DOI: 10.1007/s13367-021-0002-4
Layer-adapted meshes for solute dispersion in a steady flow through an annulus with wall absorption: Application to a catheterized artery
Nanda Poddar1, Kajal Kumar Mondal1,* and Niall Madden2
1Department of Mathematics, Cooch Behar Panchanan Barma University, Cooch Behar 736101, India
2School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway H91 TK33, Ireland (Received March 5, 2020; final revision received July 10, 2020; accepted September 21, 2020)
This paper describes the longitudinal dispersion of passive tracer molecules injected in a steady, fully devel- oped, viscous, incompressible, laminar flow through an annular pipe with a first order heterogeneous boundary absorption at the outer wall, numerically using layer-adapted meshes. The model is based on steady advection-diffusion equation with Dirichlet and Robin boundary conditions. The solutions are dis- cussed in the form of iso-concentration contours of the tracer molecules in the vertical plane. An artanh transformation is used to convert the infinite domain into a finite one. A combination of central finite dif- ference and 2-point upwind scheme is adopted to solve the governing advection-diffusion equation. It is shown that how the mixing of tracers is affected by the shear flow, aspect ratio and the first-order boundary absorption. When the flow becomes convection dominated, the monotone finite difference on a uniform mesh does not work properly, so a layer-adapted mesh, namely a “Shishkin” mesh, is used to capture the layer phenomena at the different downstream stations. The present results are compared with existing exper- imental and numerical data and we have earned an excellent agreement with them. It is observed that, due to the use of layer adapted mesh, we have achieved a better agreement with the experimental data than some other previous results available in the literature, especially in the closest downstream location. The results of this study are likely to be of interest to understand the basic mechanism of dispersion process of solute in blood through a catheterized artery with an absorptive arterial wall.
Keywords: dispersion, finite difference scheme, convection, boundary absorption, catheterized artery,
“Shishkin” mesh
1. Introduction
The research of dispersion of solute through an annulus with wall absorption has many applications in the fields of biomedical engineering, physiological fluid dynamics, environmental fluid dynamics and chemical engineering.
The concept of axial dispersion of a diffusing tracer in a fluid flowing through a circular impermeable tube was maiden introduced by Taylor (1953) and later elaborated by Aris (1956) applying method of moments by taking the axial diffusion term which was omitted by Taylor (1953).
They accepted the fact that after an adequately long time when the solute was entirely mixed any localized initial aspect of the solute produced a Gaussian distribution pass- ing with the mean speed of the flow. Applying his method of moments, Aris (1960) investigated the longitudinal dis- persion of tracer in an pulsatile flow of viscous incom- pressible fluid under a periodical pressure gradient.
However, their study of dispersion coefficients was lim- ited to asymptotically large times after injection of the sol- utes. Latini and Bernoff (2001) explored a new result based on the Fourier transformation for the advection-dif- fusion equation valid for the time afore the presence of the
Taylor regime. They demonstrated that the solute disper- sion in a pipe can be sectioned into three regimes.
Dispersion of tracers in pulsatile flows has applications in different fields such as in industry, transport matter in cardiovascular flows, the discussion of mass and heat transport within the bronchial airways by normal, abnor- mal and artificial pulmonary ventilation, and the environ- ment and estuaries. An exact result of the diffusion equation, which was linear in the axial coordinate, was discussed by Chatwin (1975) to investigate the dispersion of passive contaminant molecules along the axis of the tube, in which flow varied periodically with time. Pedley and Kamm (1988) discussed axial mass transport in an annular region that has a oscillating axial and steady sec- ondary flow. They discussed the model applying asymp- totic analysis for narrow annular intervals and large Peclet number and numerical analysis for different values of annular gap and Peclet number. Considering the time dependent pressure gradient, Mukherjee and Mazumder (1988) examined the effects of the amplitude and fre- quency of the pressure pulsations on the dispersion man- ner. Sarkar and Jayaraman (2001) analyzed the effect of wall absorption on dispersion of solute in a pulsatile flow through an annulus in case of a Newtonian fluid by apply- ing a generalized dispersion model. They explored that an
*Corresponding author; E-mail: [email protected]
increment in absorption parameter and decrement in annu- lar gap assist in the absorption of the solute across the wall. They observed that dispersion coefficient decreases with a decrement in the annular gap and the mean con- centration decreases with an increment in the absorption parameter. Possible applications of their model for the case of a catheterized artery were also showed. Since, the pioneering work of Taylor (1953), several researchers studied the great varieties of transport problems (Mazum- der and Bandyopadhyay, 2001; Mazumder and Dalal, 2000; Mazumder and Mondal, 2005; Mondal and Mazum- der, 2006; Mondal and Mazumder, 2008).
In case of environmental and physiological fluid dynam- ics, dispersion of solute with boundary absorption is a major concern. Smith (1983) applied a delay-diffusion equation to discuss the effect of boundary absorption on longitudinal dispersion in shear flows. Purnama (1988) examined the case of absorption and retention at the flow boundaries when the solute is chemically reactive. Mazum- der and Das (1992) and Jiang and Grotberg (1993) dis- cussed the effect of wall conductance on axial dispersion of solute in pulsatile flow. They established that the fre- quency parameter and wall permeability play significant roles in the transport of the tracer concentration. Jayara- man et al. (1998) analysed the solute transport phenomena in a fluid flowing through a curved tube with absorbing wall. Their outcomes, based on perturbation and spectral process, confirmed that the effect of secondary current on dispersion was reduced if the tracer was heavily soluble in the wall. Mondal and Mazumder (2005) studied the lon- gitudinal dispersion of inert solute molecules released in a steady laminar flow through an annular pipe with hetero- geneous absorption on the outer wall, which causes a depletion of contaminant in the flow. Paul and Mazumder (2011) examined the impression of nonlinear absorptions on the transport coefficients, when the flow was con- ducted by a pressure gradient comprising steady and peri- odic components. They presented a rigorous mathematical model for the evolution of the concentration of reacting solutes that travel within a fluid flowing down a pipe of circular cross-section. Later, this type of problems studied and provides some great results by Rana and Murthy (2016), Sebastian and Nagarani (2018), Jiang and Chen (2018), Debnath et al. (2019), and many other researchers.
