shape effect of cu nanoparticles in unsteady flow through curved artery with catheterized stenosis

RINP 535 No of Pages 13, Model 5G 16 January 2017 Results in Physics xxx (2017) xxx–xxx Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Ashfaq Ahmed ⇑, Sohail Nadeem Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan a r t i c l e 12 13 14 15 16 17 18 19 20 21 22 23 i n f o Article history: Received November 2016 Received in revised form January 2017 Accepted January 2017 Available online xxxx Keywords: Bricks Cu-nanoparticle Cylinders Cu-nanoparticle Platlets Cu-nanoparticle Overlapping stenosis Balloon angioplasty a b s t r a c t In this study the arterial flow of Cu-nanofluid through catheterized arteries having a balloon angioplasty with time-varying overlapping stenosis is considered The nanofluid comprises different shaped nanoparticles such as bricks, cylinders and platelets In the arteries the nature of Cu-blood nanofluid is examined mathematically by considering it as a different shaped nanoparticles inclusion in viscous fluid in toroidal coordinate system The problem is solved using a perturbation approximation in terms of a variant of curvature parameter () to achieve the axial velocity, the stream function, the resistance impedance, and the wall shear stress distribution of nanofluid Also, the results were obtained from explicit values of the physical parameters, such as the curvature parameter (), the balloon height (r⁄), the volume fraction (/) and the shape factor of Cu-nanoparticles (m) The obtained results show that there is a notable difference between curvature and non-curvature annulus flows through catheterized stenosed arteries The Platlets Cu-nanoparticle in the central portion of the tube are not sheared, and the slight velocity gradients are only found in the layers near the wall of artery than Bricks Cylinders Cu-nanoparticles Ó 2017 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/) 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Introduction It is widely known that, stenosis may aggravate due to unusual local intravascular endothelial growth at different locations in the arteries Arteries may be progressively narrowed by the atherosclerotic plaques and several investigators have suggested that this development is closely related to the hydrodynamics of blood flow through the artery [1–4] There may now take place a ‘‘coupling” between the development of the stenosis and the corresponding change in the flow characteristics It has been expressed that a localized change in shearing stress or pressure and resistances may trigger some but not all biological mechanisms whereby the endothelial cells of the intima surface of the arterial wall, and subendothelial cells of the media and adventitia, mushroom with a subsequent tapering of the lumen [5] Generally, in the artery, the impediment initiation is resulted from the fatty acid and lipoprotein deposition at the sites of the atherosclerotic lesion Consequently, the narrowing is caused by a lesion that contracts the space of lumen (e.g., stenosis, atherosclerosis) In the stenosed part the velocity gradient near the wall region is stiff due to the increased core velocity resulting in relatively higher shear stress on the wall even for mild stenosis Various researchers have stated mathematical treatment in the flow of blood in arteries subject to ⇑ Corresponding author E-mail address: qafhsam@hotmail.com (A Ahmed) numerous physiologic conditions [6–11] The recent useful contributions to that subject are referenced in the literature [12–16] Much of the research that has studied atherosclerotic vascular disease (ASVD) indicates that it deals with symmetric and nonsymmetric shaped stenoses while the stenoses may develop in a series or may be of overlapping or composite nature Blood flow analysis in the presence of an overlapping stenosis in vessel segments is available in the literature Chakravarty and Mandal [17] discussed that the problem becomes more severe in the occurrence of an overlapping stenosis in the artery instead of a mild one Chakravarty and Mandal [18] analysed the problem on a tapered blood vessel segment having an overlapping stenosis Ismail et al [19] formulated the model of blood flow through an overlapping stenosed artery in which the time-variant stenosis in the tapered arterial lumen is presented and the vascular wall deformability is taken to be elastic In medicine, a catheter is a thin tube made from medical grade materials serving a broad range of functions Catheters are medical devices that can be inserted in the body to treat diseases or perform a surgical procedure By modifying the material or adjusting the way catheters are manufactured, it is possible to tailor