MATHEMATICAL MODEL FOR THE BLOOD FLOW IN CAPILLARY VESSELS

Transcription

1 MATHEMATICAL MODEL FO THE BLOOD FLOW IN CAPILLAY VESSELS. Sonia PETILA. Bala ALBET. Tibeiu Pooviciu Infomatic Highchool Cluj-Naoca OMANIA. Univeity Babe-Bolyai Cluj-Naoca OMANIA Abtact: In thi ae we develo an oiginal model fo the blood flow in caillaie. In the fit aoach the Stoe ytem i acceted fo the blood flow in caillaie and the fluid i conideed to be incomeible. The veel wall have a linea elatic and emeable behavio. Fo the econd model a non-newtonian heological model fo the blood flow with a non-contant vicoity coefficient i ued the wall of the caillaie being linealy elatic emeable and oou. Key wod: blood flow in caillay veel Newtonian-model heological model elatic emeable oou wall.. INTODUCTION The mot imotant aect of the blood flow in caillaie i to uly with food the living cell of the ogan and to emove the byoduct fom evey cell. The caillay veel ae built o that molecule with diffeent dimenion can enetate though the tiue in the uounding of the caillaie in both way. Geneally caillaie ae conideed a tube with vey thin and oou wall though which the tanot of cetain ubtance ae ealied. The eence of thee oe and the mall diamete of the caillaie ditinguih thee tye of veel fom the othe. Due to thi educed diamete and the low chaacte of the flow we can neglect the non-tationay (ulating aect connected to the hythmical uming of the blood by the heat. Futhemoe we can neglect the inetial (convective aect connected to the vicoity of the blood. Moeove the emeable (oou chaacte of the caillaie i dominating the elaticity of the veel wall.. NEWTONIAN MODEL FO THE BLOOD FLOW IN CAPILLAIES The fit model we ooe accet fo the blood flow in thin veel the Stoe ytem fo incomeible fluid taing into conideation that the eynold numbe i mall. Imlicitly it i acceted that the blood i homogenou the vicoity i contant the flow ha a lamina chaacte and thee ae no exteio field foce. The veel wall have a linea elatic and emeable behavio and the fluid (ubtance change though thee wall vey mall in volume eect the Staling hyothei [7]. Thi claical hyothei which wa checed exeimentally late by many eeache (Mauo [3] Mechia [4] etc. maintain the fact that the ma debit though thi ind of caillay wall i ootional with the eue diffeence between the exteio and the inteio of the caillay tube. Moeove uing the eult of Beave and Joeh [] it i acceted the exitence of a li condition along the emeable uface which i coveed by a oou media an eential condition confimed alo exeimentally. Fo imlification we accet the axi-ymmetic chaacte of the flow the axi of ymmety being O. Uing the cylindical coodinate ( θ the motion domain will be at evey time t: Ω( {( θ / < + ( θ [π ( L} ( whee and L ae the (initial adiu and the length of the tube eectively i the elatic dilacement of the wall Σ( { = + ( ( L} at the conideed moment. Noting by ( u v the velocity comonent of the blood in the diection and eectively by the eue while by μ the dynamic vicoity coefficient the motion equation (Stoe and the continuity equation become 35 Tome VII (yea 9 Facicule 3 (ISSN