Catheterized arteries are now extensively used in med- ical science to measure various physiological flow char- acteristics as well as to diagnose and treat various arterial diseases. Dispersion of tracer in catheterized artery have been studied by several researchers. Sarkar and Jayaraman (2002) formulated the effect of irreversible boundary absorption on the spread of a tracer in steady annular flow with the potential application of a catheterized artery, and later Sarkar and Jayaraman (2004) investigated the impact of permeable wall characteristics in the discussion of the
dispersion in a pulsatile annular flow. Since, the lungs and the blood vessels have conductive walls, the study of mass transport events is very important to understand the indi- cator dilution techniques and other mechanisms in the bronchial zone. Work of Mazumder and Mondal (2005) was important in the exploration of dispersion through a catheterized artery in the existense of boundary absorption and the solutions may permit a remedy for the catheter- induced error based on the longitudinal dispersion of tracer by the joint activity of convection, diffusion and absorption. Some other works paid attention in this type of observation (Nagarani and Sarojamma, 2008; Nagarani and Sebastian, 2017; Nagarani et al., 2006; Nagarani et al., 2009). Sebastian and Nagarani (2018) analysed the indicator dilution techniques in the catheterized artery and more other mechanisms in the bronchial zone as the walls of the arteries and lungs include conductive walls.
The main objective of the present paper is to find out the dispersion process of tracer molecules through an annular pipe. More precisely, the results are obtained numerically using layer adapted meshes to show how the tracer dif- fusion is affected by shear flow, aspect ratio; and how the tracer molecules are consumed or protected by boundary absorption. The use of layer adapted meshes namely
“Shishkin” meshes for singularly perturbed physical mat- ters has been broadly discussed (Linß and Stynes, 2001;
Miller et al., 2012). When the flow is convection domi- nated, i.e., when the Peclet number is large, “Shishkin”
mesh is used instead of an uniform mesh to capture the layer phenomena which develops near the source of the tracer materials (Linß and Stynes, 2001; Miller et al., 2012). For validation of our model, we have compared the current results with the experimental results of Raupach and Legg (1983) and the numerical results obtained by Sullivan and Yip (1987) and have achieved a nice agree- ment with them. Also, we have given an especial attention to the closest down-stream station because the interior layer is much more stronger in that position in compare to that of the other down stream stations. In particular, at this nearest station, the steady concentration profile were not well agreed in various preceding works because they used monotone finite difference scheme on uniform mesh (Mazumder and Bandyopadhyay, 2001; Mazumder and Dalal, 2000; Mondal and Mazumder, 2006). In this current research, we have employed layer adapted mesh, a piece wise uniform mesh, to adopt the layer phenomenon which exhibits near the source of the tracers and have earned an excellent agreement with the experimental data especially in the closest dowmstream station. Even though most of the researchers have studied the phenomena of solute dis- persion in different ways, but very few have used layer adapted mesh on solving the convection-diffusion equa- tion. The results of this problem are likely to be of interest for perception of the dispersion process in a catheterized
artery in the appearance of a boundary absorption. In arteries characteristics of blood similar to a Newtonian fluid and results may give a emendation to be made for the catheter-induced error based on the longitudinal dispersion of tracers by the joint action of convection, diffusion and absorption (Fung, 1997; Leondes, 2000; Leondes, 2001).
2. Mathematical Formulation 2.1. Governing equation
We consider a steady, viscous, fully developed, incom- pressible, axial symmetric two-dimensional laminar flow through an annular pipe (Fig. 1). The length of the pipe is presumed to be much larger than it’s diameter so that the penetration effect can be ignored.
We have taken z* axis towards the flow and r* axis per- pendicular to the flow, where r* and z* denotes dimen- sional expressions.
The axial velocity satisfies the Navier-Stokes equation and it can be written as:
(1) there is no slip conditions on both walls of the annular pipe, i.e., at and , where is the fluid kinematic viscosity.
The equation of continuity in cylindrical co-ordinates is:
(2) where , , denotes the velocity components along , , directions respectively.
For the present problem, the only non-zero component of velocity is . Therefore, equation of continuity becomes , which implies that is independent of . Due to axial symmetry, all variables are independent of . Thus, equation of continuity gives is a function of only that is used in Eq. (1).
Eq. (1) represents the Navier-Stokes equation of motion along the axial direction (i.e., along z*-axis) in cylindrical co-ordinate system. Also, the equation of motion along the radial direction is which implies that the pressure is a function of z* only. Again, from Eq. (1), it can be con- cluded that the pressure gradient is constant as
; left hand side is a function of z* only and right hand side is a function of r* only, which is pos-
sible if constant = (say).
Thus, the pressure gradient is given by:
(3) where and are the density and constant pressure gra- dient of the fluid respectively.
The fluid flow along the axial direction in the annular pipe is obtained by solving Eq. (1) including no-slip boundary conditions and it is given by Bird et al. (1960) as:
(4)
where and is the aspect ratio, the ratio of the inner radius b to the outer radius a of the annular pipe. When , Eq. (4) becomes the pipe Poiseuille flow , which leads to the maximum veloc- ity at the center line of the pipe at .
