Analytical Solution to Diffusion-Advection Equation in Spherical Coordinate Based on the Fundamental Bloch NMR Flow Equations

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1 Intenatinal Junal f heetical and athematical Phsics 5, 5(5: 4-44 OI:.593/j.ijtmp Analtical Slutin t iffusin-advectin Equatin in Spheical Cdinate Based n the Fundamental Blch N Flw Equatins anladi Eli,*, Guk P.. epatment f Phsics, Nigeian efence Academ, Kaduna, Nigeia epatment f Phsics, Kaduna State Univesit, Kaduna, Nigeia Abstact A geneal slutin f tansvese magnetizatin, the nuclea magnetic esnance (N signals f diffusin-advectin equatin with spatiall vaing velcit and diffusin cefficients, which is based n the fundamental Blch N flw equatins, was btained using the methd f sepaatin f vaiable. It assumed that the velcit cmpnent is pptinal t the cdinate and that the diffusin cefficient is pptinal t the squae f the cespnding cmpnent. hee is a simple tansfmatin which educes the spatiall vaiable equatin t a cnstant cefficient. Afte sme assumptins, the 3- equatin degeneates t a - pblem. he slutin t this equatin is useful in descibing phsical phenmenn such as tanspt f mateials in a fluid. he esult btained in this stud can have applicatins in functinal magnetic esnance imaging (fi with me accuate infmatin. Kewds N diffusin advectin equatin, iffusin cefficient, Sepaatin f vaiable, agnetic esnance imaging (I. Intductin investigate the diffusin pcess f magnetizatin in a fluid mving at a unifm velcit, v, which is cnstant in time, we have t take the pcess f advectin int cnsideatin. he equatin which descibes such a pcess is knwn as the advectin equatin (Awjegbe et.,al,. he advectin equatin is the patial diffeential equatin that gvens the in f a cnseved scala as it is advected b a knwn velcit field. It is deived using the scala s cnsevatin law, tgethe with Gauss s theem, and taking the infinitesimal limit. he diffusin advectin equatin (a diffeential equatin descibing the pcess f diffusin and advectin is btained b adding the advectin peat t the main diffusin equatin. In the spheical cdinates, the advectin peat is v = v + v + v sin Whee the velcit vect v has cmpnents v, v, and v in the,, and diectins, espectivel. * Cespnding auth: danladielibak@gmail.cm (anladi Eli Published nline at Cpight 5 Scientific & Academic Publishing. All ights eseved. he N iffusin Advectin Equatin In accdance with Awjgbe et.,al, he N diffusin advectin equatin with vaiable cefficient culd be btained fm F ( v + = ( + i (, t β ( Whee is the el peat in the spheical cdinate sstem. We have t e-wite the advectin peat (the fist tem n the left hand side f Eq. ( because this fluid velcit is nw spatiall dependent. Geneall speaking, the advectin tem f the tansvese magnetizatin is ( v. Afte expansin we btained v = v + v ( ( When the fluid velcit is cnstant, v = and then ( v = v this is simila t a case f incmpessible fluid in fluid dnamics. Since pefusing substances be the advectin equatin, the apppiate equatin t accuatel descibe a flw pcess in a spheical gemet based n equatin (.

2 Intenatinal Junal f heetical and athematical Phsics 5, 5(5: Awjbe el., al is ( ( ( v v v sin = sin + sin ( FO + + β ( t, sin O If the paametes,, epesent the diffusin cefficients, then the equatin abve is the equatin f diffusin f magnetizatin as the nuclea spins mve. he F functin β ( t, is the fcing functin, which shws that the applicatin f adi fequenc (F, β field has an influence n the diffusin f magnetizatin. It is inteesting t nte that the dimensin f equatin abve exactl matches that f diffusin cefficient. In this pape, we ae inteested in getting the geneal slutin f the equatin using the methd f sepaatin f vaiables. If we define v = u v = w v = v and = u 3 = w = v (dada et.,al Whee u, v, w, O ae cnstants (. ada. et.,al In each diectin the velcit cmpnent has a linea dependence n the cdinates and the cespnding diffusin cefficient has a quadatic dependence n the cdinates. And if we assume = (E. anladi et.,al 5 then the equatin abve educes t ( ( v v + + = ( FO + sin + β ( t, sin O iffeentiating equatin ( we have v v v + v = + + (3 F + + β ( xt, he tansfmatin f equatin (3 invlves intducing a new vaiable b making sme assumptins (Spawls, u = e (3a and w = e (3b Fm (a we have = u Als, fm (b = w v = w = u (3c = w = w = 3u Equatin (3 can als be eaange as v v + v + + v + = + (4 F + β ( t, 3 Substitute (c and v = uv, = wz, = u, = w int (4 we have w u + + ( 3u + w (5 + = + + ( t, F β