2 ANNALS OF THE FACULTY OF ENGINEEING HUNEDOAA JOUNAL OF ENGINEEING. TOME VII (yea 9. Facicule 3 (ISSN and ( v v v v = μ + + ( ( u u = μ + + (3 ( v + = (4 Fo the bounday condition noting by the aveage of the eue value given in the eective ection we get: Hee = = β u = and v = fo = β u and v ν fo = + ( = fo = a v (5 (6 (7 ( = fo = L. (8 i the Beave-Joeh li condition whee β i the li aamete while i the ecific emeability of the oou media v ν i the conequence of the Staling law whee K i the contant emeability of the wall while ν (built by the intetitial and omotic eue i a given contant. Concening to and they ae the ateial and venou eue a both uoed contant. effeing to the elaticity of the caillay wall acceting the linea elatic membane model the adial comonent of the te can be exeed by the adial dilacement uch that: T he ρ mh + + σ v ef = (9 whee h i the thicne of the membane E the Young moduluσ the Poion coefficient ρ m i the denity of the caillay wall while ef i the efeence eue in the unetubed tate uoed to be contant (the above mentioned i in fact ef. It i obviou that on a thi ind of elatic wall the inematic condition fo the continuity of the eue evaluated on the defomed inteface Σ ( mut be atified namely ( = v( + ( and u( + ( =. ( Thee condition togethe with the eviou Beave-Joeh and Staling condition lead to γ and = fo = + eectively. Concening the dynamic condition it imlie the continuity of the te along the defomable inteface (wall. A the contitutive law acceted in thi cae i that of the Newtonian fluid we mut have along Σ ( [( ef [ T] μ [ D]] n e = T ( which lead to [( ef [ T ] μ[ D]] n e ( + + ( = T ( on Σ ( at any time t. coyight FACULTY of ENGINEEING - HUNEDOAA OMANIA 353

3 ANNALS OF THE FACULTY OF ENGINEEING HUNEDOAA JOUNAL OF ENGINEEING. TOME VII (yea 9. Facicule 3 (ISSN HEOLOGICAL NON-NEWTONIAN MODEL In the eviou model the blood wa invetigated a a Newtonian fluid and the ytem of equation wa the Stoe ytem. Now we accet fo the blood a heological non-newtonian eeentation with a non-contant vicoity coefficient. All the othe aumtion (non-tationay chaacte incomeibility homogeneity linea elaticity ooity of the wall ae the ame lie in the eviou model. The Staling hyothei and the Beave-Joeh li condition ae alo fulfilled. We accet again the axi-ymmetic chaacte of the blood flow in the caillay tube the axi of ymmety being O. Uing the cylindical coodinate ( θ the motion domain will be at evey time t Ω( {( θ / < + ( θ [π ( L} whee L ( and Σ( have the ame meaning a in the eviou model. In the meidian lane θ = cont if u and u ae the comonent of the velocity in and diection if i the eue (evaluated to a efeence eue ef then in the abence of the exteio foce the ma conevation incile (continuity equation can be witten a ( u + =. (3 Concening the flow equation they ae obtained fom the geneal Cauchy motion equation whee fo the te teno we accet the following eeentation (heological model fo blood α T = [ + λ ( && γ + K ] I + ( + BC D (4 & γ whee D i the ate of tain teno while I the unity teno the hyical eue while BC i given by (the Co model: = + λk( & + ( & γ γ BC n. (5 / with γ& = 4I I being the econd invaiant of the ate of tain teno D the lama vicoity and hea thining behavio n the hea thining index α the mobility aamete while the function K( & γ = ( n fo λ > λ + ( & γ (6 i the o called nomal function in the vaiable γ& which meaue the vaiation of defomation. Fo ae of imlicity we denote ( & & + + the vicoity coefficient of the blood α the elaxation time i a time contant fo the γ λ γ + αλ K L = λ && γ and & γ & γ n ( n( & γ M = o that exeing the teno D and the othe oeato ( =... etc. in n [ + ( & γ ] x i cylindical coodinate we aive to the following two equation of flow (in u and u u θ = ρ ( + u + u = ( ( u u + & γ & γ && γ & γ & γ + L + M[ + + ( + ] ρ ( + u + u = ( ( u u + & γ & γ && γ & γ & γ + L + M[ + + ( + ]. BC (7 (8 354 coyight FACULTY of ENGINEEING - HUNEDOAA OMANIA

