Peristaltic Flow Characteristics of Blood through Stenotic Artery under the influence of Magnetic Field and Slip Velocity.

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1 IOSR Joural of Egieerig (IOSRJEN ISSN (e: 5-3, ISSN (: Vol. 8, Issue 7 (July. 8, V (II PP - Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic Field ad Sli Velocity. Sushila Kumari Rajbala Rathee Jagdish Nadal 3 Assistat Professor, Dett. of Mathematics, Pt. N.R.S. Govt. College, Rohtak-4, Haryaa, Idia Assistat Professor, Dett. of Mathematics, A.I.J.H. Memorial College, Rohtak-4, Haryaa, Idia 3 Professor, Dett. of Mathematics, M. D. Uiversity, Rohtak-4, Haryaa, Idia ABSTRACT: The curret aalysis deals with a theoretical study o combied effect of both sli velocity ad magetic field o usteady geeralized o-newtoia eristaltic flow with ermeable wall of artery. A costat trasverse magetic field is alied o ulsatile blood flow ad we are treatig blood as a elasticoviscous, electrically coductig ad icomressible fluid. Aroriate trasformatio methods are adoted to solve usteady o-newtoia axially symmetric mometum equatio i cylidrical olar co-ordiate system with suitably rescribed coditios. To validate the alicability of the roosed aalysis, aalytical exressios for axial velocity, fluid acceleratio, wall shear stress ad volumetric flow rate are comuted ad obtaied umerical results are deicted grahically for differet values of flow variables of iterest, to aalyse the ifluece i axial velocity, wall shear stress ad volumetric flow rate of streamig blood. KEYWORDS: Elastico-viscous fluid, sli velocity, selective ermeability, trasverse magetic field, eristalsis, kudse umber Date of Submissio: Date of accetace: I. INTRODUCTION Today rheology of streamig blood i arteries is a very imortat issue of research i both hysiological ad cliical situatios sice oe of world- wide leadig cause of death is due to malfuctio of the cardiovascular system i huma beigs. As we kow that the major cotributio of the etire cardiovascular system is to suly oxyge rich blood to every tissue uder a sufficiet ressure gradiet for exchagig materials through the blood vessels wall by eristaltic umig of heart. It is a two- way exchage through the walls of blood vessels that the utriets are carried to tissues ad cells ad the bodily fluid returs alog with the waste from cellular metabolism. But this trasort of materials by rogressive wave of regular cotractio ad exasio of heart (termed as eristalsis is ot always so regular. May cardiovascular diseasesarticularly atherosclerosis (medically called steosis is commo disease due to arrowig of arteries lume, are very much related to the ature of blood movemet ad dyamic behaviour of vessel walls. The word, atherosclerosis has bee take from the Greek words athero (meaig aste or gruel ad sclerosis (hardess. The word steosis meas the arrowig of artery due to deositio of arteriosclerotic laque or other tyes of abormal tissue alog the walls of the chael. This results ito disorder i circulatory system by reducig or occludig the blood suly which may develo cardiovascular diseases esecially heart attack, stroke etc ad whe blood flow to a tissue becomes restricted or reduced, ecrosis will evetually occur. Therefore, i the reset sceario, the study of blood flow disorders i costricted huma arteries has of much iterest for further scietific ivestigatios ad alicatios i biomedical sciece. To examie the flow characteristics of blood through steosed vessel, Beaver ad Joseh [] emloyed boudary coditios at a aturally ermeable vessel wall ad it would hel i uderstadig that why o-sli coditio at vessel wall should be relaced by sli velocity. Lee ad Fug [] studied the blood flow at low Reyolds umber i the rage -5 i locally costricted tube by usig coformal maig method. Oka ad Murata [3] reseted hydro- dyamical theory for flow of blood through a blood vessel with ermeable wall ad ivestigated the exchage of fluid as a steady slow motio across the ermeable wall uo the motio of the fluid withi rigid circular tube. Saffma [4] further discussed boudary coditios for flow of fluid at the surface of a orous medium i suitable imroved maer (give by Beaver ad Joseh [] ad justify it. Poel et al. [5] have ivestigated a cotiuum aroach for blood flow with coule stresses. Steady flow of blood through modelled vascular steosis has bee ivestigated by McDoald [7]. Shukla et al. [8] discussed the effect of steosis o o-newtoia flow of blood i a artery but later o they took ito accout the effect of radial distributio of cells ad the existece of the eriheral lasma layer ear the wall ad studied the flow of Page