3 4 anladi Eli et al.: Analtical Slutin t iffusin-advectin Equatin in Spheical Cdinate Based n the Fundamental Blch N Flw Equatins If we then make the assumptin that F w β( t, = u + Awjgbe et.,al, hen equatin (5 becmes + + ( 3u w = + equatin (6 can als be witten as = + w ( 3u It is alwas pssible t find slutins f the equatin abve, b sepaating int time and psitin cdinates. he methd f the sepaatin f vaiables elies upn the assumptin that a functin f the fm = F(, G( t Whee is a functin f, and t, and F is a functin f, and G is a functin f t alne, will be the slutin t the patial diffeential equatin abve. Equatin (7 can nw be witten as: ( = + w ( 3u ivide thugh b = + w ( 3u Equate bth sides with a cnstant λ t have = λ (7 w + ( 3u (8 = λ Fm (7, Let (6 (7 + λ = (9 Putting ( int (9 heefe Als fm (8 Ae Ae = Ame = ( m λ Ae m + λ = m = λ + = ( Ae λ t = ( ( 3u + λ ( w = + Equatin ( can als be sepaated b equating bth sides with a cnstant µ and b e-aangement we btain ( ( λ µ 3 u + = (3 w + µ = Equatin (4 can als be witten as w + µ = Fm (3 Substitute (6 int (3 m = Ae = Am e m = Ame m ( ( λ µ m Ae m 3u m + = m Ae m ( u m ( λ µ 3 + = m = heefe, (4 (5 (6 ( 3u ( ( ± 3u 4 λ µ

4 Intenatinal Junal f heetical and athematical Phsics 5, 5(5: = Ae + Ae Fm (7 let ( 3u + ( 3u 4 ( λ µ ( 3u ( 3u 4 ( λ µ ( 3 ( 4 u λ µ = α And making A = A we have: ( 3 = e A e + e ( 3 u α α u csh α = e A Fm (5, let Put (8 int (5 = m Bme m = m = Bme m w m m + µ = mz m w m+ µ = w w 4 µ ± m = heefe, w w 4 + µ = + Fm (9, let w w 4 µ w 4 = µ w = e + If we let B = B we have: (7 (8 (9 ( Ow O O O = e + e hen the tansvese magnetizatin becmes ( 3u ( w α = 4e A csh csh Z Ae λ t Z ( he slutin abve is the N tansvese magnetizatin and signal in spheical gemet. his N signal is a functin f diffusin cefficient O. he slutin can be a tl t accuatel undestand the cmbined effect f diffusin and pefusin pcesses in human phsilgical and pathlgical flw sstems. hee seems t be evidence t suggest that in sme tanspt pcesses the velcit and diffusin cefficients ae nt cnstants but functins f time and space (Zppu and, Knight Cnclusins We have btained basic expessin f the tansvese magnetizatin (the N signals in spheical gemet based n the blch N flw equatins. his geneal slutin is quite inteesting and pmising in the cntext f sme ecent eseach wks n dnamical flw. he applicatin f this fundamental slutin t slve eal life flw pblems in which N-sensitive mateials ae tanspted will be pesented sepaatel. ACKNOWLEGEENS he auths ae gateful t bth. ada O. and. Jia hammed f thei cnstuctive cmments. EFEENCES [] Awjgbe OB, Famika OP, Flunsh ada. O, Fuwape IA, Bubake K (. athematical mdel f the Blch N flw equatins f the analsis f fluid flw in esticted gemeties using the Bubake plnmials expansin scheme, Cu. Appl. Phs. : [] Awjgbe, OB, ada O, Famika OP, ada OE (. athematical Cncept f the Blch Flw Equatins f Geneal agnetic esnance Imaging: A eview pg 85-. [3] ada O, Awjgbe, Bubake K and Ojambati OS (. BPES analses f a new diffusin-advectin equatin f fluid flw in bld vessels unde diffeent bi-phsic-gemetical cnditins, Junal f Biphsics and Stuctual Bilg Vl. (3, pp

5 44 anladi Eli et al.: Analtical Slutin t iffusin-advectin Equatin in Spheical Cdinate Based n the Fundamental Blch N Flw Equatins [4] anladi E, Hcienth O (5. Analtical slutin t iffusin-advectin Equatin with Vaiable Velcit and iffusin Cefficient in Clindical Cdinate. Submitted f publicatin in Nigeian Junal f Phsics. [6] Zppu C, Knight JH (999. Analtical slutin f spatiall vaiable cefficient advectin-diffusin equatin in upt thee dimensins. Applied mathematical mdelling. Vl 3(9: [5] Spawls P (. agnetic esnance Imaging: Pinciple, ethds, and echniques. edical Phsics Publishing: adisn, Wiscnsin pp