The non dimensional velocity profile is given by (5)
where is the dimensionless velocity and . Figure 2 shows the velocity profiles for several values of aspect ratio . The insertion of a pipe with a smaller diam- eter into another tube creates an annular region and brings the asymmetry to the flow. It is observed from the figure that increasing the aspect ratio leads to the existence of symmetry of the annular flow and the decrease in the annular gap, i.e., the increase in the aspect ratio decreases the flow velocity.
When the inert solute is released as a point source in the above mentioned flow in the annular pipe with absorption parameter on the outer wall, the concentration of the responsive solute with constant molecular diffusivity
* *( ) u r
* *
* * * *
1 p 1 r u = 0
z r r r
* *( ) = 0
u r r*=a r*=b
*
* * *
* * * * *
1 (r ur ) 1 ( )u ( ) = 0uz
r r r z
*
ur uz* u* r* z* *
*
uz
*( ) = 0uz*
z
uz* z*
* uz*
r*
* = 0 p r
*
1 p =
z
*
* * * *
1 r u
r r r
* *
* * * *
1 p = 1 r u =
z r r r
Pz*
* *
1 p P= z
z
*
Pz
*2 2 *
*
2 1
= 1 ln , 0 < <1
ln
r r
u U
a a
* 2
= 4 P az
U = b
a 0
* *2
= (1 r2)
u U a
*= 0 r
2 1 2
( ) =1 ln
u r r ln r
= u*
u U
= r*
r a
( , ) C r z Fig. 1. Schematic diagram of the flow geometry through an
annulus.
D satisfies the non-dimensional time independent advec- tion-diffusion equation of the form :
(6) on , with dimensionless expressions defined
by: , , , where U and ‘a’ are
the reference velocity and outer radius of the annular pipe respectively. Pe is the Peclet number that measures the re- lative characteristic time of the diffusion process to the convection process . The Peclet number Pe is large in typical applications such as blood flow, sewage dispersion etc. because the molecular diffusion coefficient is quite small.
We have described the point source as
at the point , where rp is the height of the injection point from the inner wall of the annular pipe and is the Dirac delta function.
2.2. Boundary conditions
We have studied the current work with both Dirichlet and Robin boundary conditions separately and they are specified respectively as:
(7) and
(8)
where is the first order absorption rate or absorption parameter.
3. Numerical Procedure 3.1. Transformation of equation
To discuss the incidence of dispersion of particles released into the laminar flow, Eq. (6) along with the boundary conditions (Eqs. (7) and (8)) and prescribed input condi- tion has been resolved numerically using a finite differ- ence method. In accordance with the construction of the problem, it is not suitable to determine the boundary con- dition at infinity. To eliminate this drawback along z direc- tion, an artanh transformation is adopted to map the
unbounded region [physical plane ]
into a bounded [computational plane ] one. The artanh transformation is given by
(9) for , where is the stretching factor connecting the physical domain to the computational domain. This form of transformation is required to avoid the loss of accuracy through discretization in diffusion and convec- tion terms.
Using the transformation (Eq. (9)), the convective-dif- fusion equation (Eq. (6)) and the corresponding boundary conditions (Eqs. (7) and (8)) into the computational field becomes
, (10)
, (11)
. (12)
We have choosen a positive real number however small to overcome the difficulty associated with singularity of the artanh transformation (Eq. (9)) at . In this research work, we have taken is of the order .
Because the flow is laminar and fully-developed, the convection in the -direction is much larger than the lon- gitudinal diffusion of particles. If the particles are injected on the line , it can not be arrived at the negative side of the line due to the predominance of the con- vection effect. Therefore, the concentration of the particles for is possessed to be zero.
When the Peclet number Pe is large, the flow becomes
2 2
( ) =1 ( , )
e C C C
Pu r r f r z
z r r r z
( ,1) ( , )
= r*
r a
*
= z
z a
= u*
u U Pe=Ua D
a2
D
a U
( , ) = f r z ( ) (z r rp)
( , ) = ( ,0)r z rp
( , ) = 0 and ( , ) = (1, ) = 0
C r C z C z
=1
=
= 0, at the outer wall of the annular pipe
= 0, at the inner wall of the annular pipe
r
r
C C
r C
r
( , ) ( ,1) (r z , ) ( , ) ( ,1) ( 1,1)r
1 1
= log and =
2 1
z r r
1 < < 1
2 2
2 2 2 2
2 2
2 2
( ) (1 ) =1 (1 )
2 (1 ) ( ) ( )
e
p
C C C C
Pu r r r r
C r r
( , 1) = ( , 1 ) = 0 and ( , ) = (1, ) = 0
C r C r ε C C
=1
=
= 0, on the outer wall of the annular pipe
= 0, on the inner wall of the annular pipe
r
r
C C
r C
r
ε
= 1
ε 105
= 0
( = 0) 1 0
Fig. 2. Velocity profiles through an annulus for several values of
the aspect ratio .
convection dominated and it is investigated that monotone finite difference method with uniform mesh is not enough to capture the layers at the near source contamination. We have used a two dimensional “Shishkin” mesh, a piece- wise uniform mesh, to solve Eq. (10) with respective boundary conditions (Eqs. (11) and (12)) separately with the Dirac-delta source term. Uniform and piecewise uni- form meshes is shown in Figs. 3a and 3b respectively.