2 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. blood through a artery with mild steosis by assumig blood as a ower law fluid. Srivastava ad Srivastava [9] had develoed a two- hase mathematical model for ulsatile flow of blood with etrace effects. Blood flow through a artery with mild steosis has bee ivestigated by Siha ad Sigh [] ad coule stresses effects o blood flow also discussed by them. Srivastava [] studied the flow through steotic blood vessel ad he cosidered blood as coule stress fluid. A umerical study o flow of fluid have bee develoed by Lee [3] ad fluid assed through tubes with double costrictios. The flow characteristics i the eighbourhood of double costrictios were studied umerically i a circular cylidrical tube. Misra et al. [6] have reseted a o-newtoia model i which they have discussed flow of blood through arteries uder steotic coditios. Haldar ad Ghosh [7] have studied blood flow uder the effect of magetic field through a ideted artery i the resece of erythrocytes takig blood as Newtoia fluid ad discussed the exressios for blood velocity, ressure ad flow rate. Murata [8] has roosed a sedimetatio model o flow roerties of aggregatig red cell susesio i costricted horizotal tubes i which he has cosidered costat hematocrit level ad Newtoia viscosity i the cetral regio circular tube. Chakravarty ad Madal [9] ivestigated a two dimesioal flow of blood uder steotic coditios through taered artery. Srivastava [] has ivestigated the flow of blood flow i a mild steosed artery takig cetral layer (core regio is as coule stress fluid as it is the susesio of erythrocytes (RBC ad a eriheral layer of lasma as Newtoia fluid. Pralhad ad Schultz [] have studied the modelig of arterial steosis ad emloyed its alicatio to blood diseases assumig blood as o- Newtoia coule stress fluid. Bali ad Awasthi [] have discussed the resistace to flow of blood havig sigle mild steosis i a artery havig effect of magetic field. A large umber of ivestigatios have bee described by several researchers to uderstad disorders i flow of blood due to develomet of steosis by cosiderig blood as Newtoia or o-newtoia fluid. Mathematical modelig of flow of blood i vertebral artery with steosis has bee reseted by Ali et al. [3]. Rathod ad Taveer [4] have emloyed the effects of eriodic body acceleratio o ulsatile blood flow through a orous medium uder the ifluece of magetic field ad they have foud that local exosure of a magetic field could relax blood vessel ad ehace blood flow. Varshey et al. [5] have also reseted the effect of magetic field o blood flow i a multile steosed artery ad they have aalysed that all the flow variables are affected by the itesity of magetic field i the resece of multile steosis. Shit ad Roy [6] have emloyed a theoretical study of hydro- magetic ulsatory blood flow i a chael which is costricted ad orous. A mathematical modelig of o- Newtoia blood flow i a steosed artery has bee reseted by Mathur ad Jai [7]. They have cosidered blood as Power- law fluid ad rovided much evideces for the vital role of hydro-dyamic factors i the develomet ad the rogressio of steosis i a artery. Rathee ad Sigh [8] have reseted the effect of exterally alied magetic field o the two- layered model of blood flow through steosed artery i orous medium ad aalysed that exact stregth of magetic field ca hel i cotrollig the bood flow i hyertesive atiets ad i steosed arteries. Kumar et al. [9] have desiged a model to ivestigate flow of o-newtoia blood through a artery with multile steosis. The usteady sli flow of blood through costricted artery has bee ivestigated by Gaur ad Guta [3] ad the obtaied results have show the variatio i flow characteristics alog the axial distace with assage of time. Siddiqui ad Geeta [3] have aalysed the usteady flow of blood with sli effects through a artery i the resece of steosis. Malek ad Horque [3] have recetly reseted a theoretical study for flow of blood through a steosed artery with ermeable wall takig cosideratio of hematocrit level. Motivated from above ivestigatios ad i order to simulate the costricted artery flow roblem, oe ste close to more realism of heomeo, we have made a attemt to reset theoretical aalysis to exlore the sigificat combied ifluece of sli velocity ad exterally alied magetic field o flow of o- Newtoia blood through steosed artery with ermeable wall. I the curret study, o-newtoia rheology of streamig blood is characterised by its elastico-viscosity. Sice the arterial wall tissue has selective ermeability, o-sli coditio at wall is ot aroriate ad we have to accout the effect of the sli velocity at the wall of arrow artery for a more realism of the heomeo. Also selective ermeability of the arterial wall ermits for alyig sli coditio i lace of o-sli coditio, it is oe more close ste to the real situatio of eristaltic flow of blood through wall of arrow artery as the arterial ermeable walls allow the fluid articles to sli at boudary. A extesive quatitative aalysis is carried out by alyig Lalace ad fiite Hakel trasformatios with secific boudary coditios to estimate effects of sli velocity ad alied magetic field with aroriate scietific discussios so as to substatiate the utility of the reset study. II. FORMULATION OF THE PROBLEM We assume oe dimesioal motio of blood i a straight ad rigid cylidrical steosed artery through a orous medium by cosiderig blood as o-newtoia elastico-viscous, icomressible ad electrically coductig fluid uder the ifluece of trasversely alied magetic field. It is well kow that whe magetic field is alied o a electrically coductig fluid like blood, a electromagetic force will roduce due to the Page