catheters for cardiovascular, urological, gastrointestinal, neurovascular, and ophthalmic applications Catheters can be inserted into a body cavity, duct, or vessel Functionally, they allow drainage, administration of fluids or gases, access by surgical instruments, and also perform a wide variety of other tasks depending on the type of http://dx.doi.org/10.1016/j.rinp.2017.01.015 2211-3797/Ó 2017 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 RINP 535 No of Pages 13, Model 5G 16 January 2017 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx catheter The process of inserting a catheter is catheterization In most uses, catheter is a thin, flexible tube (‘‘soft” catheter) though catheters are available in varying levels of stiffness depending on the application A catheter left inside the body, either temporarily or permanently, may be referred to as an indwelling catheter (for example, a peripherally inserted central catheter) A permanently inserted catheter may be referred to as a permcath [20] Nanotechnology refers to an emerging field of science that includes synthesis and development of various nanomaterials Nanoparticles can be defined as objects ranging in size from to 100 nm that due to their size may differ from the bulk material Presently, different metallic nanomaterial is being produced using copper, zinc, titanium, magnesium, gold, alginate and silver Nanoparticles are being used for diverse purposes, from medical treatments, using in various branches of industry production such as solar and oxide fuel batteries for energy storage, to wide incorporation into diverse materials of everyday use such as cosmetics or clothes [21] Ellahi et al [22] studied shape of man-sized particles with natural convection boundary layer flow incorporating entropy generation Zeeshan et al [23] studied the different shapes of nanoparticles on mixed convective steady flow over a rotating disk For nanofluid, the copper nanoparticles of disk, cylindrical, and spherical shapes of different sizes and water as base fluid are considered Nadeem [24] persued the flow of nanofluid in a curved channel with compliant walls The governing equations of the nanofluid model for curved channel are shown, including the effects of curvature Recently, several aficionados worked in the field of nanotechnology and nanoscience [25–31] Most of the workers not study the effect of the catheter that have a balloon (angioplasty) on the flow of the different shape of Cu-nanoparticles through the arteries with overlapped stenosis which occurs in many clinical applications The motivation of the present exploration is to discuss the behavior of different shaped nanoparticles such as brick Cunanoparticles, cylinder Cu-nanoparticles and platelet Cunanoparticles as a nanofluid model through catheterized curved artery with overlapping stenosis So far as we know the abstraction of the study of different shaped nanoparticles tracking through the catheterized curved artery with overlapping stenosis has not been explored yet The obtained governing equation of motion and energy for the nanofluid model have been set up and simplified under the assumptions of mild stenosis and low Reynolds number The reduced form of equations is then solved by a regular perturbation method by considering curvature parameter k is very small The graphs of parameters of interest (, d⁄, r⁄, m) that basically represents physical features of the study Streamlines are explored at the end of the article 136 Mathematical model 137 Taking an incompressible viscid copper nanofluid model flowing through an artery having length L Let (r, h, z) be the coordinates of a material point in the torodial coordinate system The z-axis is along the axis of the artery while r is along the radial direction (see Fig 1) Further we mentioning that r = is taken as the axis of symmetry of the coaxial tubes The geometry of the arterial wall with time-varying overlapping stenosis and the balloon model are defined by the functions R(z, t) and h(z), respectively and are written mathematically as follows [19]: 82 93 94 > zdịỵ 32 ðzÀdÞ À = < 11À 3L > > L20 < 4R ÀdðzÀdÞ XðtÞd z 3L0 ỵd; 32 ; : Rz;tị ẳ zdị > 3L > > : otherwise ẵR0 Xtị; ð1Þ 138 139 140 141 142 143 144 145 146 148 Fig Schematic diagram of the artery The time-varying parameter Xtị is expressed by Xtị ẳ kcosxt 1ịe ( hzị ẳ kxt ; 149 2ị 150 152 153 R0 ẵk ỵ rep zz R0 k d 0:5ị d z 3L20 ỵ d otherwise ð3Þ where R0 is the radius of the normal artery in the non-stenotic region, 3L20 is the length of the overlapping stenosis, d is the location of the stenosis, dà is taken to be the critical height (severity) of the overlapping stenosis, k is a constant, x represents the angular frequency of the forced oscillation and t is the time, rà denotes the extremum height attained by the balloon at z ẳ zd ỵ 0:5ị, kR0 is the radius of the catheter tube which keeps the balloon in position, k ( and zd represents the freedom of the axial displacement of the balloon The equations comprising the conservation of mass, momentum and energy are presented as follow [15]: r V ẳ 0; 4ị 155 156 157 158 159 160 161 162 163 164 165 166 167 169 170 DV ẳ rP ỵ lnf r2 V ỵ gðqcÞnf ðT À T Þ qnf Dt ð5Þ 172 173 DT qcp ẳ K r2 T ỵ Q Dt ð6Þ @V r V r @V # V r cos # À V # sin# b @V z þ þ þ þ ¼ 0; r @r b þ r cos # b ỵ r cos # @z @r r ð7Þ 175 176 qnf @V r @V r V # @V r V zb @V r V 2# V cos # ỵ Vr ỵ ỵ z b ỵ r cos # @t @r r @# b þ r cos # @z r ! 