International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 5 Study of odified Casson s fluid odel in odeled noral and stenotic capillary-tissue diffusion phenoena Sapna atan Shah Departent of Matheatics Harcourt Butler Technological Institute, Kanpur- 8,(India). sapna98jan@rediffail.co Abstract The study focuses on the behavior of diffusion phenoenon in the noral and stenosed capillarytissue exchange syste where the rheology of flowing blood in the capillary is characterized by the generalized Casson s fluid odel. Assessent of the severity of the disease could be ade possible through the variation of a paraeter naed as retention paraeter. The concentration profile and associated physiological diffusion variable involved in the study for noral and diseased state have been analyzed. The odel is also eployed to study the effect of shape of stenosis on flow characteristics. An extensive quantitative analysis is perfored through nuerical coputations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present odel. Keywords: Casson s fluid odel, Capillary-tissue exchange, esistance to flow, Wall shear stress, Stenosis shape paraeter.. Introduction: Stenosis is fored by substances depositing on vessel walls. A stenosis ay lead to partial or total vessel blockage in soe instances and therefore poses a serious edical proble, [, ]. The actual reason for foration of stenosis is not known, but its effect over the flow characteristics has been studied by any research workers, [3,, 5, 6]. Flow and diffusion through capillary-tissue exchange syste has been identified as one of the thrust areas of research. In narrow capillaries, at ties, the arterial transport becoe uch larger as copared to axial transport and it contributes to the developent of atherosclerotic plaques, greatly reducing the capillary diaeter. The proble of flow and diffusion becoe uch ore difficult through a capillary with stenosis at soe region. The response of blood flow through an artery under stenotic conditions has been attepted by [7, 8]. Accordingly, considerable effort has been expended studying the fluid echanics of flow through a stenosis [9,, ]. Several workers [, 3, ] proposed various representative odels for blood in narrow capillaries. Viscosity depending on the local variation of the concentration of the suspended cells has been introduced by [5, 6]. Perkkio and Keskinnen [7] studied the effect of concentration on viscosity and the effect of the concentration on blood flow through a vessel with stenosis and found it an iportant aspect fro physiological point of view. Kang and Erigen [8] have also discussed the effect of the variation of concentration of the suspended cells of blood. The theoretical study of Scott Blair and Spanner [9] pointed out that blood obeys the Casson s equation only in the liited range, except at very high and very low shear rate and that there is no difference between the Casson s plots and the Herschel-Bulkley plots of experiental data over the rang where the Casson s plot is valid. Also he suggested that the assuptions include in the Casson s equation are unsuitable for cow s blood and that the Herschel-Bulkley equation represents fairly closely what is occurring in the blood. Since the Herschel-Bulkley equation contains one ore paraeter than as copared to Casson s equation, it would be expected that ore detailed inforation about blood properties could be obtained by the use of the Herschel-Bulkley equation. Furtherore, the Herschel-Bulkley equation is reduced to the atheatical odels, which describes the behavior of Newtonian fluid, Bingha fluid and power law fluid by taking appropriate value of the paraeters. Casson s fluid odel: The Casson s relation is coonly written as: / / / du / τ τ (μ) ( ), if τ τ dr du if ττ dr () where dp c τ dz
International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 5 c is the radius of the plug-flow region, is yield stress, is wall shear stress and µ denotes Casson s viscosity coefficient. The stenosis studied in the present study is shown in figure [(a), ()b]. It is assued that the stenosis develops in the capillary wall in an axially nonsyetric but radially syetric anner and depends upon the axial distance z and the height of its growth. (z) A[L (z d) (z d) ], d z d L (), otherwise, () /() δ Where the paraeter A L ( ) where δ denotes the axiu height of stenosis at z = (d + L / / ( ) ). The ratio of the stenosis height to the radius of the noral artery is uch less than unity. (z) and are the radius of the artery with and without stenosis respectively. L is the stenosis length, d represents the location of stenosis and is stenosis shape paraeter. In the case of, stenosis shape paraeter indicates an axially syetric stenosis. r axis. Forulation of the proble: Following the report of Young [], considering the axisyetric lainar steady flow of blood, the general constitutive equation in the case of a ild stenosis subject to the additional conditions, ay therefore be written as: P r P r r, (r τ) z, where p is the fluid pressure, (- p / z) is pressure gradient in artery, where (z, r) are (axial, radial) coordinates with z easured along the axis and r easured noral to the axis of the artery, and (P,τ) are (Pressure, Shear stress). The concentration equation for the solute is expressed by C C C u D z r r r () Where C represents the concentration of the solute, u is the axial velocity and D the diffusion coefficient for the solute under consideration in the blood. Boundary conditions: To solve the above syste of equations, the following boundary conditions are introduced: (3) o Figure (a) Noral Artery r axis L Z axis u = r at r = u = at r = (z) P = P at z = P = P L at z = L C r at r = C D VNC r at r = (5) Where N is retention paraeter, C is concentration; u is the axial velocity and D the diffusion coefficient. Lo d L Figure (b) Stenotic Artery Z axis 3. Solution of the proble: The Volue rate of flow using equation (3) is defined as, du Q π r ( )dr. (6) dr
