach equation
Flow in an annular gapMHMT4
Navier Stokes equation for flow between two concentric pipes is the same and its general solution is also the same as with the circular pipe
21
2
ln4
)( CrCdz
dprruz
2 222
2 2
1( ) (1 ( ) ln )
14 lnz
R dp r ru r
dz R R
z
D2=2R2
uz(r)
D1=2R1
Only the boundary conditions are different, giving
4 2 242 (1 )
(1 )18 ln
R dpV
dz
60
70
80
90
100
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
R1/R2
f.R
e
64-circular pipe
96-parallel plates
(use per partes for integration r lnr)
2
22
64 (1 ) 64( )
1Re Re1
1ln
f f
verify that the limit for 1 is 3/2 using expansion
...32
)1ln(32
Equivalent diameterMHMT4
General cross section of a channel can be characterized by equivalent hydraulic diameter Dh, that is used in definition of Reynolds number.
4 4h
A VD
P S
Cross section surface
Volume of channel
Perimeter of cross section
Surface of wall
At turbulent flows the same correlations for pressure drop (friction factor) can be used. Correlations for circular pipe are usually used, however the cross sections with sharp corners (triangles, cusped ducts) lead to error up to 35% .
Equivalent diameter is used also in laminar flows, but different correlations for different cross sections must be used (from this point of view the laminar regime is more complicated).
Modified definitions of equivalent diameter exist for specific classes of cross sections (e.g. average distance from the point of maximum velocity in triangles, or square root of the cross section area, see next slides).
a
b
Increased f when compared with circular pipe
Equivalent diameter and fReMHMT4
Equivalent diameter for rectangular cross section
2 3
4 2
2( ) 1 /
Re 96(1 1.355 1.9467( ) 1.701( ) ...)
h
f
ab bD
a b b a
a a a
b b b
Equivalent diameter for excentric inner tube is independent of excentricity!
2
2 2
64(1 )Re
11 (1 ) / ln
f
2 22 1
2 12 1
4 ( )2( )
2 ( )h
R RD R R
R R
e
However fRe decreases with the increasing
excentricity!
Re 28f
fRe varies from the values about 30 (corners) to about 130 (bundle of pipes)
Non-Newtonian fluid flowMHMT4
Non-Newtonian fluid flowMHMT4
How to calculate the volumetric flowrate in a circular pipe as a function of pressure drop in the case of non-Newtonian fluids?
In 1D case (fully developed unidirectional axial flows) the constitutive equations for incompressible generalised Newtonian fluids (GNF) can be expressed as
)( fdr
duz
shear rate
shear stress
p
y
n
f
Kf
f
)(
)()(
)(
/1
for Newtonian fluid
for Power law fluid
for Bingham fluid
RR
zR
zRz
R
z drrfdrrdr
rdudrr
dr
rduru
rdrrruV
0
2
0
2
0
20
2
0
)()(
))(
2
1)](
2([2)(2
Volumetric flowrate for quite arbitrary radial velocity profile is
Non-Newtonian fluid flowMHMT4
2 2 3 2
0 0 0
( ) ( )( ) ( ) ( )wR R
w w w
R R RV f r dr f d f d
Radial shear stress profile follows from the equilibrium of forces
wzRpR
zrpr
2
22
2
w
Rr Rr
w
(r)
This linear shear stress profile holds for any fluid and can be used for replacement of the integration variable r
This is Rabinowitsch Mooney Weissenberg (RMW) equation
23 3
0
1( )
w
w
Vf d
R
giving volumetric flowrate regardless of
specific model as a function of wall shear stress
z
pRw
2
Non-Newtonian fluid flowMHMT4
Application RMW equation for power law fluid
2 1/ 1/ 1/3 1/ 3
0
1( ) ( )
3 1 3 1 2
w
n n nwn
w
V n n R pd
R K n K n LK
Friction factor
2
2 64
Refg
D p
Lu
232
Reg
Lu
D p
3 1 2
( )nu n LK
pR n R
2
Re 8( )2(3 1)
n nn
g
n D u
n K
Modified Reynolds number (reduces to standard Re for n=1)
1/( ) ( ) nfK
Non-Newtonian fluid flowMHMT4
Application RMW equation for Bingham fluid ( ) y
p
f
4)(3
1)(
3
41
Re
64
w
y
w
y
f
4 4 3 32 4
3 3 3
1 1 4 1( ) (1 ( ) )
4 3 4 3 3
w
y
y w y w y y ywy
w p p w p w w
Vd
R
Introducing the friction factor and expression for the wall shear stress
2 2
82 wf
D p
Lu u
we obtain
Rep
uD
Remark: Given pressure drop (therefore w) it is quite easy to calculate flowrate. Reversely: given flowrate the pressure drop must be calculated by solution of algebraic equation of the 4 th order (but there exists graphical representation of the previous equation).