178 179 @P @ V r @V r @ V r b @2V r ỵ ỵ lnf ỵ þ 2 2 @r r @r r @# @r b ỵ r cos #ị @z   V r @V # @V r V # sin # sin # @V r cos # À À 2 ỵ ỵ r @# b ỵ r cos # r r @# r @r ! 2bsin# @V z cos # ðV r cos # À V # sin#Þ ; 8ị ỵ 2 b ỵ r cos #ị @z b ỵ r cos #ị ẳ Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 181 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx 182 qnf @V # @V # V # @V # V zb @V # V r V # V cos # ỵ Vr ỵ ỵ ỵ ỵ z b þ r cos # @t @r r @# b þ r cos # @z r !  @P @ V # @V # @ V # b @2V # ẳ ỵ ỵ lnf ỵ þ r @# r @r r @#2 ðb þ r cos #ị2 @z2 @r 184 b ỵ r cos #Þ ! ðV r cos# À V # sin#ị ; ẳ b @P @ V z @V z @ V z ỵ ỵ lnf ỵ b ỵ r cos# @z @r r @r r @#2 ð9Þ ðqcp Þnf b 185 @2V z Vz 2 b ỵ r cos #ị @z b ỵ r cos#ị   @V z sin# @V z ỵ cos # b ỵ r cos# r @# @r  2b @V r @V # cos # sin# ị ỵ gqcịnf T T ị; ỵ b ỵ r cos# @z @z þ V # @V r cos# @V # À 2ỵ ỵ r @# b ỵ r cos # @r r   sin# V r @V # 2bsinu @V z ỵ b ỵ r cos # r r @# b ỵ r cos#ị2 @z sin# qnf @V z @V z V # @V z V zb @V z V z ðV r cos# À V h sin#ị ỵ Vr ỵ ỵ ỵ b ỵ r cos # @t @r r @# b ỵ r cos# @z  @T @t ỵ V r @T ỵ @r Vzb @T bỵr cos # @z ! ẳ K nf @ @2 T @r ỵ 1r @T @r # ỵ bỵrcoscos # b2 @2 T bỵr cos #ị2 @z2 @T @r ỵ 10ị 187 188 A ỵ Q 0: ð11Þ Fig Variation of velocity profiles V z for different values of Cu nanoparticle volume fraction with shape factor of bricks, cylinder and platelets for both catheter r ẳ 0ị and balloon height r ẳ 0:1ị (Panels (a), (b) and (c) respectively) Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 190 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx 196 Here V r in radial and V z in axial directions are the velocity components of the nanofluid, T is the temperature of the nanofluid, Q is the heat absorption/generation constant For the developed nanofluid model qnf is the density, lnf is the viscosity, knf is the thermal conductivity, cnf is the thermal expansion coefficient and ðqcp Þnf is the heat capacitance We utilize the thermophysical prop- 197 erties are given by [32]: 191 192 193 194 195 198 knf lnf ẳ 1/ị 5=2 ; anf ẳ qc ị ; qnf ẳ /ịqf ỵ /qs ; qcịnf p nf ẳ /ịqcịf ỵ /qcịs ; qcp ịnf ẳ /ịqcp ịf ỵ /qcp ịs ; knf kf ẳ ks ỵ1ỵmịkf /1ỵmịks kf ị : ks ỵ1ỵmịkf þ/ðks Àkf Þ ð12Þ 201 In the above expressions qf is density, lf is viscosity, cf is ther- 202 mal expansion coefficient, ðqcp Þf is heat capacitance and kf is ther- 204 205 206 207 mal conductivity of base fluid qs is density, cs is thermal expansion coefficient, ðqcp Þs Is heat capacitance, ks is thermal conductivity of the materials constituting Cu-nanoparticles and / is the nanofluid volume fraction The dimensional boundary conditions are V z ¼ at r ¼ hðzÞ; V z ¼ at r ¼ Rðz; tị; 209 210 211 212 T ẳ at r ¼ hðzÞ; T ¼ T ð13Þ at r ¼ Rðz; tÞ:b In order to achieve the simplified governing equations, we bring into use the following nondimensional quantities: r ¼ R0 r ; z ¼ L0 z0 ; V r ¼ du V 0r ; V z ¼ u0 V 0z ; L0 p¼ u0 L0 lf p0 R20 ; Gr ẳ q2f cf R20 gTT ị ; lf u0 Q R2 srz ¼ uR0 l0 f s0rz ;  ¼ Rb0 ; qu R 0 b ¼ ðT ÀT ; Re ¼ f l0 ; h ẳ TTT : ịkf ÀT f 215 216 217 Substitute the nondimensional quantities (14) into Eqs (5)–(9), we get   @V r V r V r cos #0 @V 0z ỵ d ỵ ỵ ẳ 0; 0 0 @r r ỵ r cos # ỵ r cos # @z0 219 @ V 0r dà e2 @V 0r dà e2 @ V 0r Vr ỵ ỵ d e @r 02 r0 @r0 r 02 @#02 r 02   à à de @ Vr de sin #0 @V 0r @V r ỵ ỵ cos # 0 02 @r r @# ỵ r cos # ỵ r cos #0 ị @z ! 0 à 0 2e2  sin # @V z d e2 V r cos2 # ; ð16Þ À þ 2 @z0 ð1 þ r cos # ị ỵ r0 cos #0 ị ẳ @P0 lnf ỵ @r lf 220 d e d e ỵ  sin #0 ỵ r cos#0 222 223 qnf V z sin #0 @P lnf à @V 0r R e e ỵ de ẳ 0 r @#0 lf qf ỵ r cos # @#0 Vr0 r0 ! @V 0z dà e2 2 V 0r sin#0 cos #0 þ ; ð17Þ 2 ð1 þ r cos #0 ị @z ỵ r cos #0 Þ 2e2 sin#0  qnf @V 0r V 0z @V 0z dà V 0r V 0z cos #0 à @V r Re e ỵ ỵ 0 ỵ d Vr qf @t @r ỵ r cos # @z0 ỵ r cos #0 @P lnf @ V 0z @V 0z @ V 0z ỵ ỵ ỵ ẳ 0 lf @r02 r0 @r0 r02 @#02 ỵ r cos # @z0 225 226  @ V 0z 2 V 0z  cos #0 @V 0z ỵ 02 ỵ r0 cos #0 @r ỵ r cos # ị @z ỵ r cos # Þ ! ðqcÞnf  sin #0 @V 0z 2e2 cos #0 @V 0r ỵ þ Gr h; ð18Þ ðqcÞf r ð1 þ r cos #0 ị @#0 ỵ r cos #0 ị2 @z0 e2 ỵ 14ị 214 !  