4 ANNALS OF THE FACULTY OF ENGINEEING HUNEDOAA JOUNAL OF ENGINEEING. TOME VII (yea 9. Facicule 3 (ISSN Thee evolution ytem ae comleted by the bounday condition which exe both the eence of a eue gadient along the O axi (in accod with the hythmical uming of the blood in veel and the elatic chaacte of the emeable oou wall moe eciely = and u = fo = (9 β ( = u and u ν fo = + ( K (the fit elation in ( exee the Beave-Joeh li condition with the li aamete β while K i the ecific emeability of the oou media meantime u ν i the conequence of the Staling law with K the contant emeability of the wallν built by the intetitial and omotic eue uoed to be fixed and a a given contan while fo the eue we have co(ω = + m fo = whee a > ( a co(ω = + m fo = L whee a > ( a + L f ( d whee m = f ( ξ f i a imitivable and deivable function accoding to with a maximum fo = and a minimum fo = +. It can be emaed that > at any time = = L of the motion ( T. Obevation: Thee bounday condition on the edge = and = L of the caillay ae in co( ω accod with the accetance of a eeentation fo the eue of the tye = + f ( a + namely of a eue gadient (in the cylindical efeence e e unde the fom co( ω gad = e + f ( e. If f ( = and f ( + ( = we have ( a + co( ω gad = = gad O ( t ( a in accod with the motion of the eue in the inteio of = + + the caillay. On the othe hand acceting fo the caillay wall the linea elatic membane model the adial comonent of the membane te i exeed by the adial dilacement a follow he T = ρ mh + + (3 ef σ whee h i the thicne of the membane E the Young moduluσ the Poion coefficient ρ m i the denity of the caillay wall while ef i the efeence eue in the unetubed tate. It i evident that thi te mut coincide with the te geneated by the blood on the ame adial diection namely T = T n e = T which eeent the elation fo detemining f (the eue o (. At the ame time the inetic condition mut be atified on the elatic wall = u ( + ( but alo u ( + ( = what lead to ( ν and = fo = + eectively. It can be emaed that the lat elation togethe with u ( + ( = imlie u = in the whole a uounding of the elatic wall while the condition = and u = fo the axi = how that u u e = deend only on and t o that we have a ulating flow along the axi O coyight FACULTY of ENGINEEING - HUNEDOAA OMANIA 355

5 ANNALS OF THE FACULTY OF ENGINEEING HUNEDOAA JOUNAL OF ENGINEEING. TOME VII (yea 9. Facicule 3 (ISSN which calm down on the elatic wall ( u = whee the exteio imoed eue will have a minimum. At the ame time fom ( ν = + ( we obtain -ωin( ω. Thi lat evaluation fo emit u to mae ecie the a + = + ( t condition on the caillay wall (linea elatic membane namely the exeion of the equilibium condition T n e = T in cylindical coodinate. Moe eciely if we note by α P = [ + λ ( & γ + K ] (4 & γ the equilibium condition become P ( & γ + u u ( + = + ( + ( + ( (5 mh Kωin( ω he ρ ( + + a + σ ef what ovide an equation to detemine the defomation of the caillay wall namely ( o that the whole et of unnown of ou oblem can be detemined. 4. CONCLUSIONS In thi ae we elaboated an oiginal mathematical model fo the blood flow in caillay veel. Fit we eented a model whee the blood wa acceted a a Newtonian fluid. In the econd aoach we extended the model to a moe geneal heological (non-newtonian blood behavio which tand cloe to the ealitic henomena. The eviou model will be aoached numeically in anothe ae whee we will alo conide a moe geneal behavio fo the blood. EFEENCES [] Beave G.S. Joeh D.D.: Bounday condition at a natually emeable wall J. Fluid Mech [] Landi E.M. Paenheime J..: Handboo of Phyiology ed. W.F. Hamilgton and P. Dow (Am. Phyiol. Soc. Wahington [3] Mauo A.: Natue of olvent tanfe in Omoi. Science [4] Mechia G. Setnia I.: Exeimental tudy of omoi though a collodion membane J. Gen. Phyiol [5] Oa S. Muata T.: A theoetical tudy of the flow of blood in a caillay with emeable wall Ja. J. Al Phy [6] Petila T. Tif D.: Baic of Fluid Mechanic and Intoduction to Comutational Fluid Dynamic Singe Science New Yo 5. [7] Staling E.M.: On the abotion of fluid fom the convective tiue ace J. Phyiol coyight FACULTY of ENGINEEING - HUNEDOAA OMANIA