D. Srikanth

D. Srinivasacharya

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 336 (2008) 820–827http://france.elsevier.com/direct/CRAS2B/Effect of couple stresses on the pulsatile flowthrough a constricted annulusD. Srinivasacharya a,∗, D. Srikanth ba Department of Mathematics, National Institute of Technology, Warangal-506 004, Indiab Department of Mathematics, Gokaraju Rangaraju Institute of Engineering & Technology, Hyderabad-500 050, IndiaReceived 19 April 2007; accepted after revision 8 September 2008Available online 14 October 2008Presented by Évariste Sanchez-PalenciaAbstractIn this Note, the pulsatile flow of an incompressible couple stress fluid through an annulus with mild constriction at the outerwall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheteris inserted into it. An analytical expression in terms of Bessel functions of the first and second kind is obtained for the velocitycomponent. The impedance (resistance to the flow) and wall shear stress are calculated and their variation with respect to thecouple stress fluid parameter, height of the constriction and size of the catheter on the impedance and wall shear stress is studiedgraphically. It is observed that increase in the catheter size increases the resistance to the flow as well as the wall shear stresswhile the trend is reversed in case of couple stress fluid parameter. To cite this article: D. Srinivasacharya, D. Srikanth, C. R.Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméEffet de couples de contrainte sur un flux pulsatoire à travers d’un tube annulaire contraint. Dans cette Note on considèrele flux pulsatoire d’un fluide incompressible à couple de contrainte à travers d’un tube annulaire dont la paroi externe est légèrementétranglée. Cette configuration sert de modèle simplifié du flux sanguin dans une artère sténosée pendant l’insertion d’un catétère.La composante de vitesse a été obtenue sous la forme d’une expression analytique en termes des fonctions de Bessel de premièreet seconde espèce. L’impédance de résistance et la tension de cisaillement de la paroi est également évaluée, ainsi que la variationde ces paramètres en fonction du couple de contrainte relatif au fluide. On étudie graphiquement l’influence de la hauteur de laconstriction et de la dimension du catétère sur l’impédance et le couple de cisaillement agissant sur la paroi. On observe quel’augmentation des dimensions du catétère fait augmenter la résistance au flux ainsi que la tension de la paroi ; cependant, latendance est inverse dans le cas du paramètre du couple de cisaillement du fluide. Pour citer cet article : D. Srinivasacharya,D. Srikanth, C. R. Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Fluid mechanics; Pulsatile blood flow; Catheterized artery; Constriction; Annulus; Couple stress fluidMots-clés : Mécanique des fluides ; Flux sanguin pulsatoire ; Artère catérisée ; Constriction ; Tube annulaire ; Fluide à couple de contrainte* Corresponding author.E-mail addresses: dsc@nitw.ac.in, dsrinivasacharya@yahoo.com (D. Srinivasacharya).1631-0721/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2008.09.008D. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827 8211. IntroductionThe study of flow characteristics in a constricted tube has received much attention as it represents a mathematicalmodel for the blood flow in an artery with stenosis (i.e. abnormal and unnatural growth, in the lumen of the artery).Several researchers [1–3] have studied the flow of blood in stenosed artery by considering it as a Newtonian fluid.It is well known that blood, at low shear rates and during its flow through narrow blood vessels, behaves like a non-Newtonian fluid. The couple stress fluid theory developed by Stokes [4] represents the simplest generalization of theclassical viscous fluid theory that sustains couple stresses and the body couples. The important feature of these fluids isthat the stress tensor is not symmetric and their accurate flow behavior cannot be predicted by the classical Newtoniantheory. The main effect of couple stresses will be to introduce a size dependent effect that is not present in theclassical viscous theories. This model has been widely used because of its relative mathematical simplicity comparedwith other models developed for the couple stress fluid. The fluids consisting of rigid, randomly oriented particlessuspended in a viscous medium, such as blood fluids, lubricants containing small amount of high polymer additive,electro-rheological fluids and synthetic fluids are examples of these fluids. Chaturani has analyzed the problems ofpulsatile flow of couple stress fluid with application to blood flow [5]. An analysis of the effects of couple stresseson the blood flow through thin artery with mild stenosis has been carried out by Sinha and Singh [6]. Srivastava [7]considered the flow of couple stress fluid through stenotic blood vessels.The flow through an annulus with mild constriction at the outer wall can be used as a model for the blood flowthrough the catheterized stenotic artery. The insertion of a catheter (a long flexible cylindrical tube) into a constrictedtube (i.e. stenosed artery) results in an annular region between the walls of the catheter and artery. This will alter theflow field, modify the pressure distribution and increase the resistance. Even though the catheter tool devices are usedfor the measurement of arterial blood pressure or pressure gradient and flow velocity or flow rate, X-ray angiographyand intravascualar ultrasound diagnosis and coronary balloon angioplasty treatment of various arterial diseases, a littleattention has been given in the literature to the flow in catheterized arteries. MacDonald [8] considered the pulsatileblood flow in a catheterized artery and obtained theoretical estimates for pressure gradient corrections for catheters,which are positioned eccentrically, as well as coaxially with the artery. The effect of catheterization on various flowcharacteristics in an artery with or without stenosis was studied by Karahalios [9]. Daripa and Dash [10] consideredthe steady and pulsatile flow of the Casson fluid in a narrow artery when a catheter is inserted into it and estimated theincrease in frictional resistance in the artery due to catheterization.In the present study, pulsatile flow of an incompressible couple stress fluid through a constricted annulus is investi-gated. The velocity, resistance to the flow (impedance) and shearing stress are calculated. The variation of impedanceand shearing stress is analyzed for various values of couple stress and geometric parameters.2. Formulation of the problemConsider the flow of an incompressible couple stress fluid between an artery, an axisymmetric rigid tube of radius‘a’ with a mild constriction (stenosis) and a catheter, a coaxial tube of radius ‘ka’ (k  1). Let the length of the tubebe L, the magnitude of the distance along the artery over which the stenosis is spread out be L0, the location of thestenosis be indicated by d and the maximum height of the stenosis be h. The schematic diagram is shown in Fig. 1.The equations governing the flow of an incompressible couple stress fluid in the absence of body force and bodycouple are [4]div q̄∗ = 0 (1)ρ[∂q̄∗∂t+ (q̄∗.