200 203 qnf à @V 0r à @V 0r dà V 0z @V r e2 V 2z cos #0 Re e d ỵ 0 þ d Vr qf @t @r0 þ r cos # @z0 ỵ r cos #0 @h @t Re P r e ỵ d V 0r @h þ @r ! sin #0 @h 1þr0 cos #0 @z ẳ K nf qcp ịf K f qcp ịnf B B B @ @2 h ỵ 1r @r2 cos #0 1ỵr cos #0 e2 @h ỵ @r @h @r 1ỵr0 cos #0 ị 15ị 229 C qcp ịf ỵC Cỵb A qcp ịnf @ h @z2 ð19Þ 93 82 < 11À 94 ðzÀ dịỵ32z   2= > dị > < 41d z dị   z dỵ  3; 5Xtị d ; : Rzị ẳ  32 ðzÀ dÞ > > : XðtÞ otherwise ð20Þ ( hzị ẳ 228 231 232 234 235 p2 zz0d 0:5ị  kỵr e  ỵ 3: d6z6d k  ẳ d, where d L0 21ị otherwise r ẳ Rr0 and zd ẳ 2zd ỵ1L0 2L0 237 238 In the above equations, Re is the Reynolds number and u0 is the averge velocity of the tube with a radius R0 The legitimate equations denoting the unsteady flow of nanofluid in the form of a mild     stenosis Rd0 ¼ dà , subject to the further condition RL00 ’ oð1Þ [33] 239 We obtain the dimensionless form of constituting equations (11)–(17) after releasing the dashes; 243 @p ẳ 0; @r 22ị @p ẳ 0; @# ð23Þ 240 241 242 244 245 247 248 250 251 @p lnf ẳ ỵ r cos # @z lf Fig Variation of velocity profiles V z for different values of of shape factor of bricks, cylinder and platelets @ V z @V z @ V z in V z ỵ ỵ r @r r @#2 @r þ r cos #Þ2  ðqcÞnf  cos # @V z  sin # @V z ỵ Gr h; 24ị þ À þ r cos # @r rð1 þ r cos #Þ @# ðqcÞf Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 253 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx Fig Variation of wall shear stress srz for different values of Cu nanoparticler volume fraction / with shape factor of bricks, cylinder and platelets for both catheter r ẳ 0ị and balloon height r ẳ 0:1ị (Panels (a), (b) and (c) respectively) 254 256 257 258 259 @ h @h  cos # @h ks ỵ 2kf ỵ 2/kf ks ị ỵ ỵ ỵ b ẳ 0: @r2 r @r ỵ r cos # @r ks ỵ 2kf À 2/ðkf À ks Þ ( ð25Þ The boundary conditions are defined as follows after dropping dashes; V z ¼ at r ẳ hzị; V z ẳ at r ẳ Rz; tị; 26ị 261 h ẳ at r ẳ hzị; h ẳ at r ẳ Rz; tÞ: 262 With 93 82 94 <   2= > > < 41 À dà ðz À dÞ  11 z dị ỵ 32z dị 5Xtị d 6z6d  ỵ 3; : ; Rz; tị ẳ  32 z dị > > : XðtÞ otherwise ð27Þ 263 265 hðzÞ ẳ 266 z zd 0:5ị2 k ỵ r ep k 6z6d  ỵ 3: d otherwise 28ị 268 Solutions development 269 We use perturbation expansion of dependent variables in Eqs (24) and (25) as follows: 270 V z ẳ V z0 ỵ  cos #V z1 ỵ 2 cos2 #V z2 ỵ h ẳ h0 þ  cos #h1 þ 2 cos2 #h2 þ 271 272 ð29Þ Zeroth order system is given by Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 274 275 276 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx Fig (a) Variation of wall shear stress srz for different values of shape factor of bricks, cylinder and platelets for catheter r ẳ 0ị and balloon height r ẳ 0:1ị (b) Variation of wall shear stress srz for different values of shape factor of bricks, cylinder and platelets dP d V z0 dV z0 ẳ ỵ dz /ị52 dr r dr ! ỵ Gr qcịnf d h0 dh0 kf h0 ; ỵ ỵ b ẳ 0; r dr knf ðqcÞf dr ð30Þ 278 279 280 Corresponding boundary conditions are V z0 ¼ at r ẳ hzị; V z0 ẳ at r ẳ Rðz; tÞ 282 283 284 d V z1 ð1 /ị dr 2 ỵ dV z1 V z1 d V z0 dV z0 ỵ2 ỵr r dr r dr dr 287 ỵ Gr ! ðqcÞnf d h1 dh1 d h0 dh0 kf ðrh0 ỵ h1 ị ẳ 0; ỵ b ẳ 0; 31ị ỵr ỵ2 ỵ r dr qcịf dr knf dr dr rs2 logðhr ÞÀ Corresponding boundary conditions are 293 294 295 ỵ ỵ r ỵ s4 ỵ r s5 À ð4r s2 À s3 ÞlogðrÞ 256  cos # r3 s10 ỵ s14 r ỵ r s12 ỵ s7 r5 ỵ s6 r6 ỵ s8 kf qcịnf ỵs7 r ỵ r4 s8 ỵ r3 s9 ỵ s13 r ịlogrịị Gr b; knf qcịf 32ị h1 ẳ at r ẳ hzị; h1 ¼ at r ¼ Rðz; tÞ: 292  r  cos# dp B C7 ỵ 4r À R À s2 log @ A5 dz R ðr Àh2 Þð3r ÀR2 Þ 4ð1 À /ị V z1 ẳ at r ẳ hzị; V z1 ẳ at r ẳ Rz; tị; 290 13 291 2r First order system is given by 5=2 288 V z z; tị ẳ h0 ẳ at r ẳ hzị; h0 ẳ at r ẳ Rz; tÞ: 286 After solving the Zeroth and First orders systems of equations together with the corresponding boundary conditions Expressions (25) produce the solutions of the axial velocity, the temperature distribution is given by ðh 25 À R5 ịs11 5s12 ịỵ 11 B CC B B B CC B B B B B ðh À R3 Þðs13 À 3s14 ÞÀ CC B CC B B B CC B B B B B s7 ðh6 À R6 Þ À ðh À RÞs À CC B CC B B B CC B B B CC B B 4 C B B B ! ðh À R Þð4s10 À s9 ịỵ C 16 B 2h4 3s3 þ 6s4 þ 3h2 ðs2 þ s5 ÞÀ CC B B 2R 3s ỵ 6s ỵ 3R s ỵ s ÞÀ Bh @ CC qcịnf 5=2 B R  cos # B A ỵ 12 7201 /ị B1 ỵ B12 B CC ðqcÞf s6 B CC B B ðh À R7 ÞÀ 6ð2R2 s2 À s3 ÞlogðRÞ 6ð2h s2 À s3 ÞlogðhÞ B CC B B B CC B B B CC B B ỵ 20s 13 B CC B Bh ịloghịỵ CC B B B 60 B CC B B 3h4hs11 ỵ 5s9 ị B CC B B B CC B B B CC B B 20s ỵ 13 @ AA @ @ R3 ịlogRị 60 dP 3R4Rs11 ỵ 5s9 ị ẳ dz 2h 3s3 ỵ 6s4 ỵ 3h s2 ỵ s5 ị h2 @ Aỵ 12 62h s2 À s3 ÞlogðhÞ C C C C C C C C C C C kf C Gb knf r C C C C C C C C C C C A ð33Þ Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 297 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx 298 299 Z where  r  h r À R2 À s2 log hz; tị ẳ R ỵ ỵ  cos # log 302 303 ỵ 2r3 3r 3rs2 ỵ 2h 3h ỵ 2rịr 3hs2 ỵ 3rs2 35ị rV z dr: Since the volume flow rate F is uniform between all the sections of the artery ð0 z d; d z d ỵ L0 and d ỵ L0 z L0 ỵ 2dị ! logðhÞ h R 3hs2 À 3Rs2 R h Rð3 À 2RịR ỵ 3s2 ! 