Non-Newtonian fluid flowMHMT4
Exercise: Derive the RMW equation for the Herschel Bulkley model
1/( ) ( )y nfK
1 3 1 2 1 12
3
2( ) [ (1 ) (1 ) ( ) (1 ) ]
3 1 2 1 1
n n ny y y y yw n n n n
w w w w w
V n n n
R K n n n
K-consistency, n-power law index, y-yield stress
….verify that the n=1 reduces to the previously derived Bingham model.
Non-Newtonian fluid flowMHMT4
Practical importance of RMW equation is in the fact that it enables generalization of experiments with arbitrary liquid, without necessity to identify a specific rheological model.
L
2R
differential pressure transducer
flow meterpump
p
V
2w
R p
L
3
V
R
T
Diagram of consistency variables
Experimentally determined curves w, are independent of the pipe dimensions, therefore can be used for design of pipelines (with the same liquid and at the same temperature).
Remark: different w, curves recorded at different diameters of pipes indicate anomalies, for example wall slip (Mooney analysis), different flow regime (turbulent flow) or experimental errors (insufficient stabilization length of pipe, Bagley correction).
Thixotropic fluid in a pipeMHMT4
flow meter
pump
L
2R
differential pressure transducer
pV
fluid X
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
flowrateV
pressure dropp
Research described in the following pages was motivated by experimentally determined strange behaviour of a secret fluid X:
How is it possible that one and the same fluid at the same temperature and at the same flowrate exhibits different pressure drops ?
Thixotropic fluid in a pipeMHMT4
Thixotropic fluids are characterised by viscosity, which depends upon the deformation history (structure and consistency is affected by shear rate at previous times). Example of constitutive equation of a thixotropic liquid was presented as the HZS model (see previous lecture). Problem of pressure drop of a thixotropic fluid in laminar flow in a circular pipe is usually solved numerically, see the list of relevant papers
Sestak J., Zitny R., Houska M.: Změny vlastností tixotropních látek v průběhu zpracování. In 21.konf.SSChI, Vyhne °1994 (HZS model, evaluation of structural parameter change in a continuous system of series of ideally mixed vessels)
Sestak J., Zitny R., Houska M.: Dynamika tixoropnich kapalin. Rozpravy CSAV Praha 1990
Sestak J., Zitny R., Houska M.: Simple rheological models of food liquids for process design and quality assessment. Journal of Food Engineering, 1983, pp.35-49 (thixotropic integral model Zitny, and differential model HZS)
Sestak J., Houska M., Zitny R.: Mixing of thixotropic fluids. Journal of Rheology, 1982, pp. 459-475
Zitny R.: Nestacionární tok tixotropní kapaliny v trubce. Acta polytechnica, 1977, pp.95-102 (integral model of thixotropy solved by assuming that the structural parameter depends only upon time and axial coordinate)
Zitny R.: Vliv dissipace a tixotropie na tok nenewtonskych kapalin v trubce. Disertation CVUT 1977
Sestak J., Zitny R.: Tok tixotropni kapaliny v trubce. Acta Polytechnica, 1976, pp.45 (integral model of thixotropy, pressure drop calculated from correlation where is Deborah number, ratio of relaxation and process time)
0
1/
structural number (K / )Re Reynolds for De Deborah numberpower law fluids
Re1 ( 1) (1 )
64m
f m De
SN K
SN De e
-time constantof thixotropy
uDe
L
Ahmadpour A., Sadeghy K.: An exact solution for laminar, unidirectional flow of Houska thixotropic fluids in a circular pipe. J. of Non-Newtonian Fluid Mechanics, 194 (2013), pp.23-31
Corvisier P., Nouar C., Devienne R., Lebouché M.: Development of a thixotropic fluid flow in a pipe. Experiments in Fluids, 31 (2001), pp.579-587