3.2. Discretization of the equation
In order to resolve Eq. (6) subject to the fixed boundary and input conditions, a coupled scheme of central differ- encing and two-point upwind scheme is taken. The diffu- sion terms are discretized conducting central differencing technique, however, for the convection terms two-point upwind scheme are used. So, a two-point upwind method
are used for the terms , , and
.
indicates the value of C at ith grid point along the -axis, i.e., along the z-axis in the physical domain, and the jth grid point along the r-axis. The mesh point, where the particles are injected, is described as where i = 1, i = M + 1 correspond to , ;
, correspond to , r = 1 and corresponds to r = rp. M and N represents the maximum number of grid spacings respectively along and r direc- tions.
The following discretization is used for at (i, j) grid point (Hirsch, 1997; Roos et al., 2008):
(13) where
i.e.
(14)
and the discretized form of is detailed as:
(15) where
i.e.
(16)
Also, the discretized form of diffusion terms and at grid point are given below.
The forward difference is given by
(17) from Taylors expansion and the backward difference is given by
(18)
where and are the grid spacings
along the direction.
Eqs. (17) and (18) produces the three point central dif- ference representation for the terms and at ith grid point. They are respectively as:
( ) (1 2)
e C
Pu r
1 C r r
2 2
2 (1 ) C
( , ) C i j
(1, )jp
= 0
= 1 ε
= 1
j j N = 1 r = j= jp
C
, 1,
1 1
= i j i j ( )
i
C C
C O
1= i i1
1, , 1
1 1
1 1
= i j i j ( )
i
C C C O
C r
, , 1
1 1
= i j i j ( )
j
C C
C O
r
1=r rj j1
, 1 , 1
1 1
1 1
= i j i j ( )
j
C C C O
r
2 2
C r
2 2 2 2
(1 ) C2
( , )i j
2 2 3
1, , 2 2 3
2 3 2
2
= ( )
2 6
i j i j
i
C C
C C C O
2 2 3
, 1, 1 1 3
2 3 1
1
= ( )
2 6
i j i j
i
C C
C C C O
2= i1 i
1= i i1
C
2 2
C
Fig. 3. (a) Uniform and (b) piecewise uniform meshes for M N= = 25.
(19)
and
(20)
Similarly, the discretized form of and at the grid point j are respectively as:
(21) and
(22) where and are the grid spacings along the r direction. The layer-adapted mesh reduces to the uniform mesh when we take and . The steady-state concentration (Eq. (12)) is solved for ,
for each row , for each column . In
addition, an inverse transformation is used to return from the computational plane to the physical plane.
3.3. Discretization of boundary condition
The discretized form of the Dirichlet boundary condition is given by:
(23) and the given input condition is .
The two point central difference scheme for Robin boundary condition can be specified as:
, (24)
(25)
where and are the
grid spacings along r direction.
Since, is outside from the grid point of upper boundary, we replace by j in Eq. (24) and since, is outside from the grid point of lower boundary, we replace by j in Eq. (25). Thus, the discretization form of the boundary conditions are reduces to
, (26)
. (27)
4. Discussion of Computed Results
4.1. Results using Dirichlet boundary condition 4.1.1. A comparison for validation of present results
with earlier data
To validate our numerical scheme, we have compared our results with the experimental results of Raupach and Legg (1983) and numerical results of Sullivan and Yip (1987). Raupach and Legg conducted their experiment on passive tracer by affixing a rough surface of 7 mm gravel glued to a wooden base board where a heat source was located 60 mm above the ground surface and the depth of the carrier fluid was 0.54 m. The approximate small veloc- ity profile from their measured data yielded , with roughness height mm and . The vertical and downstream distances were normalized by the heat source height h for comparison with measured and calculated concentration values. Following Raupach and Legg (1983), concentration C has been normalized by a temperature scale of the form: , where is the constant flux of tracer material through a plane per- pendicular to the flow and u(h) is the velocity of the annu- lar flow at the height of injection point of the tracer material. Assessment of the concentration are obtainable at four down-stream positions: x = 0.2778, 0.8333, 1.6667, 3.3333.
Parallelism of the current normalized concentration pro- files with the experimental data of Raupach and Legg (1983) and the numerical results of Sullivan and Yip (1987) for the steady state concentration Eq. (10) at four
downstream positions are shown
in Fig. 4. For better assent of the current result between Raupach and Legg (1983) and Sullivan and Yip (1987), we have taken the value of the Peclet number , stretching factor , aspect ratio and the injection height under Dirichlet boundary condition. To capture the layer phenomena near the source and different downstream stations, we have used layer adapted meshes namely “Shishkin” mesh because mono- tone finite difference method on uniform mesh does not work properly. Even though, the velocity profile in the current research is different from the works of Raupach and Legg (1983) and Sullivan and Yip (1987), yet the comparison of the present result with the results of
2 1
2 1 2 1
1, , 1,
1 1 2 1 2 2 1 2
1 2 3 3
= ( ) ( )
(truncation error) 6
i j i j i j
i
C C C C
C
2
1, , 1,
2 1 1 2 1 2 1 2 2 1 2
3
1 2
3
2 1 1 2 2
= ( ) ( )
(truncation error) 3
i j i j i j
i
C C C C
C
C r
2 2
C r
2 1
2 1 2 1
, 1 , , 1
1 1 2 1 2 2 1 2
= ( ) i j i j ( ) i j
j
C C C C
r
2
, 1 , , 1
2 1 1 2 1 2 1 2 2 1 2
2 1 1 2 2
= ( ) i j i j ( ) i j
j
C C C C
r
1=r rj j1
2=rj1rj
1= 2
1= 2
( , ) C i j
= 1, ,
i …M j= 1, ,… N
( , ) = 0 for 1 1,1 1 except at = p
C i j j M k N j j
(1, ) = 1p
C j
, 1 , 1
, = 0
2
i j i j
i j
C C
r C
, 1 , 1= 0
2
i j i j
C C
r
1 2
= r
1= (r rj j1), = (2 rj1rj)
( ,i rj1)
1 j ( ,i rj1)
1 j
, , 2
, 1= 0 2
i j i j
i j
C C
r C
, 2 , = 0
2
i j i j
C C
r
* * *
*0
=u lnz
u z
0*= 0.12
z = 0.38
*= ( )
hu hF F
( / = 2.5, 7.5, 15, 30)x h*
= 1000 Pe
= 0.051
= 0.0001
= 0.11 rp
Raupach and Legg (1983) and Sullivan and Yip (1987) is remarkably good especially in the closest downstream sta- tion and it is displayed in Fig. 4. It is seen from Fig. 4 that the elongation in the concentration profiles of the passive tracer material reduces as it proceeds towards down- stream. This is because the mixing of the injected particles with the shear flow increases along the four downstream stations.