3h 2h R2 2Rịỵ 301 Fẳ loghị ! logRịị The total flux in the artery (tube) is 305 306 307 308 kf b Š knf 309 ð34Þ In this study, the shear stress is calculated as follows [26]: 311  l @V  srz ¼ À nf z  lf @r rẳR 310 36ị The expression V z ẳ 1r @w with w ¼ at r ¼ h provides the corre@r sponding stream function Fig Variation of resistance impedance k for different values of Cu nanoparticler volume fractions with shape factor of bricks, cylinder and platelets for both catheter r ẳ 0ị and balloon height r ẳ 0:1Þ (Panels (a), (b) and (c) respectively) Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 313 314 315 316 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx w¼ dP r4 À h ðr À h Þð2R2 À s2 Þ À dz /ị5=2 16 16 where ỵ À h rðr À R2 Þ h 3r h R2 r R2 ỵ ỵ 16 40 80 24 48 !  ! ðr3 À h Þs2 r s2 h ðr À R2 ịs2 logRị ỵ ỵ log R 72 24 r6 À h ðr À h Þðs2 À s5 Þ ðr2 À h Þðs3 À 2s4 Þ þ þ 256 256 384 þ s2 s3 ðr logðrÞ À h logðhÞÞ À ðr logðrÞ À h logðhÞÞ 64 128 ðr À h Þð4s10 À s9 Þ ðr À h Þðs11 5s12 ị 2048 3200 ỵ cos # À ðr3 À h Þðs13 À 3s14 Þ r7 h ịs6 r h ịs7 ỵ þ 1152 896 768 þ ðr À hÞs8 s11 5 À ðr logðrÞ À h logðhÞÞ 128 640  ðqcÞ k nf f Àh logðhÞÞ Gr b: ðqcÞf knf 318 319 320 323 rP ¼ Z Là   Z L @P dz ẳ F vzịdz; @z 2h 3s3 ỵ6s4 ỵ3h s2 ỵs5 ị 1 325 ð37Þ The change in pressure over the length of the time-varying overlapping stenosis is 321 15h R2 hRị s9 s13 ỵ r logrị h loghịị ỵ r logrị 512 384 Aỵ C B h2 @ C B C B 12 C B C B C B s Às ÞlogðhÞ 6ð2h C B C B C B C B C B C B C B C B C B R2 2R 3s3 ỵ6s4 ỵ3R s2 ỵs5 ị C B A C B @ C B C B 12 C B C B C B 6ð2R s2 Às3 ÞlogðRÞ C B C B C B C B 1C B C B 5 C B C B h R ịs11 5s12 ịỵ 25 C B C B CC B B C B C B C B C B 3 C B C B C B ðh ÀR Þðs13 À3s14 ÞÀ CC B B C B CC B B C B C C ðqcÞnf kf Gr bC B B 5=2 B B s7 ðh6 ÀR6 ÞÀðhÀRÞs À C C ðqcÞ k F C 7201/ị B1ỵ B nf f C B C B C B CC B B C B CC B B C B CC B 4 B C B h R ị4s10 s9 ịỵ CC B B C B 16 CC B B C B CC B cosu B C B CC B B C B 7 s6 C B ðh ÀR ÞÀ C B C B CC B B C B C B C B C B ! C B C B C B 20s ỵ CC B 13 B C B Bh C B C B loghịỵ C CC B 60 B C B C B C B C B ỵ5s ị 3h4hs 11 CC B B C B CC B B C B ! CC B B C B CC B 20s13 ỵ B C B AA @ R @ A @ logRị 60 3R4Rs11 ỵ5s9 Þ vðzÞ ¼ 2 45ðh ÀR2 Þðh þR2 Às2 ÞÀ C B B 1C B 9ðh ÀR5 ÞÀ5ðh ÀR3 Þs2 À C C B AA @ 2cosu@ s2 À ðr logðrÞ À h loghịị ỵ cos # 324 0 ð39Þ 327 The resistance impedance or resistance to flow experienced by the nanofluid in each of the parts of the artery under observation using Eq (38) may be defined as 328 kẳ rP F Z kẳ 38ị ; d 329 330 331 333 Rzịdz ỵ Z d ỵL0 d Z vzịdz ỵ L d ỵL0 334 Rzịdz; 40ị where Rzị ẳ vzịjRẳ1 : Fig (a) Variation of resistance impedance k for different values of shape factor of bricks, cylinder and platelets (b) Variation of wall shear stress srz for different values of shape factor of bricks, cylinder and platelets Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 336 337 338 340 RINP 535 No of Pages 13, Model 5G 16 January 2017 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx 349 Table Copper- nanoparticle shape factor (m) Nanoparticles Shapes Shape factor Cu À Bricks 3.7 Cu À Cylinders Cu À Platelets 4.9 5.7 Fig Variation of temperature profile h with r for different values of shape factor of bricks, cylinder and platelets 341 Graphical results and discussion 342 Computer codes in Mathematica are so arranged and processed to investigate the results obtained through using a perturbation technique for the velocity V z in axial direction, the temperature profile h, the resistance impedance k and the wall shear stress srz in order to discuss the results obtained in Eqs (32), (34), (36), (37) and (40) graphically and quantitatively By considering the data: 343 344 345 346 347 348 k ¼ 0:1; L0 ¼ 1; d ¼ 0:75; w ¼ 7:854; Re ¼ 1: 351 To see the effects of shape factor m of different shapes of nanoparticles (Bricks, Cylinders and Platelets) on the axial velocity for catheter r ẳ 0ị and balloon height r ẳ 0:3ị, we prepared the Fig 2(a)(c) for various of parameters ðm ¼ 3:7; 4:9; 5:7; Gr ¼ 1:8; b ¼ 1:6; z ¼ 2:2; dà ¼ 0:1; k ¼ 0:02; F ¼ 0:01Þ Here identified that the axial velocity step up by successive values of the volume fraction parameter / ẳ 0; 0:02; 0:04ị for bricks Cunanoparticles m ¼ 3:7Þ, cylinders Cu-nanoparticles ðm ¼ 4:9Þ and platelets Cu-nanoparticles m ẳ 5:7ị for both curvature  ẳ 0:1ị and non-curvature  ẳ 0ị arteries in the region 0:02 r 0:63Þ while obverse behavior is in the region ð0:63 r 1Þ It is also paying attention towards that the axial velocity is depressed and shifted towards the center of the artery for balloon height ðrà ¼ 0:3ị than for the catheter r ẳ 0:0ị and it increases in the region ð0:15 r 0:7Þ and inverse