Schmitt L., Ghnassia G., Bimbenet J.J., Cuvelier G.: Flow properties of stirred Youghurt: Calculation of the pressure drop for a thixotropic fluid. J.Food Eng. 37 (1998), pp.367-388
Escudier M.P., Presti F.: Pipe flow of a thixotropic liquid. J. Non-Newtonian Fluid Mech., 62 (1996), pp.291-306
Billingham J., Fergusson J.W.J.: Laminar unidirectional flow of a thixotropic fluid in a circular pipe. J. of Non-Newtonian Fluid Mech., 47 (1993), pp.21-55
Kemblowski Z., Petera J.: Memory effects during the flow of thixotropic fluids in pipes. Rheol.Acta 20, (1981), pp. 311-323
Thixotropic fluid in a pipeMHMT4
The HZS model of thixotropic fluids is represented by the Herschel Bulkley constitutive equation (a combination of power law liquid with a consistency coefficient K and power law index n and Bingham liquid with a yield stress y )
1
2(( ) 2 : )2 :
ny yK K
Structural parameter =1 describes fully recovered inner structure (and high consistency of liquid), while =0 corresponds to completely destroyed structure (and minimum consistency K and yield stress). Time changes of are described by
regenerationdiffusion structure decay (usually neglected)
(1 ) 2 :m
DD a b
Dt
There exist many different modifications and interpretations, for example the latest work, Ahmadpour (2013), assumes only partial and not the material time derivative on the left side, the diffusion term on the right side is considered only by Billingham and Fergusson (1993) in a generalized Moore’s model of thixotropy.
Thixotropic fluid in a pipeMHMT4
( ) ny y K K
The tensorial form of the HZS model can be simplified for the special case of unidirectional simple shear flow to scalar equations for the shear stress and the shear rate
2
decay regenerationdiffusion
(1 ) mDD a b
Dt
u
r
Complete solution of a creeping laminar flow in a pipe should calculate axial as well as radial profiles of velocity and structure parameter based upon linearity of radial shear stress profile
Ru(t,r,x) (t,r,x) =w r/R
A great simplification would be assumption that the depends only upon the axial coordinate and time, (t,x). This assumption can be accepted only if the flow is so slow that the diffusion in the radial direction has enough time to equalize the radial profile of and that the problem of -transport with a nonuniform radial velocity profile can be substituted by a model with “plug flow” (constant velocity) and modified diffusion in the axial direction (model of axial dispersion, which will be discussed later – lecture on mass transport)
2
2
decay regenerationconstant dispersion velocity
(1 )a
mu D a bt x x
Thixotropic fluid in a pipeMHMT4
For independent of radial coordinate it is possible to apply RMW (Rabinowitsch, Mooney, Weissenberg) equation with
*1/ 2
3 3 *
* * * * *1 3 1 2 1 12
*
1( )
2( ) ( (1 ) (1 ) ( ) (1 ) )
3 1 2 1 1
w
y
y n
w
n n ny y y y yw n n n n
w w w w w
Vd
R K
n n n
K n n n
yyyKKK ** ,
We need to calculate pressure drop (dp/dx=2w/R) for given flowrate therefore it is necessary to invert the previous equation
))()(()1)(12(
2
12
11(
13
))(
(
3*
2**
3**
w
y
w
y
w
y
n
wyw
nnn
n
nn
n
R
VK
Remark: Iterative evaluation is necessary, but is fast and convergent for arbitrary n,K,y
Thixotropic fluid in a pipeMHMT4
If the structural parameter depends only upon x and time and if we neglect the axial dispersion term the evolution of is described by hyperbolic partial differential equation
(1 ) mu a bt x
which can be integrated analytically along characteristic dx=u.dt as soon as the velocity and shear rate are constant
rateshear sticcharacteri
32 ,
R
V
R
Vu
]))(([1
)( ))((0
0ttbamm
m
ebaaaba
t
(t) is value of structural parameter of a fluid particle having value 0 at time t0 (assuming that the particle is under action of constant shear rate).
