Two Dimentional Flow in Renal Tubules with Linear Model

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1 Adances in Applied Mathematical Biosciences. ISSN Volume, Numbe (), pp Intenational eseach Publication House Two Dimentional Flow in enal Tubules with Linea Model L.N. Achala and K.. Sheenias Post Gaduate Depatment of Mathematics and Cente in Applied Mathematics, MES College, 5 th Coss, Malleswaam, Bangaloe-, Kanataka, India. anagund@hotmail.com, sheenias_kisu@yahoo.co.in Abstact Two dimensional flow in enal tubules is studied by consideing adial component of elocity as a linea function of by steam function appoach. We hae shown that the olume ate of flow is a quadatic function of. We also calculated the tube adius equied fo minimum adial elocity. We hae also calculated the ange fo pessue dop fo unifom eabsoption ate. Keywods: enal flow; Tubules; Nephon; adial elocity; eabsoption; Axial elocity; Steam function. AMS Subject Classification: 76Zxx Intoduction The functional unit of the kidney is called the Nephon o enal tubule, each kidney as about million of these tubules. One majo pat of a nephon is the glomeula tuft tough which blood coming fom the enal atey and affeent ateioles is filteed. The glomelecula filtate is essential identical to plasma and no chemical sepaation occus up to this point. If the kidney delies this filtate fo excetion, the body loses many aluable mateials, including wate, at a ate faste than the one at which they can be supplied by feeding. Thus 8 pecent of filtate is eabsobed in the poximal tubule, and of the emaining about 95 pecent is futhe eabsobed by the end of the collection of the collecting duct. This eabsob ion ceates a adial component of the elocity in the cylindical tubule, which must be consideed along with the axial component of the elocity. Due to loss of fluid fom the walls, both the adial and axial elocities deceases with. Mathematically, we hae to sole poblem of flow of a iscous fluid in a cicula cylinde when thee ae axial and adial components of elocity and the adial

2 8 L.N. Achala and K.. Sheenias elocity at all points on the suface of the cylinde is pescibed and is a deceasing function φ () of. Figue : Two-dimensional flow in enal tubule. Duing last decades an extensie eseach wok has been done on the fluid dynamics of biological fluids in pesence and absence of magnetic fluid due to bioengineeing and medical applications [-]. Mathematical models of the enal tubule ae classified as epithelial, tubula o multitubula. The tubula models allow pediction of the effect of epithelial tanspot to modify the luminal solution and conesely the effect of alteed luminal composition on tansepithelial fluxes. Hee the conseation equation constitute asset of ODE with initial data that must be integated along tubule length. The mulitubula models ae mathematically the most complex and hae chaacte of BVP, whee tanspot along an entie tubule contibutes to the intestitial composition. Fank [] hae analyed the model in which a foce of unspecified oigin dies the fluid fom descending thin limb (DTL) to intestitial ascula space, thus concentating the solution in DTL Layton [5] hae soled hypebolic patial diffeential equations of enal model explicitly fo both dynamic and steady state. Layton. et. al. [6] hae done Numeical Simulation of popagating concentation pofiles in enal tubules. Layton. et. al. [7] gae a dynamic numeical method fo models of the uine concentating mechanism Pitman [8] hae studied mass conseation in a dynamic numeical method fo a model of the uine concentating mechanism. Sands. et. al. [9] explained in detail uine concentating mechanism and its egulation in thei book. Macano. et. al [] gae an inese algoithm fo a mathematical model of an aian uine concentating mechanism. Anita et. al. [] hae studied the dynamics of coupled nephons. Thus many eseaches hae studied the mathematical model of enal fluid flows. In this pape we study enal fluid flow by using linea model neglecting the inetial tem and consideing fluid as noniscous in cylindical pola coodinate system.

3 Two Dimentional Flow in enal Tubules with Linea Model 9 Basic Equations and Bounday Conditions At the outset, we may note that the equation of motion can be simplified since the inetial tem in elation to the iscous tems can be neglected. The aeage tubula adius is about, cm the aeage elocity is about sec, cm / and fluid elocity about sec/ 7 cm dynes. This gies a eynolds numbe of about and, since this is ey much less than, we neglect the inetial tems to get the following equations of continuity and motion. ) (, () ) ( ( p μ, () ) ( p μ. () The bounday conditions ae,, finite, at. (), ( ) φ at, (5) p p at p L p. (6) Patially diffeentiating equations () and () with espect to & espectiely and eliminating p we get ( ). (7) Patially diffeentiating equation (7) with espect to and using equation (), we get ( ). (8) Intoducing the steam function we can satisfy the equation () defined by, ψ ψ. (9) Substituting (9) in (7), we get