3 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. iteractio of curret with magetic field. The electromotive force is give by the roortioality relatio to the seed of motio ad the magetic field itesity. The by Ohm s law, we have J ( E u B ( where E is the electric field itesity vector, is the electrical coductivity, u is the velocity vector, B B B is the total magetic flux itesity vector i which B is the iduced magetic field vector which is of very small amout, hece assumed to be egligible i comariso with the exteral alied magetic field vector B for MHD (mageto-hydrodyamic flow. We also assume that the electric field itesity vector E, due to the olarizatio of charge is also egligible. Therefore i the mometum equatio, the electromagetic force is icluded ad is defied as F J B B u, where B B ( Further we assume that the blood flow is lamiar, usteady, axially-symmetric ad fully develoed. Now the geometry of artery with symmetric shae steosis as roosed by Haldar ad Ghosh (994 Rz ( s s Al z d ( z d, R d z d l (3 where, s is a steosis shae determiig arameter, d deotes the ositio of steosis, l is exressig the legth of steosis, R is radius of ormal artery, R(z is radius of steosed circular cylidrical artery, z deotes the axial ositio ad A is a arameter give by s/( s s A s, ε is the maximum height of develoed steosis at z d l with Rl ( s /( s s / R. III. GOVERNING EQUATIONS Now the equatio of motio which is Navier-Stokes equatios for flow of blood, icludig Lorez s force ad with above assumtios, i cylidrical olar co-ordiates is u ( u u B u, (4 t z t k Where ( ( r (5 r r r Ad 3 Page

4 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. o cos( t, o is the steady-state art of the ressure gradiet or costat amlitude of ressure z gradiet, is the amlitude of the ulsatile comoet that arises the systolic ad diastolic ressure, f ad f is the frequecy of heart ulse, z is the axial distace ad t is time variable. u (r,t is velocity comoet i axial directio is the desity of blood is the viscosity of blood is elastic viscosity coefficiet of blood σ is the electrical coductivity B is the exteral magetic field K is the ermeability of the isotroic orous medium r is the radial coordiate. The dimesioless quatities are defied below: * u * r * R * R * R * z * K u, r, t t,,, a a, z, K (6 R R R R O droig stars, we have u cos ( ( u t u ( H M u, (7 t t r r r where R is the Womersley arameter, H B R is the Hartma umber, M is the K ermeability arameter ad. It is suosed that for t, oly heart s umig actio is reset ad whe t, artery s blood flow leads to the istat ressure gradiet i.e. (8 z IV. THE INITIAL AND BOUNDARY CONDITIONS Whe itesity of ermeability is low, the boudary coditio roosed by Beaver ad Joseh [], which was du further exlaied by Saffma [4] (also kow as the Saffma s sli coditio as u where is a dr K costat deedig o the roerties of the orous material ad o its structure ad K is take as the ermeability arameter (or Darcy umber of orous material of the wall. This coditio has effective i case of usteady flows ad eve whe MHD effects o fluids is take. Now iitial ad boudary coditios are rescribed below as: u hu r ( Sli coditio or u hu Or (i u hu whe r a ad t (9 where h R K ad Rz ( a R 4 Page