In Figs. 5a and 5b, the contours of iso-concentration are depicted for the results of steady dispersion in the com- putational plane and the physical plane respec- tively when (Latini and Bernoff, 2001). The slug material is discharged from the point and we have used uniform mesh in this particular case. It is observed that when the Peclet number is large enough, the flow becomes convection dominated and the uniform mesh fails to capture the layer phenom- ena throghout the solution space starting from the source.
It is noticed that a dense black region appears near the source because the iso-concentration contours become agglomerated due to domination of convection and the use of uniform mesh.
Figures 6a and 6b show the lines of equi-concentration of the steady dispersion in the vertical computational and physical plane respectively, when the slug of the solute is released from the point and when . The strength of the concentration of the slug material at the outermost contour is 0.00025 and it increases with the value of 0.00025 towards the source. In Figs. 6c and 6d, the contours of iso-concentration are drawn for
with the same releasing point. Both the cases, we have employed layer adapted meshes.
From the figures, it is seen that as Peclet number Pe
increases, the solute expands more along the longitudinal as well as in the upright directions. Also, it is observed from the figures that the concentration contours become more elongated in the longitudinal direction in compare to that of the vertical direction, because dispersion due to the longitudinal convection is much more stronger than trans- verse diffusion. From Figs. 6c and 6d, it is noticed that the expansion of the iso-concentration lines is much clear due
( , )r ( , )r z
= 1.25 106
Pe
= 0.11 rp
= 1.25 106
Pe
= 0.11
rp Pe= 1000
=1.25 106
Pe
Fig. 4. Comparison of normalized steady concentration profiles at different downstream distances between the experimental data of Raupach and Legg , the solution scheme of Sullivan and Yip , and the present solution (—–) for
using non-uniform mesh.
(***)
( ) Pe= 1000
Fig. 5. Iso-Concentration contours for steady dispersion when the tracer injected at a height , (a) in the computational plane (b) in the physical plane when using uniform mesh. The strength of the concentration of tracer material is 0.025 at the outermost contour.
= 0.11
rp ( , )r
( , )r z Pe= 1.25 10 6
to the use of non-uniform mesh in compare to that of the contours in Figs. 5a and 5b where the uniform mesh is used.
The lines of iso-concentration of steady dispersion in the vertical computational and physical planes are plotted in Figs. 7a, 7b, 7c, and 7d for the releasing heights rp= 0.61 (near the upper wall) and (near the inner wall) respectively, when and . It is seen from the figures that if the height of the releasing point increases the sprading of the solute along the longitudinal direction increases significantly. Also, a slight bending of the con- centration contour towards the inner wall of the annular pipe is observed in the physical plane when [see Fig. 7b].
The variation of the equi-concentration lines of steady dispersion in a vertical computational [ ] plane for different values of aspect ratio ( = 0.01, 0.15, 0.35, 0.46)
are depicted in Fig. 8. The computation of iso-concentra- tion lines for different aspect ratios is performed when the solute is released at a height with Peclet number using layer adapted meshes. The values of the aspect ratio are increased from Fig. 8a to Fig. 8d grad- ually. It is engrossing to remark that as increases, the steady dispersion of solute pointedly decreases. This is because as increases, the annular gap decreases and consequently the velocity of the flow decreases. Thus, we can conclude that the increment of , i.e., decrease in the annular gap prevents the dispersion process of the solute.
4.2. Results using Robin boundary condition
In Fig. 9 normalized steady concentration profiles for the Peclet number , stretching factor , aspect ratio and injection height under Robin boundary condition in a non-uniform mesh is shown.
= 0.11 rp
= 1000
Pe = 0.01
= 0.61 rp
( , )r
= 0.51 rp
=1000 Pe
= 1000
Pe = 0.051
= 104
rp= 0.11
Fig. 6. Iso-concentration contours for steady dispersion using layer adapted meshes when the tracer injected at a height , (a) in the ( , )r plane, (b) in the ( , )r z plane for Pe= 1000, (c) in the ( , )r plane, (d) in the ( , )r z plane for Pe= 1.25 10rp= 0.116.
It is clear from the figures that the elongation in the con- centration profiles of the injected particles decreases as it proceeds towards downstream. This is because the mixing of the tracer material with the shear flow increases along the four downstream stations. Excellent qualitative agree- ment is observed between the data concentration measure- ments of experimental data of Raupach and Legg (1983), numerical results of Sullivan and Yip (1987) and the cur- rent numerical scheme in all four downstream locations.