0,46
0,47
0,48
0,49
0,50
0,51
0,52
0,53
0,54
0,00E+00 5,00E-05 1,00E-04 1,50E-04 2,00E-04 2,50E-04 3,00E-04
flowrate
lam
bda
Thixotropic fluid in a pipeMHMT4
In our experiment the fresh fluid at the inlet to pipe has (pressumably) fully recovered structure, therefore 0=1. At a distance x from inlet depends of a fluid particle upon the time of action (t=x/u) and upon the intensity of action (m). The time of action decreases with the increasing flowrate, while the intensity of decomposition increases with flowrate:
flow meter
pump
2R
differential pressure transducer
pV
fluid X
x=?=1
2
( )1( , ) ( )
m x Ra BVm V
mx V a BV e
a BV
It is interesting that there exists a flowrate when is minimum, see graph. This extreme (maximum of thixotropy effect) is determined by equation
0),,,,,(
V
mRxbaV
…that must be solved numerically.
Thixotropic fluid in a pipeMHMT4
Resulting expressions (for example expressions for friction factor, for structural parameter, etc) are usually formulated in terms of dimensionless parameters, Debora number, thixotropy number, Bingham number and other, see for example Billingham, Fergusson (1993) .
1 1( (1 ) )De Tx
e m m mDe De
))(( 32 m
RV
baxR
VDe
3( )m
b VTx
a R
Deborah number De is the ratio of thixotropic time scale and the process time. The thixotropic number Tx is just the ratio of decay and bildup terms.
The „optimum flowrate“ condition can be expressed generally as0V
…and for the special case m=1, the Debora number at maximum tixotropy goes to infinity. Nontrivial optimum exists only for m<1.
Example: a=b=0.01, m=0.9, R=0.01, x=1 optimum at De=1, Tx=5.3 (therefore flowrate 2.10 -5 m3/s)
Thixotropic fluid in a pipeMHMT4
flow meter
pump
L
2R
differential pressure transducer
pV
fluid X
=1
The following MATLAB program calculates pressures p corresponding to the stepwise increase and decrease of flowrate
timeVmin=V1
Vmax=Vnstep/2
t1=0 tnstep
duration of time steps tistep is determined according to flowrate (velocity u) and constant x
x
x
t
characteristics
x
2RV
xt
istepistep
Thixotropic fluid in a pipeMHMT4
% simulation (flowrate up and down)l=3; %length of piper=0.01; %radiusnx=50;nstep=150;k=50; dk=500; ty=3000; dty=3000; n=0.8 % Herschel Bulkleya=0.000001; b=0.01; m=0.9; % HZS thixotropy modelvmin=1e-5; vmax=1e-4; % min,max.flowratex=linspace(0,l,nx);dx=l/(nx-1);% time sequence of flowrate and corresponding timesnstep2=nstep/2;dv=(vmax-vmin)/nstep2; v(1)=vmin;time(1)=0;for i=2:nstep2 v(i)=v(i-1)+dv; time(i)=time(i-1)+dx*3.141*r^2/v(i-1); endfor i=nstep2+1:nstep v(i)=v(i-1)-dv; time(i)=time(i-1)+dx*3.141*r^2/v(i-1); endlamx(1:nx)=1;[dp(1),taux]=dpa(k,dk,ty,dty,n,v(1),r,lamx,nx,dx);[dt,lam]=lamnew(a,b,m,v(1),r,lamx,nx,dx);figure(3)hold offfor istep=2:nstep lamx(1:nx)=lam(1:nx); [dp(istep),taux]=dpa(k,dk,ty,dty,n,v(istep),r,lamx,nx,dx); [dt,lam]=lamnew(a,b,m,v(istep),r,lamx,nx,dx); if mod(istep,40)==0 plot(x(1:nx),lam(1:nx)) hold on endendfigure(1)plot(time(1:nstep),v(1:nstep))figure(2)hold offplot(v(1:nstep),dp(1:nstep))hold onplot(v(1),dp(1),'rd')
function [dt,lam]=lamnew(a,b,m,vdot,r,lamx,nx,dx)% vector of structural parameters lam(1..nx) at time tnew% inlet value is always lam(1)=1 (fresh fluid)% a,b,m model parameters (see model HZS)u=vdot/(3.141*r^2);gamma=vdot/(3.141*r^3);lam(1)=1;dt=dx/u;gam=gamma^m;for i=1:nx-1lam(i+1)=(a-(a-(a+b*gam)*lamx(i))*exp(-(a+b*gam)*dt))/(a+b*gam);end