4 5 L.N. Achala and K.. Sheenias ( D ψ ), D () whee D is defined by ( D ). () f g, () If ()() then the fom of (8) suggests that an analytical solution may be possible. If ( ) A A g Fom equation (5) since φ( ) f ( ) g( ) φ( ) () when, we get. () This suggest that we may get an analytical solution when the adial component of elocity on the suface of the cylinde is gien by ( ) a a φ. (5) adial Velocity at wall Deceases linealy with Fo (), let the solution be (, ) F( )( a a ) G( ) ψ (6) So that using (9), we get F ()( a a ) (7) ' ()( a a ) G () F ' (8) d d d d D ( D ψ ) ( ) F() ( a a G() ) ( ) d d d d (9) d d a ( ) F() d d Fom equation () we get d d ( ) F d d d a( d () ( a a ) ( ) G() d d ) F () d d d d ()

5 Two Dimentional Flow in enal Tubules with Linea Model 5 Fom aboe equation we get ( d d Equation () gies d d ) F d d d ( ) G d d d d d ( d d ) H () ( a a ) () a ( ) F() () d d () () () Whee d d H () ( ) F() () d d Soling () we get d F d df d A B (5) Soling aboe diffeential equation we get F A B (6) 8 () C D ln Fom equation () we get d d d d ( ) ( ) G d d d d () a F() (7) Soling equation (7) we get d d ( ) G() af () M N (8) d d Now fom bounday conditions () and (5) we hae d [ F'( )] at d (9) d [ G' ( )] at d () F( ) at ()

6 5 L.N. Achala and K.. Sheenias F' ( ) and G' ( ) ae finite at () F '( ) G '( ) F ( ) () Fom (6), () and () we get C, B () Fom (6), () and () we get D A, D A, (5) 8 Soling aboe equation we get 8 A, D, (6) Equation (6) becomes [( ) ( ) ] (7) ( ) F Substituting (7) in (8), we get d G d dg d M N a a (8) Integating (8), we obtain 6 M N a a G () N M ln (9) 8 Fom () and (9) N () Fom () and (9) M a M () Equation () can detemine only one of the two unknown constants M and M, to detemine both of these, we need one moe elation. This elation can be found in tems of Q which is the total flux at. Using Q ( ) π (, ) d ()

7 Two Dimentional Flow in enal Tubules with Linea Model 5 5 M a a π ( )( a a ) M d () Q π M 8 a 8 Q a M ( π M ) () (5) Q a (6) π Fom (9), (), (5) and (6) we get a Q Q a a a G π π () ( ) N ( ) The constant ψ, can always contain an abitay constant without affecting the elocity components. Fom (7), (7) and (7) we get N need not be detemined since ( ) 6 (7) (, ) [ ( ) ]( a a) (8) Q a (, ) ( )[ ( a a ) ( )] π (9) Diffeentiating equation () we get dq d 8π ( a a ) ( ) d π ( a ) a (5) Thus the decease of flux is equal to the amount of the fluid coming out of the cylinde pe unit length pe unit time. Integating equation (5) we get Q ( ) Q π (a a ) (5) Fom equation (9) and (5) we get Q( Z) a ( )[ ( )] π Using (), (), (8), (9) and (5), we get p 8 μ ( a a ) (5) (5)

8 5 L.N. Achala and K.. Sheenias p aμ [ Q( ) a π ] (5) Integating equation (5) we get aμ p (, ) ( a a) k( ) (55) whee Diffeentiating (55) patially with espect to and using (5) we get aμ Q( ) k'( ) [ ] (56) aπ Integating aboe equation we get π aμ Q( ) k ( ) [ ] k (57) a Q ( ) Q( ) d The aeage pessue p() at any section is gien by p ( ) p(, )πd π d (58) a 8 Q ( ) a μ[ ( ) ] (59) π Thus the pessue dop oe tube length L is Δ p p() p( L) Using equation (59) we get 8 QL ( ) a μ L (6) π ( ) ( ) p The Maximum and Minimum of adial Velocity We hae a condition that the Maximum and Minimum of an equation is gien by

9 Two Dimentional Flow in enal Tubules with Linea Model 55 If f '( c ) and f ''( c ) < Then f ( x ) has a maximum at xc (6) If f '( c ) and f ''( c ) > Then f ( x ) has a minimum at xc Diffeentiating equation (8) with espect to we get d [ ]( a a) (6) d Fom the condition (6) equation (6) becomes [ ] (6) Fom equation (6) we get ± (6) Diffeentiate again equation (6) we get d d 6 ( a a ) (65) Clealy d d < when Then Maximum adial elocity exists at Then Maximum adial Velocity is gien by ( ) [ ( ) ]( a a ) max ( ) ( a a ) (66) max Then fom equation (65) it is clea that d d > when. Then the Minimum adial elocity exists at