Also, it is found that as the downstream distance increases the present numerical solution deviates a bit from the experimental data of Raupach and Legg (1983) and the numerical results of Sullivan and Yip (1987). The flow geometry used by Raupach and (1983) and Sullivan and Yip (1987) was an open channel flow and in the cur-
rent research work, annular flow is considered. Since for an open channel flow, the mixing of solute with the flow is faster than that of an annular flow, the deviation in the steady concentration profiles in the third and fourth down- stream locations is observed, although they look the same shape.
4.2.1. Effect of boundary absorption
In this section, we have discussed the effect of first order boundary absorption on the dispersion of solute through an annular pipe. In the numerous previous literature (Mazumder and Das, 1992; Mondal and Mazumder, 2005;
Rana and Murthy, 2016; Sebastian and Nagarani, 2018;
Smith, 1983), absorption parameter had been taken as as of first order. First order absorption process is the one
*/ x h
Fig. 7. Iso-Concentration contours for steady dispersion using non-uniform mesh when the tracer injected at a height , (a) in the plane (b) in the plane and when the tracer injected at a height , (c) in the plane, (d) in the plane.
Here, and the strength of the concentration of tracer material is 0.00025 at the outermost contour.
= 0.61 rp
( , )r ( , )r z rp= 0.11 ( , )r ( , )r z
= 1000 Pe
whose rate is directly propotional to concentration of undergoing reaction that is increase in concentration of reactants faster the absorption. In the present problem, we have used the first order boundary absorption at the outer wall, because we want to study the effects of absorption parameter to the tracer materials throughout the annular flow. In Figs. 10a, 10b, 10c, 10d, 10e, and 10f, the lines of equi-concentration of the steady dispersion are plotted in the vertical plane for various values of boundary absorption ( = 0, 3, 7, 10, 20, 50), when the solute is injected at the height in all cases. It turns out from Fig. 10a that the equi-concentration contours are spreading along the outer and inner walls of the annular pipe symetrically when there is no absorption effect at the boundary walls. It is noticed from Figs. 10b, 10c, 10d, and
10e that as increases iso-concentration lines show the tendency to drop from the outer wall. It is because the boundary absorption counteracts the solute dispersion pro- cess through the annulus. Figures 10e and 10f elucidate that after a certain value of behaviour of iso-concen- tration contours are more or less similar. As a result effect of the absorption parameter becomes insignificant.
Again, a first order boundary absorption at the outer wall causes a depletion of the tracer molecules towards the boundary. Since, the boundary is the region of minimum velocity and strong shear, the rest of the tracers experience increased average convection and diminished rate of shear dispersion, the contours have the tendency to drop from the outer wall.
( , )r
= 0.51 rp
Fig. 8. Iso-Concentration contours for steady dispersion in the plane using layer adapted meshes when the tracer released at a height and with different aspect ratios: (a) , (b) , (c) , (d) . The strength of the concentration of tracer material is 0.00025 at the outermost contour.
( , )r
= 0.51
rp Pe= 1000 = 0.01 = 0.15 = 0.35 = 0.46
5. Application to a Catheterized Artery
In this section, we discuss the possible application of our results to dispersion in the blood stream. It is observed from the available literatures that the blood flowing through catheterized arteries is taken as a single phase Newtonian or non-Newtonian fluid (Fung, 1997; Leondes, 2000; Leondes, 2001). Many researchers have treated the blood as a two phase fluid where the core region is mod- eled as non-Newtonian fluid while the cell depleted layer is assumed as a Newtonian fluid. It is generally agreed that for a large portion of the cardiovascular system, blood may be treated as a homogenous, Newtonian fluid. In this work, we have considered a steady, viscous, fully devel- oped, incompressible, axial symmetric two-dimensional laminar flow through an annular pipe The basic fluid equations with suitable boundary conditions are used to describe the flow. In our analysis, the blood is treated as a Newtonian fluid to demonstrate the impact of fluid rhe- ology. In the arteries, characteristics of the blood similar to that o f a Newto nian fluid, that is, a fluid in which the shear stress is linearly proportional to the rate of defor- Fig. 9. Comparison of normalized steady concentration profiles
at different downstream distances between the experiments of Raupach and Legg , the solution scheme of Sullivan and Yip , and the current solution (—–) under Robin boundary condition for using non-uniform mesh.
(***) ( )
= 1000 Pe
Fig. 10. Iso-Concentration contours for steady dispersion in the computational plane using layer adapted meshes when the tracer released at a height for with different absorption parameters: (a) , (b) , (c) , (d) , (e)
, (f) . The strength of the concentration of tracer material is 0.00025 at the outermost contour.
( , )r
= 0.51
rp Pe= 1000 = 0 = 3 = 7 = 10
= 20
= 50
mation. Also, we have taken the value of Peclet number and the blood under this condition behaves as a Newtonian fluid as non-Newtonian effects are negligible.
The shear thinning viscosity of blood and its effect on the boundary absorption correlated to Peclet number, so we have indirectly used this fact. In the indicator dilution technique, it is common to introduce the solute into the blood stream and measure it’s concentration at some point as it dissolves with the blood.
Furthermore, since many intravenous drugs are thera- peutic at low concentration but toxic at high concentra- tion, it is important to know the rate of spread of drugs in the blood stream. Therefore, the study may be relevant to understand the various physiological processes involved in the above situations. Analysis of current mathematical model can be used to understand the technique of indica- tor dilution in a catheterized artery. Due to the conductive nature of the arterial walls, it is important to study the dil- atation mechanism considering the characteristics of the wall. The catheter is used to inject the dye and withdraw the blood sample for the purpose of measurement. The results reported in Section 4 are important in understand- ing the dispersion process through a catheterized artery with the reactive arterial wall. A tube of radius a can be thought of as a blood vessel, and insertion of another tube of radius , (i.e., a catheter into a blood ves- sel) causes the formation of an annular region between the walls of the catheter and artery.