International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 53 By integrating equation (6), using equations (3) and (5) we have, π dp 6 c / c c Q ( )[ ( ) ( ) ( ) ], 8μ dz 7 3 (7) Equation (7) can be rewritten as; π dp Q ( )f(y), 8μ dz where 6 / f(y) [ (y) (y) (y) ], 7 3 with c y =. Fro above equation pressure gradient is written as follows, dp ( ) dz 8μ Q π f(y) (8) Integrating equation (8) using the condition P = P at z = and P = P L at z = L. We have, 8μQ L ΔP P P dz L π (z)/ f(y(z)) (9) The resistance to flow (resistive ipedance) is denoted by λ and defined as follows [Young, ()], λ= P -P L Q () The resistance to flow fro equation () using equations (9) is written as, L f dz d+l λ =- + L L d ( (z)/ ) f(y(z)) () where f is given by / 6 c c c f. 7 3 Following the apparent viscosity (µ app ) is defined as follows; μ app (z)/ f(y) The shearing stress at the wall can be defined as; / / du dr r(z) τ [τ μ ] To solve the Eq. () takes the for: ν C C C DL x η η η The boundary conditions are: () (3) () C at η =, η C D VNC at η = η (5) dp 3 3 8 u= + η- η - η μ L dx 5 c c 3 c (6) On using Eq. (6) the solution for concentration subject to the boundary conditions is given as: η 8 3 η 3-3 C 5 5 η 5 η5/ dp C = - - dx x 5 6 μ L D η η +η c - c c 3 5 C u - η c +M z 5D L 5 (7)
esistance to flow International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 5 Where 3 3 V N - D (n+) dp 8 5 VN M= - - η - dz c μ L D N ηc D V V N η + c - - u - 3 D L D. esults discussion: In order to have estiate of the quantitative effects of various paraeters involved in the analysis coputer codes were developed and to evaluate the analytical results obtained for resistance to blood flow, concentration profile and associated physiological diffusion variables for noral and diseased syste associated with stenosis due to the local deposition of lipids have been deterine. The results are shown in Fig -9 by using the values of paraeter based on experiental data in capillary. Fig - shows the results for resistance to flow for different values of stenosis shape paraeter, stenosis length, stenosis size and yield stress. esistance to flow decreases as stenosis shape paraeter increases and increases as stenosis size, stenosis length and yield stress increases. esistance to flow increase as stenosis grows or radius of artery decreases. This referred to as Fahraeus-Lindquist effect in very thin tubes. The present results are therefore consistent with the observation of Haldar [5,, 5]. Fig 5-7 shows the results for apparent viscosity for different values of stenosis shape paraeter, stenosis length, stenosis size and yield stress. Apprent viscosity increases as stenosis size, stenosis length increases and yield stress increases and decreases as stenosis shape paraeter increases and results are copared with [7, 9]. It is clear that apparent viscosity increases as stenosis grows. But the sae is not true in the absence of stenosis. In capillary flow, the viscosity of blood flow found to vary with the radius of the capillary. The developent of stenosis accelerates the velocity of plasa between the cells. This in turn increases the concentration of red cell and viscosity of blood in stenotic region, therefore increases. Fig (8) shows the diffution of large and sall olecular weight nutrients within the capillary region for different values of stenosis size. Large olecular weight nutrients within the capillary region face ore resistance to diffuse into the tissue and therefore the cells of the the deeper region are deprived of getting sufficient nutrition. This result is consistent with result of Tandon et al. [5]. Fig (9) represents the effects of retention paraeter (N) on concentration in blood flow capillary region. Increasing values of retention paraeter described the increase in retention of solute within the blood flow in the capillary region. The value of retention paraeter (N=) iplies the coplete retention. No solute or fluid diffuses and as retention paraeter decreases fro to. ore solute diffuses, which in turns, decreases the solute concentration in the capillary region. The variation of the values of retention paraeter in the stenotic region ay also be associated with the type of plaques deposited on the walls: calcified, fibrous or fatty plaque. 8 6 3 5 6 7 8...3..5.6 Stenosis length Fig (3) Variation of resistance to flow with stenosis length for different values of stenosis shape paraeter
Apparent viscosity Apparent viscosity esistance to flow International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 55 3.5E+7 3.E+7.5E+7.E+7.5E+7 3.E+7 5.E+6 5.E+...3..5.6.7 Fig () Variation of resistance to flow with yield stress for different values of stenosis Yield stress shape paraeter.7.6 / o..5 / o...3. / o. 3 / o. / o. 5 / o. 6 / o. 7. 3 5 6 7 8 Stenosis shape paraeter Fig (5) Variation of apparent viscosity with stenosis shape paraeter for different values of stenosis size.8.7.6.5..3....3..5.6.7.8 Yeild stress Fig (6) Variation of apparent viscosity with yield stress for different values os stenosis shape paraeter.