occurs in the region ð0:7 r 1Þ Fig shows the variation of velocity profiles V z of different values of shape factor of bricks Cu-nanoparticles, cylinders Cu-nanoparticles and platelets Cu-nanoparticles We have observed that the velocity for platelets Cu-nanoparticles is higher from both bricks Cu-nanoparticles and cylinders Cunanoparticles Arterial curvature  ẳ 0:1ị is responsible to reduce the flow, but non-curvature (straight arteries)  ẳ 0ị supports the velocities of the flow for all types of Cu-nanoparticles Fig is related to the wall shear stress srz for different quantities of volume fraction ð/ ¼ 0; 0:02; 0:04Þ with shape factor of bricks Cu-nanoparticles, cylinders Cu-nanoparticles and platelets Cu-nanoparticles for both catheters r ẳ 0ị and balloon height r ẳ 0:1ị These graphs are drawn for uniform artery d ẳ 0; r ẳ 0ị, stenosed catheterized artery d ẳ 0:1; r ẳ 0ị and stenosed balloon angiopalstic artery d ẳ 0:1; r ẳ 0:3ị We find out that the wall shear stress intensify for uniform artery, stenosed catheterized artery and stenosed balloon angiopalstic artery by increasing volume fraction / ẳ 0; 0:02; 0:04ị The wall shear stress remains higher in uniform artery than stenosed catheterized artery and stenosed balloon angiopalstic artery for bricks Cunanoparticles ðm ¼ 3:7ị, cylinders Cu-nanoparticles m ẳ 4:9ị and platelets Cu-nanoparticles m ¼ 5:7Þ Fig 5(a) belongs to the wall shear stress srz for different values of shape factor of bricks Cu-nanoparticles, cylinders Cunanoparticles and platelets Cu-nanoparticles for catheter r ẳ 0ị and balloon height r ẳ 0:1ị Fig 5(b) is also belonging to the wall shear stress srz for three values of shape factor of bricks Cunanoparticles, cylinders Cu-nanoparticles and platelets Cunanoparticles We perceived that the wall shear stress unalterably remains higher in respect of platelets Cu-nanoparticles from bricks Cu-nanoparticles and cylinders Cu-nanoparticles for uniform 352 Fig Plot shows streamlines for noncurvature artery k ẳ 0ị with different values of the shape factor at z ¼ with catheter ðm ¼ 3:7Þ, (b) ðm ¼ 4:9Þ, (c) ðm ¼ 5:7Þ Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 RINP 535 No of Pages 13, Model 5G 16 January 2017 10 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx Fig 10 Plot shows streamlines for curvature artery k ẳ 0:1ị with different values of the shape factor at z ¼ with catheter m ẳ 3:7ị, (b) m ẳ 4:9ị, (c) m ẳ 5:7Þ Fig 11 Plot shows streamlines for noncurvature artery ðk ¼ 0Þ with different values of the shape factor at z ẳ with balloon angiplasty m ẳ 3:7ị, (b) m ẳ 4:9ị, (c) m ẳ 5:7ị Fig 12 Plot shows streamlines for curvature artery k ẳ 0:1ị with different values of the shape factor at z ¼ with balloon angioplasty m ẳ 3:7ị, (b) m ẳ 4:9ị, (c) m ẳ 5:7ị Table Thermophysical characteristics of base fluid and Cu-nanoparticles cp (J/kgK) q ðkg=m3 Þ k (W/mK) c  10À5 ð1=KÞ Fluid phase (Blood) Cu 3594 1063 0.492 0.18 385 8933 400 1.67 artery ðdà ¼ 0; rà ¼ 0Þ, stenosed catheterized artery à ðd ¼ 0:1; rà ¼ 0Þ and stenosed balloon angiopalstic artery ðdà ¼ 0:1; r ẳ 0:3ị The curvature  ẳ 0:1ị of the artery influence the wall shear stress to increase for pure fluid / ẳ 0ị and all types of Cu-nanoparticles (Bricks, Cylinders and Platelets) in uniform artery, stenosed catheterized artery and stenosed balloon angiopalstic artery than the non-curvature  ẳ 0ị artery Fig depicted the resistance impedance k variation for non-identical values of Cu nanoparticles volume fractions Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 398 399 400 401 402 403 404 405 406 RINP 535 No of Pages 13, Model 5G 16 January 2017 11 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx Table Computational values of velocity corresponding to the critical points of the stenosis for Gr ¼ 1:8; b ¼ 1:6; d ¼ 0:75; dà ¼ 0:1 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Axial velocity for m ¼ 3:7 z ẳ 1:03 k ẳ 0ị z ẳ 1:03 k ¼ 0:1Þ z ¼ 1:5 ðk ¼ 0Þ z ¼ 1:5 k ẳ 0:1ị z ẳ 1:97 k ẳ 0ị z ẳ 1:97 k ẳ 0:1ị 0.0166939 0.0289712 0.0339808 0.0350119 0.0331504 0.0289805 0.0229382 0.0154235 0.0068444 0.0170366 0.0294674 0.0344290 0.0353168 0.0332743 0.0289315 0.0227645 0.0152088 0.0067021 À0.039633 À0.004598 0.0140053 0.0245615 0.0296574 0.0304885 0.0277794 0.0220749 0.0138563 À0.040972 À0.004730 0.0143291 0.0249810 0.0299775 0.0306187 0.0277102 0.0218651 0.0136235 0.0166939 0.0289712 0.0339808 0.0350119 0.0331504 0.0289805 0.0229382 0.0154235 0.0068444 0.0170366 0.0294674 0.0344290 0.0353168 0.0332743 0.0289315 0.0227645 0.0152088 0.0067021 Table Computational values of velocity corresponding to the critical points of the stenosis for Gr ¼ 1:8; b ¼ 1:6; d ¼ 0:75; dà ¼ 0:1 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Axial velocity for m ẳ 4:9 