Parameter , the ratio of the catheter radius b to the arterial radius a is varied from 0.01 to 0.46 to analyze the effect of catheter size on the dispersion of solute (Figs. 8a, 8b, 8c, and 8d). It is seen that as the size of the catheter increases, the frictional resistance to the flow through the artery also increases. To make an accurate pressure read- ing in the annular region, it is necessary to understand Catheter-induced error. It is observed that increasing the
size of the catheter prevents the dispersion process of the tracers. It is also noticed that when a catheter with large size [ , (see Fig. 8d)] is introduced into the artery, the longitudinal dispersion of the solute reduced signifi- cantly in compare to that of a catheter of smaller size [ , (Fig. 8a)]. Lack in the annular gap, i.e., the increase in the aspect ratio decreases the flow velocity. As a result, if we increase the size of the catheter in a blood vessel, blood pressure drops and a correction on it is highly needed.
To describe the current discussion with arterial blood flow, Peclet number Pe is restricted to the order of 1000 for diffusion in blood otherwise it is convection dominated [see Figs. 5 and 6]. Figures 11a and 11b show that the con- centration of the tracer molecules in the transversal as well as in the longitudinal directions reduces with the increase of the first order boundary absorption . These figures depict that the presence of absorption parameter at the exterior wall of the annulus inhibits the process of disper- sion and thus the pressure at the upper wall diminishes.
This result may be applied for controlling the pressure at the blood vessels. However, after some certain valuea of the absorption parameter , iso-concentratation contours are become similar. This result may be corrrelated to some real life situation, for example, after a particular limit of a certain medicine (absorption) human/animal disease become nonresponsive.
6. Conclusions
The dispersion of a solute in a steady flow inside an annulus is analyzed numerically, considering the wall absorption at the outer wall. The main objective of the present paper is the accurate numerical modelling of the dispersion tracer molecules through an annular pipe. We have shown how to combine a standard finite difference
= 1000 Pe
=
b a ( < 1)
= 0.46
= 0.01
Fig. 11. Iso-Concentration contours for steady dispersion in the plane using layer adapted meshes when the tracer injected at a height rp= 0.51 for Pe= 1000 with various absorption parameters (a) = 0, 3, 7, 10, (b) = 0, 10, 20, 50.( , )r
scheme, with specially designed layer-adapted mesh. The mesh is based on the well-known piecewise uniform meshes of Shishkin (Miller et al., 2012), which are pop- ular in the theoretical study of the numerical solution of convection-diffusion problems since they are easy to implement (and analyse) and, moreover, resolve layers that are present while guaranteeing an error bound irre- spective of how the diffusion coefficient is (equivalently, how large the Peclet number is); this is the so-called
“parameter robustness” property. However, this quality is often regarded as only having theoretical interest, and most studies of Shishkin meshes feature “toy” problems (by which we mean that numerical solutions are not com- pared directly with experimental data). So we have shown how this mesh can be use to solve a meaningful applied problem. The steady state concentration equation is solved by using a combination of central finite difference and 2- point upwind scheme with layer adapted meshes and acheived an excellent agreement with the experimental data especially in the nearest dowmstream location. More precisely, at this station, the steady concentration profiles were not well agreed in various previous efforts. Layer adapted meshes is used through out the problem because finite diffference method on uniform mesh is inadequate to capture the layer phenomena near the source of the sol- ute. It is shown that the convection process increases with an increament in Peclet number. It is also investigated that the longitudinal dispersion of the tracer material decreases with a reduction of the annular gap or an increase in the catheter size. It is found that as absorption rate increases at the outer wall of the annulus, the equi-concentration lines show the tendency to fall from the outer wall. More- over, the transverse dispersion of the tracer material dec- creases with an increment in the boundary absorption parameter. It is also observed that for a large catheter size the longitudinal dispersion of the solute reduces consid- erably.
Acknowledgements
Nanda Poddar is grateful to University Grants Commis- sion, India for funding to accomplished the work under Junior Research Fellowship grant 1164/(CSIR-UGC NET DEC. 2017). Dr. Kajal Kumar Mondal honestly acknowl- edges UGC, India for partial financial support for pursu- ing this research work under project grant number F.PSW - 192/15-16 (ERO). We express our sincere thanks to the referees for their constructive comments and suggestions to improve the quality of the papar.
References
Aris, R., 1956, On the dispersion of a solute in a fluid flowing through a tube, Proc. R. Soc. London Ser. A-Math. Phys. Eng.
Sci. 235, 67-77.
Aris, R., 1960, On the dispersion of a solute in pulsating flow through a tube, Proc. R. Soc. London Ser. A-Math. Phys. Eng.
Sci. 259, 370-376.
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, 1960, Transport Phenomena, John Wiley and Sons, NewYork.
Chatwin, P.C., 1975, On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes, J. Fluid Mech. 71, 513-527.
Debnath, S., A.K. Saha, B.S. Mazumder, and A.K. Roy, 2019, Transport of a reactive solute in a pulsatile non-Newtonian liq- uid flowing through an annular pipe, J. Eng. Math. 116, 1-22.
Fung, Y.C., 1997, Biomechanics: Circulation, 2nd ed., Springer, New York.
Hirsch, C., 1997, Numerical Computation of Internal and Exter- nal Flows, Vol. 1, John Wiley & Sons, New York, 195-197.
Jayaraman, G., T.J. Pedley, and A. Goyal, 1998, Dispersion of solute in a fluid flowing through a curved tube with absorbing walls, Q. J. Mech. Appl. Math. 51, 577-598.