5 (ii Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. h J ( r ur (, whe r a ( a ( H M ( h J ( a (iii u(, t is fiite. V. REQUIRED INTEGRAL TRANSFORMATIONS If g (t is cotiuous fuctio ad is of exoetial order for t the its Lalace trasformatio is defied as st g( s e g t dt, s ( If,a the its fiite Hakel trasformatio defied f (r satisfies the Dirichlet coditios i fiite iterval as a f ( r f r J r dr ( where are the roots of the equatio J ( a hj( a, ad J ( r ad J ( r are Bessel s fuctios of first kid the f (r is give by J r f ( r a h J a o f VI. ANALYSIS Alyig Lalace trasformatio o eq. (7 i light of eq. (, we obtai; s H M u r s, s s s ( r u ( r, s su ( r, s u ( r, u r, (4 r r r Now alyig fiite Hakel trasformatio of first kid ad zeroth order o eq. (4 i light of eq. (, we obtai; s a,, s s s H M u s J a s u s, u, u (3 Or s a u, s J a H where l s( s l s l ( s ( ( s l( H M M Alyig iverse Lalace trasformatio o eq. (5, we obtai; (5 5 Page

6 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. lt l l e cos t l l ( l ( H M ( l ( a u, t J( a sit ( l ( (6 Alyig iverse fiite Hakel trasformatio o eq. (6, we obtai the exressio for fluid velocity which is u r, t lt e h J ( ( ( r l l l H M a ( h J ( a l cost sit ( l ( ( l ( The exressio for fluid acceleratio is give by u F( r, t t or lt l l( e h J r ( l ( ( H M F r, t a h J a l sit cos t ( l ( ( l ( Similarly the exressio for the volumetric flow rate is give by a Q( r, t r u( r, t dr or Q r, t l lt e 4 h J ( a l l ( l ( H M a ( h J ( a l cost sit ( l ( ( l ( ad the exressio for the shear stress is give by ( rt, u r or l lt e ( h l l ( l ( H M ( r, t J ( r a ( h J ( a l cost sit ( l ( ( l ( ( l (7 (8 (9 VII. GRAPHICAL RESULTS AND DISCUSSIONS 6 Page

7 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. I this aalysis, the exressio for axial velocity, volumetric flow rate, fluid acceleratio ad wall shear stress are obtaied ad lotted i resect of axial distace z for differet value of Hartma umber H, ermeability arameter M, sli arameter h ad time variable t. The flow characteristics of the blood ca be discussed i two differet ways; either the effect of idividual factor like artery radius, ressure gradiet (average steady ressure gradiet which is take as amlitude for the ulsatile ressure gradiet, various arameters which are related to the roblem, is studied or we ca calculate the values of flow arameters at a articular site i cardiovascular system. I ormal artery, radius R =.8 cm, = dye /cm, = 4 dye /cm, i coroary artery R =.5cm, = dye /cm, femoral artery R=.5cm, = 3 dye /cm, ad.35, =, d = cm, l = 4cm. I the reset aalysis, we have followed the secod method. The velocity rofile from the eq. (7 for differet values of Hartma umber H, ermeability arameter M, sli arameter h ad time variable t have bee show through Fig., Fig. ad Fig. 3 for give value of Womersley arameter ad give value of ad Fig. 4 resets a three dimesioal view for velocity rofile. It is observed that uto some extet, as Hartma umber H icreases, axial velocity icreases ad attais its maximum value ear ceterlie of the artery ad attais miimum value at the walls but after that it starts decreases. The effect of ermeability arameter causes to stregthe the fluid velocity with icrease i value of ermeability arameter M ad reveals its maximum value at the maximum height of the steosis. Also sli arameter ehaces the blood velocity as comare to the o-sli coditio at arterial wall. It is clear from the figures Fig. 9, Fig., ad Fig. that volumetric flow rate is havig miimum value at steosis eds, while it is havig a maximum value at steosis throat ad it is clear from the three dimesioal Fig. that it icreases with icreasig the value of the sli arameter. It is a commo fact that the wall shear stress has a sigificat role i arterial diseases develomet. So it is ecessary to study the effects of these variables o wall shear stress. Fig. 5, Fig. 6 ad Fig. 7 deicts the variatios of wall shear stress for various values of Hartma umber H, ermeability arameter M, sli arameter h ad time variable t ad it is ascertaied from Fig. 8 that the wall shear stress decreases with icrease i the value of ermeability arameter M ad sli arameter h. Also it is ivestigated that the effect of ressure gradiet o steosis site decreases with icrease i value of Hartma umber H uto some extet. 7 Page