function tauw=hb(ty,k,n,vdot,r)% ty-yield stress, k-consistency, n-flow index, vdot-flowrate, r-radiuskappa=n/(3*n+1);eps=1e10;iter=0;while (iter<50 & eps>1e-4) iter=iter+1; tauw=ty+k*(vdot/(kappa*3.141*r^3))^n; t=ty/tauw; kappan=n/(3*n+1)*(1-t/(2*n+1)-2*n/((n+1)*(2*n+1))*(t^2+n*t^3)); eps=abs(kappa-kappan); kappa=kappan;end
function [dp,taux]=dpa(k,dk,ty,dty,n,vdot,r,lamx,nx,dx)% pipe length (nx-1).dx , radius r, flowrate vdot, strict parameter% lam(1),...lam(nx)% result: pressure drop dp, vector of wall stresses taux(1)....taux(nx)for i=1:nx taux(i)=hb(ty+lamx(i)*dty,k+lamx(i)*dk,n,vdot,r);enddp=0;for i=1:nx-1 dp=dp+taux(i)*2*dx/r;end
]))(([1
)( ))((0
0ttbamm
m
ebaaaba
t
))()(()1)(12(
2
12
11(
13
))(
(
3*
2**
3**
w
y
w
y
w
y
n
wyw
nnn
n
nn
n
R
VK
Thixotropic fluid in a pipeMHMT4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
0 10 20 30 40 50 60 70 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-4
time [s]
V f
low
rate
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
3
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-4
time [s]
V f
low
rate
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-4
time [s]
V f
low
rate
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
Very slow (180 s) Medium (70 s) Fast changes (25 s)
k=50 dk=500 ty=3000 dty=3000 n=0.8 a=0.000001 b=0.01 m=0.9
red diamond is starting point (zero time,
minimum flowrate)
Thixotropic fluid in a pipeMHMT4
k=50 dk=500 ty=3000 dty=3000 n=0.8 medium rate of flowrate changes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
7
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
b=0.001
a=0.000001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
b=0.01
a=0.000001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
b=0.05
a=0.000001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
5
6
7
8
9
10
11
12
13
14x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
a=0.01
b=0.01
a=0.1
b=0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
7
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
a=1
b=0.01
Thixotropic fluid in a pipeMHMT4
k=50 dk=500 ty=3000 dty=3000 n=0.8 a=0.001 b=0.01 m=0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
4
5
6
7
8
9
10
11
12x 10
6
V flowrate
pres
sure
dro
p
n=0.8, m=0.9
m=0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
0.5
1
1.5
2x 10
7
V flowrate
pres
sure
dro
p
m=0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10-4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10
7
V flowrate
pres
sure
dro
p
m=0.5
EXAMMHMT4
Navier Stokes equations
What is important (at least for exam)MHMT4
Navier Stokes equations (2D incompressible)
2 2
2 2
2 2
2 2
( ) ( )
( ) ( )
x x x x xx y x
y y y y yx y y
u u u u upu u g
t x y x x y
u u u u upu u g
t x y y x y
2 2
2 2
2 2
2 2
( ( )) ( ) ( )y yx xx y
u gu gu u
t x y x y x y x y
x y
Formulation with vorticity and stream function
What is important (at least for exam)MHMT4
Drag flow between parallel plates (steady and transient)
Penetration depth( )t t
22( ) (1 ( ) )
4z
R dp ru r
dz R
Flow in a circular pipe
21
2 f
Lp u
D
Darcy Weisbach, Reynolds number and Mooney diagram
Re
64f Re
uD
What is important (at least for exam)MHMT4
Non Newtonian flows
RMW equation
23 3
0
1( )
w
w
Vf d
R
z
pRw
2