10 56 L.N. Achala and K.. Sheenias ( ) [( ) ( ) ]( a a ) min ( ) ( a a ) (67) min The Maximum and Minimum of Axial Velocity Diffeentiating equation (9) we get d d ( )(a a) (68) Fom the condition (6), equation (68) gies a a (69) a (7) a And diffeentiating equation (68) with espect to we get d d a ( ). (7) We get the fou conditions fo the Maximum and Minimum Case: If a >, and Then d d Case: If a >, and Then d d Case: If a <, and Then d d Case: If a >, and < a <, clealy has Maximum at a > a >, clealy has Minimum at a < a >, clealy has Minimum at a >

11 Two Dimentional Flow in enal Tubules with Linea Model 57 Then d d a <, clealy has Maximum at a The Pessue dop Fo Unifom eabsoption ate oe the Tube Length L The pessue dop oe the tube length L is 8 QL ( ) a Δ p μ( ) L (7) π Fo the unifom eabsoption ate, the pessue dop oe the tube length L is gien by 8 QL ( ) Δ p μ (7) π Whee Q ( ) Q( ) d Fo the unifom eabsoption ate Q ( ) is gien by Q( ) Q a π (7) Then Q( ) is gien by Q( ) Q a π (75) Fom equation (7) we get [ Q Q( )] Q ( ) Q Q ( ) [ Q Q( )] (76) And Q() Q At Fom equation (7) we get 8 μ [ Q() Q( L)] Δ p L (77) π

12 58 L.N. Achala and K.. Sheenias Fom equation (77) it is clea that μq() Δp L (78) π Fom equation (7) at L Q ( ) Q aπ L (79) Fom equation (77) and (79) 8μ L Δ p Q() π a (8) π Clealy fom Equation (8) we get 8μL Δp Q() (8) π Fom equation (78) and (8) we get μq() 8μL L Δp Q() (8) π π The pessue dop fo the unifom eabsoption if μ cm, Q 7 cm /sec, 7 dynesec/ cm, L cm, QL ( ). Q() is gien by 8 μ [ Q() Q( L)] Δ p L π Substituting the gien alues we get Δ. p dyne / cm π Discussion In this pape we study two dimensional flows in enal tubules with linea model. We conside adial component of elocity as a linea function of. We calculate the steam function and axial elocity. Using this we calculate olume ate of flow which is a quadatic function of. We also calculate aeage flux and pessue dop oe the tube length. Using the condition of maximum and minimum of a function, adial

13 Two Dimentional Flow in enal Tubules with Linea Model 59 elocity will be minimum at and axial elocity depending on the constants a and a. Fo unifom eabsoption ate we calculate the pessue dop and flux of the flow, which is gien by the elation (8). efeences [] Haik, Y., Pai, V. and Chen, C. J.: Deelopment of magnetic deice fo cell sepaation, Jl of Magnetism and magnetic Mateials, 9, 5-6 (999). [] uuge, E. K. and usetski, A. N.: Magnetic fluid as dug caies: Tageted tanspot of dugs by magnetic field, Jl. Of magnetism and magnetic mateials,, 5-9 (99) [] Plains, J and Laua, M.: Study of colloidal magnetite binding Eythocytes: Pospects fo cell sepaation, Jl. Of magnetism and magnetic mateials,, 9-5 (99) [] Fank. Jen and John. L. Stephenson.: Extenally dien countecuent multiplication in mathematical model of the uinay concentating mechanism of the enal inne medulla, Bull. Math. Bio., Vol. 56, No., 9-5, (99) [5] H. E. Layton and E. Buce Pitman.: A dynamic numeical method fo models of enal tubules, Bull. Math. Bio., Vol. 56, No., , (99) [6] H. E. Layton, E. Buce Pitman and L. C. Moo.: Numeical Simulation of popagating concentation pofiles in enal tubules, Bull. Math. Bio., Vol. 56, No., , (99) [7] Layton, H. E., E. Buce Pitman, and Mak A. Kneppe.: A dynamic numeical method fo models of the uine concentating mechanism, SIAM Jounal on Applied Mathematics 55(5): 9-8, Octobe, 995. [8] Pitman, E. B., and H. E. Layton.: Mass conseation in a dynamic numeical method fo a model of the uine concentating mechanism, Confeence poceedings of The Thid Intenational Congess on Industial and Applied Mathematics, appeaing in Zeitschift fue Angewandte Mathematik und Mechanik 76(S): 5-8, 996 [9] Sands, Jeff M., and Haold E. Layton, Uine concentating mechanism and its egulation, Chapte 5 in: The Kidney: Physiology and Pathophysiology (thid edition), edited by D. W. Seldin and G. Giebisch. Philadelphia: Lippincott Williams & Wilkins,, p [] Macano-Velaque, M., and Haold E. Layton.: An inese algoithm fo a mathematical model of an aian uine concentating mechanism, Bulletin of Mathematical Biology 65: , [] Anita T. Layton, Leon C. Mooe, Haold E. Layton, Multistable dynamics mediated by tubuloglomeula feedback in a model of coupled nephons, Bulletin of Mathematical Biology. 7():55-555, 9. (Apil, 9)

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