8 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. 8 Page

9 volumetric flow rate Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. -6 x Variatio of Flow Rate alog z h = -.5 h = -.35 h = -.55 h = -.75 x -7.6 Variatio i volumetric flow rate.5 Flow Rate h = z Fig. Fig. r z VIII. CONCLUSION I this aer, the combied effects of sli coditio ad exteral alied magetic field o ulsatile blood flow i a costricted orous artery are aalysed. It is observed that the sli velocity has a imortat role i decreasig wall shear stress ad effective elastic viscosity of blood. We ca also measure the extet of stregth of magetic field from which we ca cotrol ad icrease the blood flow i atiets of hyertesio ad havig blockage i their arteries. The reset model ca also be used as a tool to reduce the blood viscosity by usig sli velocity with magetic field at the costricted wall ad it also rovides scoe to estimate the effect 9 Page

10 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. of the various arameters rovided above o various flow characteristics ad to esure which arameter is havig the most sigificat role ad ca be used for better uderstadig of blood circulatio i huma body. The mai cocer of this study is cosideratio of sli velocity at ermeable wall ad the coclusio of the study are give below: ( The ulsatile axial velocity u( r, t of the blood at the maximum height of the steosis icreases with icreasig values of sli arameter h, Hartma umber H ad ermeability arameter M while it decreases with higher values of Hartma umber H. ( The volumetric flow rate Q( r, t of the blood at the maximum height of the steosis icreases with icreasig values of sli arameter h, Hartma umber H ad ermeability arameter M while it decreases with higher values of Hartma umber H. (3 ( r, t decreases at the maximum height of the steosis whe there is a icrease i the values of sli arameter h, Hartma umber H ad ermeability arameter M while it icreases with higher values of Hartma umber H. (4 This model rovides a most commo form of fluid velocity, wall shear stress ad volumetric flow rate ad the other mathematical models related to this study ca be simly ossessed from this model by alyig suitable values of arameters. REFERENCES []. Beaver G. S. ad Joseh D. D. Boudary coditios at a aturally ermeable wall. Joural of Fluid Mechaics.967; 3: []. Lee J. S. ad Fug Y. C. Flow i locally costricted tube at low Reyolds umber. Joural of Alied Mechaics. 97; 37: 9-6. [3]. Oka S. ad Murata T. A theoretical study of flow of blood i a caillary with ermeable wall. Jaaeese Joural of Alied Scieces. 97; 9: [4]. Saffma P. D. O the boudary coditios at the surface of a orous medium. Studies i Alied Mathematics. 97; 5: 93-. [5]. Poel A. S., Regirer S.A. ad Usick P.I. A cotiuum model of blood flow. Biorheology. 974; : [6]. Yua S. W. Foudatio of fluid mechaics. Pretice Hall of Idia Pvt. Ltd. New Delhi [7]. McDoald D. A. O steady blood flow through modelled vascular steosis. Joural of Biomechaics. 979; :3-. [8]. Shukla J. B., Parihar R. S. ad Rao B. R. P. Effect of steosis o o-newtoia flow of blood i a artery. Bulleti Of Mathematical Bioscieces. 98; 4: [9]. Srivastava L. M. ad Srivastava V. P. O two-hase model of alatial blood flow with etrace effects. Biorheology. 983; : []. Siha P. ad Sigh C. Effects of coule stresses o the blood flow through a artery with mild steosis. Biorheology. 984; : []. Fug Y. C. Biodyamics Circulatio. Sriger Verlag. 984; New York. []. Srivastava L. M. Flow of coule stress fluid through steotic blood vessels. Joural of Biomechaics. 985; : [3]. Lee T. S. Numerical studies of fluid flow through tubes with double costrictios. Joural of Numerical Methods i Fluids. 99; : 3-6. [4]. Fug Y. C. Biodyamics Motio, flow, stress ad growth. Sriger Verlag. New York 99. [5]. Mazumdar J. N. Biofluid mechaics. World Scietific, Sigaore 99. [6]. Misra J. C., Patra M. K. ad Misra S. C. A o-newtoia model for blood flow through arteries uder steotic coditios. Joural of Biomechaics. 993; 6; 9-4. [7]. Haldar K. ad Ghosh S. N. Effect of a magetic field o blood flow through a ideted tube i the resece of erythrocytes. Idia Joural Pure Alied Mathematics. 994; 5: [8]. Murata T. Theoretical aalysis of flow roerties of aggregatig red cell susesios i arrow horizotal tubes. Cliical Hemorh. 998; 4: [9]. Chakravarty S. ad Madal P. K. Two-dimetioal blood flow through taered arteries uder steotic coditios. Iteratioal Joural of No-Liear Mechaics. ; 36: []. Srivastava V.P. Flow of a coule stress fluid reresetig blood through steotic vessels with a eriheral layer. Idia Joural of Pure ad Alied Mathematics. 3; 34: Page