Concentration profile Concentration Apparent viscosity International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): 3-7893 www..org 56..8.6....8.6.....6.8....6 Stenosis length 3 Fig (7) Variation of apparent viscosity with stenosis length for different values of stenosis shape paraeter 6 5 3 / o. / o. 3 / o. / o...3..5.6.7.8.9 r Fig (8) Concentration profile for diffrent values of stenosis size 9 8 7 6 5 3 N= N=.8 N=.6 N=...3..5.6.7.8.9 r Fig (9). Concentration profile for different values of retention paraeter (N)
International Journal of Coputational Engineering & Manageent, Vol., January ISSN (Online): ****-**** www..org 57 5. Concluding rearks: The present study incorporates the ore realistic representation for blood in sall diaeter blood vessels and siultaneous dispersion of solute in capillary in noral and stenotic depending on various paraeters including retention paraeter. Casson s fluid odel appears to be realistic in the sense that the equations are fairly closely to the blood flow and the central core region is easily represented and one ore paraeter index behavior (non-newtonian nature of this fluid) is given in the odel. The results are ore encouraging and correlating well with the experiental observation that deeper region cells are deprived of the nutrients in the stenotic region. More experiental results are required for further developent fro clinical point of view. eferences:. H. Mogens "An Evaluation of a Micropolar A. Harry Model for Blood Flow through an Idealized Stenosis", J. Bioechanics, Vol., No.3. pp. -8. (989). A. Ogulu, J. Fiz, Mal Vol.5, pp.9. (99) 3. J.B. Shukla,.S. Parihar and B..P. ao, "Effect of Stenosis on Non-newtonian Flow of Blood in an Artery". Bull. Math. Biol. Vol., pp.83-9. (98-a).. J.B. Shukla,.S. Parihar and B..P. ao. "Biorheological Aspects of Blood Flow through Artery with Mild Stenosis: Effects of Peripheral Layer." Biorheology vol. 7, pp.3-. (98-c). 5. K. Haldar, Blood flow and red blood cell deforation, Math. Biol.Vol-755. (985) 6. S. Chakraborty and A. Chowdhury, Gosh: heol. Acta Vol. 7, pp.8. (988) 7. T. Fukushia, T. Azua and T. Matsuzawa: "Nuerical Analysis of Blood Flow in the Vertebral Artery." J. Bioech. Engng. Vol., pp.3-7. (98). 8. D.F. Young: "Fluid Mechanics of Arterial Stenoses". J. Bioech. Engng. Vol-, pp.57-75. (979). 9. J.S. Lee and Y.C. Fung: "Flow in Locally Constricted Tubes at Low eynolds Nubers". J. App. Mech., Vol. 37, pp. 9-6. (97).. V., cristini.& Ghassan, S. Kassab, Coputer Modeling of red blood cell heology in the Microcirculation: A Brief Overview. A.M.B.E. 33 (5) -.. Sugihara- Seki, M. Motion of a sphere in a cylindrical tube filled with a Brinkan ediu. Fluid Dyn. es. 3 (3) 59-76.. T.W Secob Pries. A, Blood flow and red blood cell deforation in non-unifor capillaries: effect of endothelial surface layer. Microc. 9, (5) 89-96 3. SE SE, Kurland GS: Tube flow behavior and shear stress rate characteristics of canine blood: A. J. Physiol. 3(96)7-.. AC Erigen: Siple icrofluids, Int. J. Eng. Sci. () (96) 5-7. 5. PN Tandon and Agrawal : Astudy on nutritional transport in synovial joint, Int. J. of Coputer and Matheatics with applications:7(7) (989)3-. 6. PN Tandon and Pal TS: On tranural fluid exchange and variation of viscosity of blood flowing through pereable capillaries: Med. & Life Sci. Eng. 5()(979)8-9. 7. J. Perkkio: Locally Constricted Tubes at Low Bull. Math. Biol. 9 (983) 59-69. 8. KC Kang, Nuerical Analysis of Blood Flow Bull. Math. Biol. 38 (976) 35-. 9. GW Scott GW and Spanner DC, An introduction to biorheology, Elsevier Scientific Publishing Copany Asterda,.,(97).5.. DF Young and Tsai FY, Flow characteristics in odels of arterial stenosis I, steady flow, J. Bioech., Vol. 6, pp. 395-, (973).