∇)q̄∗]= −gradP ∗ − μ curl curl q̄∗ − η curl curl curl curl q̄∗ (2)where ρ is the density, q̄∗ is the velocity vector, η is the couple stress fluid parameter, P ∗ is the fluid pressure and μis the fluid viscosity.The force stress tensor τ and the couple stress tensor M that arises in the theory of couple stress fluids are given byτ = (−P ∗ + λ1 div q̄∗)I + μ[grad q̄∗ + (grad q̄∗)T ] + 1I × [divM + ρC] (3)2822 D. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827Fig. 1. Schematic diagram of catheterized stenosed artery.Fig. 1. Schéma d’une artère sténosée avec catétère.andM = mI + 2η1 grad(curl q̄∗) + 2η1(grad(curl q̄∗))T (4)where m is 1/3rd trace of M and ρC is the body couple tensor. The quantity λ1 is the material constant and η′ is theconstant associated with couple stresses. The dimensions of the material constant λ1 is that of viscosity whereas thedimensions of η and η′ are those of momentum. These material constants are considered by the inequalities [4].μ  0, 3λ + 2μ  0, η  0, η′  η (5)The problem has been studied in cylindrical coordinate system (r, θ, z). Since the flow is axisymmetric all thevariables are independent of θ . Hence for this flow the velocity is given by q̄ = (u∗(r, z, t),0,w∗(r, z, t)). Assumethat the stenosis over a length of the artery being assumed to have developed in an axisymmetric manner. The stenosedwall of the artery is defined as [1]rs(z) =⎧⎨⎩a −h2[1 + cos 2πL0(z − d − L02)], d  z  L0 + da otherwise(6)It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow ina tube with mild stenosis. In this case Eqs. (1) and (2) become−∂P∗∂r= 0 (7)ρ∂w∗∂t= −∂P∗∂z+ μD2w∗ − ηD4w∗ (8)where D2 = ∂2∂r2+ 1r∂∂r.Since the flow is pulsatile of circular frequency Ω , we seek a solution of the form w∗(r, z, t) = w(r, z)eiΩt , P ∗ =P(r, z)eiΩt .Taking w0 as a typical axial velocity and introducing the following non-dimensional variablesr = ar̃, w = w0w̃, P = w0μp̃a, z = Lz̃, rs = ar̃s, t = t̃Ω(9)into Eqs. (6)–(8) and dropping tildes, we getrs(z) =⎧⎨⎩1 −ε2[1 + cos 2πγ(z − d1L− γ2)], d1  z  d1 + γ1 otherwise(10)∂p = 0 (11)∂rD. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827 823α2∂w∂t= −∂p∂z+ D2w − 1ϕ2D4w (12)where ε = h/a, γ = L0/L, d1 = d/L, α2 = a2Ωρ/μ is the Womersely number and ϕ = a(μ/η)1/2 is the couplestress fluid parameter. The term − 1ϕ2D4w in Eq. (12) gives the effect of couple stresses. Hence, as ϕ increases, theeffect of couple stresses decrease.The corresponding non-dimensional boundary conditionsw = 0 at r = rs(z) and r = k (13)∂2w∂r2− σr∂w∂r= 0 at r = rs(z) and r = k (14)where σ = η′/η, another couple stress fluid parameter. Boundary condition (14) shows that couple stresses (4) vanishat the tube wall and catheter wall. All the effects of couple stresses will be absent in a material for which η = η′ i.e.,σ = 1. This is equivalent to requiring that the couple stress tensor be symmetric. If couple stress tensor is symmetric,then all its effects will be absent.In addition to the above boundary conditions, we have the condition that the pressure p is p1 at z = 0 and p0 atz = L.3. Solution of the problemIt can be noted from (11) that p is a function of z only. Eq. (12) can be simplified to the form(D2 − α21)(D2 − α22)w = −ϕ2 dpdz(15)whereα21 + α22 = ϕ2 and α21α22 = iα2Ωϕ2 (16)Using the separation of variables, the solution of (15) isw(r, z) = C1(z)I0(α1r) + C2(z)K0(α1r) + C3(z)I0(α2r) + C4(z)K0(α2r) − 1iα2Ωdpdz(17)where I0(αir) and K0(αir) for i = 1,2 are the modified Bessel functions of the zeroth order first and second kind re-spectively. C1(z), C2(z), C3(z) and C4(z) are arbitrary functions of z. Using the non-dimensional boundary conditions(12) and (13), we can obtain the values of C1(z), C2(z), C3(z), and C4(z).The dimensionless flux, defined as Q = ∫ rs (z)k2rw dr can be obtained from Eq. (17) in the following formQ = 2dpdzF[rs(z), k](18)whereF[rs(z), k] = d1(z)α1[rs(z)I1(α1rs(z)) − kI1(α1k)] − d2(z)α1[rs(z)K1(α1rs(z)) − kK1(α1k)]+ d3(z)α2[rs(z)I1(α2rs(z)) − kI1(α2k)] − d4(z)α2[rs(z)K1(α2rs(z)) − kK1(α2k)]− ((rs(z))2 − k2)(α21 + α22)α21α22(19)with dpdz di(z) = Ci(z), i = 1,2,3,4.The pressure drop p across the tube is obtained from Eq. (18) asp = Q21∫dzF [rs(z), k] (20)0824 D. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827The dimensionless resistance to the flow (resistive impedance), λ is given byλ = pQ= 121∫0dzF [rs(z), k] (21)which can be written asλ = 12[ d1∫0dzF [rs(z), k] +d1+γ∫d1dzF [rs(z), k] +1∫d1+γdzF [rs(z), k]](22)Since rs(z) = 1, in the regions 0  z  d1 and d1 + γ  z  1, the resistance to the flow (λ) simplifies toλ = 1 − γF [rs(z), k]rs (z)=1+ γ1∫0dξF [rs(ξ), k] (23)where ξ = (z − d1)/γ .The stress Trz can be written asTrz = T srz + T Arz (24)where T srz is the symmetric part and TArz is anti-symmetric parts of the stress and are given byT srz = μ∂w∂rand T Arz =12(∂mrθ∂r+ mrθ + mθrr)where mrθ = 12 (−η ∂2w∂r2+ η′r∂w∂r) and mθr = 12 (−η′ ∂2w∂r2+ ηr∂w∂r).Hence the total stress is given byTrz = α1(1 − α214(α21 + α22))[C1(z)I1(α1r) − C2(z)K1(α1r)]+ α2(1 − α224(α21 + α22))[C3(z)I1(α2r) − C4(z)K1(α2r)](25)4. Results and discussionsThe system of equations in terms of C1(z), C2(z), C3(z), and C4(z) obtained by using the boundary conditions(13) and (14), the dimensionless impedance (23) and the shearing stress (25) are evaluated numerically using MATH-EMATICA for various values non-dimensional parameters like ε (the maximum height of the stenosis, which is verysmall because of the mild stenosis), k (the catheter size) and ϕ and σ (the parameters which represent the effect ofcouple stresses) and the results are graphically presented in Figs. 2–7.It is interesting to notice from these figures that the impedance and stress distribution curves are intersecting at aparticular value of t . Further, it is observed that the impedance or the stress distributions are decreasing as t increasesand then starts increasing after certain value of t . The distributions are negative for some period of time and becomepositive. The time for which this change takes place corresponds to separation of flow [3].Under a given pressure gradient a greater resistance implies less flow of fluid. Thus the impedance gives themeasure of the volume of the fluid transported by the artery. Fig. 2 shows the variation of impedance with t fordifferent values of the catheter size k when ϕ = 1.5, γ = 0.5, α = 1.5, σ = 0.5, ε = 0.1. k → 0 corresponds to thecase when there is no catheter. It can be observed from this figure that the presence of catheter in a stenosed arteryincreases the impedance. Further, as the size of the catheter (k) increases the impedance increases significantly. Fig. 3shows the effect of ϕ, the couple stress parameter, on the impedance for the fixed values of k = 0.1, γ = 0.5, α = 1.5,σ = 0.5, ε = 0.15. As ϕ increases impedance decreases rapidly. ϕ → ∞ corresponds to Newtonian fluid case. It canbe noted that the impedance in case of couple stress fluid is more than that of a Newtonian fluid case i.e., the presenceof couple stresses in the fluid increases the impedance. Fig. 4 shows the effect of σ on impedance. Here it is worthD. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827 825Fig. 2. Effect of k on impedence with ϕ = 1.5, γ = 0.5, α = 1.5; σ = 0.5, ε = 0.1.Fig. 2. Effet de k sur l’impédance, avec ϕ = 1,5 ; γ = 0,5 ; σ = 0,5 ; ε = 0,1.Fig. 3. Effect of ϕ on impedence with k = 0.1, γ = 0.5, α = 1.5, σ = 0.5, ε = 0.15.Fig. 3. Effet de ϕ sur l’impédance avec k = 0,1, γ = 0,5 ; α = 1,5 ; σ = 0,5 ; ε = 0,15.Fig. 4. Effect of σ on impedence with k = 0.1, ϕ = 2.5, γ = 0.5, α = 1.5, ε = 0.15.Fig. 4. Effet de σ sur l’impédance avec k = 0,1, ϕ = 2,5 ; γ = 0,5 ; α = 1,5 ; ε = 0,15.noting that as σ increases impedance also increases. The couple stresses will be absent for the case σ = 1. From thisfigure also it can be observed that the impedance is more for couple stress fluids.826 D. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827Fig. 5. Effect of k on stress with ϕ = 2, α = 2, σ = 0.5, ε = 0.1.Fig. 5. Effet de k sur la tension avec ϕ = 2 ; α = 2 ; σ = 0,5 ; ε = 0,1.Fig. 6. Effect of ϕ on stress with k = 0.1, α = 2, σ = 0.5, ε = 0.1.Fig. 6. Effet de ϕ sur la tension avec k = 0,1, α = 2 ; σ = 0,5 ; ε = 0,1.Fig. 7. Effect of σ on stress with k = 0.1, ϕ = 2, α = 2, ε = 0.1.Fig. 7. Effet de σ sur la tension avec k = 0,1, ϕ = 2 ; α = 2 ; ε = 0,1.Figs. 5–7 depict the variation of shear stress at the maximum height of the stenosis for various values of the fluidand geometric parameters. The effect of the catheter size on the shear stress with ϕ = 2, α = 2, σ = 0.5, ε = 0.1 isshown in Fig. 5. It can be observed that in the absence of the catheter, shear stress is very low and when the catheteris introduced and its size increases from 0.05 to 2, the shear stress increases by about 50%. From Fig. 6, it is seen thatD. Srinivasacharya, D. Srikanth / C. R. Mecanique 336 (2008) 820–827 827as the couple stress parameter ϕ increases stress decreases. For the fixed values of k = 0.1, ϕ = 2, α = 2, ε = 0.1 theeffect of σ on stress is given in Fig. 7. It can be noted from this figure that as σ increases stress decreases.5. ConclusionsThis work presents an analytical solution for the couple stress fluid through an annulus with mild constriction atthe outer wall. The resistive impedance and wall shear stress are calculated and the effects of geometric parametersand couple stress fluid parameters on the impedance and wall shear stresses are studied. The presence of catheter in astenosed artery and the couple stresses in the fluid increases the impedance and shear stress.References[1] D.F. Young, Effect of time dependent stenosis on flow through a tube, J. Engrg. Ind., Trans. ASME 90 (1968) 248.[2] J.S. Lee, Y.C. Fung, Flow in locally constricted tubes at low Reynolds numbers, J. Appl. Mech., Trans. ASME 37 (1970) 9.[3] N. Padmanabhan, Mathematical model of arterial stenosis, Med. & Biol. Eng. & Comput. 18 (1980) 281.[4] V.K. Stokes, Couple stresses in fluids, Phys. Fluids 9 (1966) 1710.[5] P. Chaturani, Upadhya, Pulsatile flow of a couple stress fluid through circular tubes with applications to blood flow, Biorheology 15 (3–4)(1978) 193.[6] P. Sinha, C. Singh, Effects of couple stresses on the blood flow through an artery with mild stenosis, Biorheology 21 (3) (1984) 303.[7] L.M. Srivastava, Flow of couple stress fluid through stenotic blood vessels, J. Biomech. 18 (7) (1985) 479.[8] D.A. MacDonald, Pulsatile flow in a catheterized artery, J. Biomech. 19 (1986) 239.[9] G.T. Karahalios, Some possible effects of a catheter on the arterial wall, Med. Phys. 17 (1990) 922.[10] P. Daripa, R.K. Dash, A numerical study of pulsatile blood flow in an eccentric artery using a fast algorithm, J. Engrg. Math. 42 (2002) 1.

Effect of couple stresses on the pulsatile flow through a constricted annulus

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Effect of couple stresses on the pulsatile flow through a constricted annulus

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