z ẳ 1:03 k ẳ 0ị z ¼ 1:03 ðk ¼ 0:1Þ z ¼ 1:5 ðk ¼ 0ị z ẳ 1:5 k ẳ 0:1ị z ẳ 1:97 k ẳ 0ị z ẳ 1:97 k ẳ 0:1ị 0.0166993 0.0289838 0.0339969 0.0350269 0.0331603 0.0289828 0.0229326 0.0154130 0.0068357 0.0170389 0.0294764 0.0344424 0.0353303 0.0332839 0.0289342 0.0227599 0.0151994 0.0066942 À0.0396094 À0.0045968 0.0140049 0.0245643 0.0296626 0.0304934 0.0277814 0.0220729 0.0138516 À0.0409469 À0.0047291 0.0143281 0.0249832 0.0299822 0.0306235 0.0277124 0.0218634 0.0136192 0.0166993 0.0289838 0.0339969 0.0350269 0.0331603 0.0289828 0.0229326 0.0154130 0.0068357 0.0170389 0.0294764 0.0344424 0.0353303 0.0332839 0.0289342 0.0227599 0.0151994 0.0066942 Table Computational values of velocity corresponding to the critical points of the stenosis for Gr ¼ 1:8; b ¼ 1:6; d ¼ 0:75; dà ¼ 0:1 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 Axial velocity for m ẳ 5:7 z ẳ 1:03 k ẳ 0ị z ¼ 1:03 ðk ¼ 0:1Þ z ¼ 1:5 ðk ¼ 0ị z ẳ 1:5 k ẳ 0:1ị z ẳ 1:97 k ẳ 0ị z ẳ 1:97 k ẳ 0:1ị 0.0167029 0.0289923 0.0340077 0.0350370 0.0331671 0.0289843 0.0229288 0.0154058 0.0068298 0.0170405 0.0294824 0.0344514 0.0353395 0.0332904 0.0289361 0.0227568 0.0151930 0.0066888 À0.0395934 À0.0045960 0.0140046 0.0245662 0.0296661 0.0304968 0.0277828 0.0220715 0.0138485 À0.0409293 À0.0047281 0.0143274 0.0249846 0.0299855 0.0306268 0.0277138 0.0218622 0.0136162 0.0167029 0.0289923 0.0340077 0.0350370 0.0331671 0.0289843 0.0229288 0.0154058 0.0068298 0.0170405 0.0294824 0.0344514 0.0353395 0.0332904 0.0289361 0.0227568 0.0151930 0.0066888 / ẳ 0; 0:01; 0:02ị with shape factor of bricks Cu nanoparticle, cylinders Cu nanoparticle and platelets Cu nanoparticle for both catheter r ẳ 0ị and balloon height r ẳ 0:1ị in (Panels (a), (b) and (c) respectively) Clearly, resistance impedance k increases in values of increasing stenosis critical values dà for all shape factors of bricks Cu nanoparticle, cylinders Cu nanoparticles and platelets Cu nanoparticle Curvature of the artery and balloon height is responsible for enhancing the resistance to flow Platelets Cu nanoparticles exhibit more resistance than bricks Cunanoparticle and cylinders Cu-nanoparticle It is of the essence to mention that the volume fraction / abated the resistance impedance Fig sorted out the resistance impedance k variation with time nearly three cardiac cycles have an oscillation profile decaying as time t increases where the oscillation has the reverse trend with respect to the corresponding shape of oscillation in Fig 7(a)–(b) Resistance impedance k for different values of shape factor of bricks Cu-nanoparticle, cylinders Cu-nanoparticle and platelets Cu-nanoparticle is shown Fig (a) and wall shear stress srz variation for shape factor of bricks Cu-nanoparticle, cylinders Cunanoparticle and platelets Cu-nanoparticle is in Fig (b) The resistance impedance k is lower for platelets Cu-nanoparticle as compared to bricks Cu-nanoparticle and cylinders Cunanoparticle with the time increases In the 1st cycle of time t the resistance impedance of nanoparticles remains lower for the curvature artery than non-curvature artery, but resistance impedance in 2nd and 3rd cycles exhibits reverse behavior until it decays and inverse behavior is observed for wall shear stress Curvature  ẳ 0:1ị of the artery produces slightly more resistance and wall shear stress than non-curvature  ẳ 0ị artery for all forms of nanoparticle shapes (see Table 1) Fig is developed for the temperature profile h variation with r for distinct quantities of shape factor of bricks Cu-nanoparticle, cylinders Cu-nanoparticle and platelets Cu-nanoparticle We can Please cite this article in press as: Ahmed A, Nadeem S Shape effect of Cu-nanoparticles in unsteady flow through curved artery with catheterized stenosis Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.015 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 RINP 535 No of Pages 13, Model 5G 16 January 2017 12 A Ahmed, S Nadeem / Results in Physics xxx (2017) xxx–xxx 476 observe that the platelets Cu-nanoparticle generating more temperature than bricks Cu-nanoparticle and cylinders Cu-nanoparticle Clearly, curvature  ẳ 0:1ị is responsible for enhancing the temperature of nanofluid in the artery than a non curvature artery  ẳ 0ị In the fluid flows trapping unveil an interesting phenomenon This phenomenon and appearance capture more attraction in the being of nanoparticles in blood flows In the wave curves, streamlines undergo a particular spatial arrangement to confine a bolus which goes as a whole with the wave movement The phenomenon of continuous internally circulating bolus or boluses of the flowing substance by closed streamlines is known trapping The bolus defined as a small rounded mass of a substance closed by streamlines in the wave frame