Jiang, W.Q. and G.Q. Chen, 2018, Solution of Gill’s generalized dispersion model: Solute transport in Poiseuille flow with wall absorption, Int. J. Heat Mass Transf. 127, 34-43.
Jiang, Y. and J.B. Grotberg, 1993, Bolus contaminant dispersion in oscillatory tube flow with conductive walls, J. Biomech.
Eng.-Trans. ASME 115, 424-431.
Latini, M. and A.J. Bernoff, 2001, Transient anomalous diffusion in Poiseuille flow, J. Fluid Mech. 441, 399-411.
Leondes, C., 2000, Biomechanical Systems: Techniques and Appli- cations, Volume II: Cardiovascular Techniques, CRC Press, Boca Raton.
Leondes, C., 2001, Biomechanical Systems: Techniques and Appli- cations, Volume IV: Biofluid Methods in Vascular and Pulmo- nary Systems, CRC Press, Boca Raton.
Linß, T. and M. Stynes, 2001, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput.
Meth. Appl. Mech. Eng. 190, 3527-3542.
Mazumder, B.S. and S. Bandyopadhyay, 2001, On solute disper- sion from an elevated line source in an open-channel flow, J.
Eng. Math. 40, 197-209.
Mazumder, B.S. and D.C. Dalal, 2000, Contaminant dispersion from an elevated time-dependent source, J. Comput. Appl.
Math. 126, 185-205.
Mazumder, B.S. and S.K. Das, 1992, Effect of boundary reaction on solute dispersion in pulsatile flow through a tube, J. Fluid Mech. 239, 523-549.
Mazumder, B.S. and K.K. Mondal, 2005, On solute transport in oscillatory flow through an annular pipe with a reactive wall and its application to a catheterized artery, Q. J. Mech. Appl.
Math. 58, 349-365.
Miller, J.J.H., E. O’Riordan, and G.I. Shishkin, 2012, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore.
Mondal, K.K. and B.S. Mazumder, 2005, On the solute disper- sion in a pipe of annular cross-section with absorption bound- ary, ZAMM-Z. Angew. Math. Mech. 85, 422-430.
Mondal, K.K. and B.S. Mazumder, 2006, On dispersion of set-
tling particles from an elevated source in an open-channel flow, J. Comput. Appl. Math. 193, 22-37.
Mondal, K.K. and B.S. Mazumder, 2008, Dispersion of fine set- tling particles from an elevated source in an oscillatory turbu- lent flow, Eur. J. Mech. B-Fluids 27, 707-725.
Mukherjee, A. and B.S. Mazumder, 1988, Dispersion of contam- inant in oscillatory flows, Acta Mech. 74, 107-122.
Nagarani, P. and G. Sarojamma, 2008, Effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery, Korea-Aust. Rheol. J. 20, 189-196.
Nagarani, P. and B.T. Sebastian, 2017, Effect of flow unsteadi- ness on dispersion in non-Newtonian fluid in an annulus, J.
Appl. Math. & Inform. 35, 241-260.
Nagarani, P., G. Sarojamma, and G. Jayaraman, 2006, Exact anal- ysis of unsteady convective diffusion in Casson fluid flow in an annulus - Application to catheterized artery, Acta Mech.
187, 189-202.
Nagarani, P., G. Sarojamma, and G. Jayaraman, 2009, Effect of boundary absorption on dispersion in Casson fluid flow in an annulus: Application to catheterized artery, Acta Mech. 202, 47-63.
Paul, S. and B.S. Mazumder, 2011, Effects of nonlinear chemical reactions on the transport coefficients associated with steady and oscillatory flows through a tube, Int. J. Heat Mass Transf.
54, 75-85.
Pedley, T.J. and R.D. Kamm, 1988, The effect of secondary motion on axial transport in oscillatory tube flow, J. Fluid Mech. 193, 347-367.
Purnama, A., 1988, Boundary retention effects upon contaminant
dispersion in parallel flows, J. Fluid Mech. 195, 393-412.
Rana, J. and P.V.S.N. Murthy, 2016, Solute dispersion in pulsatile Casson fluid flow in a tube with wall absorption, J. Fluid Mech. 793, 877-914.
Raupach, M.R. and B.J. Legg, 1983, Turbulent dispersion from an elevated line source: Measurements of wind-concentration moments and budgets, J. Fluid Mech. 136, 111-137.
Roos, H.G., M. Stynes, and L. Tobiska, 2008, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin.
Sarkar, A. and G. Jayaraman, 2001, Nonlinear analysis of oscil- latory flow in the annulus of an elastic tube: Application to catheterized artery, Phys. Fluids 13, 2901-2911.
Sarkar, A. and G. Jayaraman, 2002, The effect of wall absorption on dispersion in annular flows, Acta Mech. 158, 105-119.
Sarkar, A. and G. Jayaraman, 2004, The effect of wall absorption on dispersion in oscillatory flow in an annulus: Application to a catheterized artery, Acta Mech. 172, 151-167.
Sebastian, B.T. and P. Nagarani, 2018, Convection-diffusion in unsteady non-Newtonian fluid flow in an annulus with wall absorption, Korea-Aust. Rheol. J. 30, 261-271.
Smith, R., 1983, Effect of boundary absorption upon longitudinal dispersion in shear flows, J. Fluid Mech. 134, 161-177.
Sullivan, P.J. and H. Yip, 1987, Near-field contaminant disper- sion from an elevated line-source, Z. Angew. Math. Phys. 38, 409-423.
Taylor, G.I., 1953, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. London Ser. A-Math. Phys.
Eng. Sci. 219, 186-203.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