11 Peristaltic Flow Characteristics of Blood through Steotic Artery uder the ifluece of Magetic.. []. Pralhad R. N. ad Schultz D. H. Modelig of arterial steosis ad its alicatios to blood diseases. Joural of Mathematical Bioscieces. 4; 9: 3-. []. Bali R. ad Awasthi U. Effect of magetic field o the resistace to blood flow through steotic artery. Alied Mathematics ad Comtatio. 7; 88: [3]. Ali R., Kaur R., Katiyar V. K. ad Sigh M. P. Mathematical modelig of blood flow through vertebral artery with steosis. Idia Joural of Biomechaics : Secial Issue. 9; (NCBM 7-8. [4]. Rathod V. P. ad Taveer S. Pulsatile flow of coule stress fluid through a orous medium with eriodic body acceleratio ad magetic field. Bulleti of Malaysia Mathematical Scieces Society. 9; 3: [5]. Varshey G., Katiyar V. K. ad Kumar S. Effect of magetic field o the blood flow i artery havig multile steosis. A umerical study. Iteratioal Joural of Egieerig, Sciece ad Techology. ; : [6]. Shit G. C. ad Roy M. Hydro-magetic ulsatory flow of blood i a costricted orous chael: A theoretical study. Proceedigs i World Cogress Eg. ; : [7]. Mathur P. ad Jai S. Mathematical modelig of o-newtoia blood flow through artery i the resece of steosis. Iteratioal Joural of Advaces i Alied Mathematics ad Bioscieces. 3; 4: -. [8]. Rathee R. ad Sigh J. Aalysis of two- layered model of blood flow through comosite steosed artery i orous medium uder the effect of magetic field. Joural of Rajastha Academy of Physical Scieces. 3; : [9]. Kumar H., Chadel R. S., Kumar S. ad Kumar S. A o Newtoia arterial blood flow model through multile steosis. Iteratioal Joural Latest Research i Sciece ad Techology. 4; 3: 6-. [3]. Gaur M. ad Guta M. K. Usteady sli flow of blood through costricted artery. Advaces i Alied Scieces ad Research.5; 6: [3]. Siddiqui S. U. ad Geeta. Aalysis of usteady blood flow through steosed artery with sli effects. Iteratioal Joural of Bioscieces ad Biotechology. 6; 8(5: [3]. Malek A. ad Horque A. Hematocrit level o blood flow through a steosed artery with ermeable wall. Iteratioal Joural of Alied Mathematics. 7; (: Page