is conveyed at the wave movement Figs and 11 reveal that the bulk of trapping bolus reduces by increasing the numerical amount of shape factor m for bricks Cunanoparticle, cylinders Cu-nanoparticle and platelets Cunanoparticle for non-curvature artery reverse behavior is exhibited for curvature artery in Figs 10and 12 the balloon position is at z ¼ This is obviously perceived that the curvature of the artery hinders the flow while non-curvature enhances the flow Moreover, the platelets Cu-nanoparticle moves rapidly than bricks Cunanoparticle and cylinders Cu-nanoparticle in non-curvature  ẳ 0ị artery than curvature  ẳ 0:1Þ artery Tables 2, and reveal computative values of velocity corresponding to the stationary point heights of the stenosis (z ¼ 1:3, z ¼ 1:5 and z ¼ 1:97) for curvature and non-curvature arteries with fixed nondimensional parameters Gr ¼ 1:8; b ¼ 1:6; d ¼ 0:75; dà ¼ 0:1 We perceived that the values of axial velocities of bricks Cu-nanoparticle, cylinders Cu-nanoparticle and platelets Cunanoparticle remain same at critical heights z ¼ 1:03; z ¼ 1:97 for curvature  ẳ 0:1ị and non-curvature  ẳ 0ị arteries respectively The axial velocity near the catheter is higher for curvature artery than the axial velocities for non-curvature artery, but the reverse trend of the axial velocities is discerned near the wall of the considered lumen of the artery (see Table 5) 477 Conclusions 478 The effects of Cu-nanoparticles such as bricks, cylinders and platelets as blood model through catheterized curved artery with time-varying overlap stenosis is examined Graphical results are supplied for the temperature profile, wall shearing stress distribution, and resistance to flow (resistance impedance), axial velocity and bolus trapping of the concerned parameters Summarizing the necessary finding which is as follows: 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502  The order-of-magnitude estimates of the quantities of interest with the conditions Re e < 1, dà ( and e $ 0ð1Þ  The Platlets Cu-nanoparticle in the central portion of the tube are not sheared, and the slight velocity gradients are only found in the layers near the wall of artery than Bricks Cylinders Cunanoparticles  The balloon height enhances shear stresses and velocity gradients  The wall shear stress remains higher for the uniform artery than the stenosed artery and stenosed angioplasty for all shapes of bricks cu-nanoparticles, cylinders cu-nanoparticles and platelets cu-nanoparticles  The wall shear stress remains higher for the platelets cunanoparticles for catheter, angioplasty and for stenotic severity  The resistance impedance vanishes for the uniform artery and it rapidly increases as stenosis critical height increases for catheter and angioplasty practices for bricks cu-nanoparticles, cylinders cu-nanoparticles and platelets cu-nanoparticles  The resistance impedance and wall shear stress profiles with the time exhibits oscillation form for bricks cu-nanoparticles, cylinders cu-nanoparticles and platelets cu-nanoparticles and this oscillation decaying as time progresses  Platelets cu-nanoparticles assumes minimum resistance as compared to bricks cu-nanoparticles and cylinders cunanoparticles  The trapping appear near the wall of the catheter and the streamlines split to trap a bolus towards the wall of the stenosis for bricks cu-nanoparticles, cylinders cu-nanoparticles and platelets cu-nanoparticles respectively while the size of the trapping bolus near the wall of the stenosis decreases for noncurvature artery but opposite behavior is recognized for curvature artery  The trapping appear near the wall of the balloon angioplasty the streamlines split to trap the boluses towards the wall of the stenosis for bricks cu-nanoparticles, cylinders cu-nanoparticles and platelets cu-nanoparticles respectively, while the size of the trapping boluses near the wall of the stenosis decreases for non-curvature artery but opposite behavior is recognized for curvature artery 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 References 525 [1] Fox JA, Hugo AE Localization of Atheroma: a theory based on boundary laver separation British Heart J 1906;28:388 [2] Stephenson Jr SE, Mann GV, Younker R, Scott Jr HW Factors influencing the segmental deposition of atheromatous material Arch Surg 1962;84:49 [3] Texon M A hemodynamic concept of atherosclerosis, with particular reference to coronary occlusion Arch Intern Med 1957;99:418 [4] Texon M The hemodynamic concept of atherosclerosis Bull New York Acad Med 1960;36:263 [5] Rodbard S Dynamics of blood flow in stenotic lesions Am Heart J 1966;72:698 [6] Fung YC Biodynamics circulation New York: Springer-Verlag; 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