FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE

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1 FLUID DYNAMICS FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE S. A. ODEJIDE 1, Y. A. S. AREGBESOLA 1, O. D. MAKINDE 1 Obafemi Awolowo Uniersity, Department of Mathematics, Faculty of Science, Ile-Ife, Nigeria Applied Mathematics Department, Priate Bag X1106, Soenga 077, South Africa Receied February 13, 007 In this paper, we examined an incompressible iscous fluid flow and heat transfer in a collapsible tube. The non linear equation arising from the model were soled using perturbation series. An increasing or decreasing in the Prandtl number leads to increasing or decreasing in the rate of heat transfer across the wall. AMS Subject Classification: 78M0, 74G15, 76E5. Key words: incompressible iscous fluid, collapsible tube, perturbation series, heat transfer. 1. INTRODUCTION Fluid flow through collapsible tubes is a complex problem due to the interaction between the tube-wall and the flowing fluid, Heil (1997). Collapsible tubes are easily deformed by negatie transmural pressure owing to marked reduction of rigidity. Hence, they show a considerable nonlinearity and reeal arious complicated phenomena. They are usually used to simulate biological flows such as blood flow in artries or eins, flow of urine in urethras and air flow in the bronchial airways. They can also be used to study and prediction of many diseases, as the lung disease (asthma and emphysema), or the cardioascular diseases (heart stroke), Makinde (005). In this work, laminar flow of an incompressible iscous fluid through a collapsible tube of circular cross section is considered. Our objecties are to study the effect of temperature along the tube as the fluid Prandtl number and Reynolds number increases. Also to examine the rate of heat transfer across the wall. In Section, we establish the mathematical formulation of the problem. Graphical interpretation and discussion of the pertinent results were presented in Section 3.. MATHEMATICAL FORMULATION Let consider the transient flow of a iscous incompressible fluid in a collapsible tube. We take a cylindrical polar coordinate system (r,, z) where oz Rom. Journ. Phys., Vol. 53, Nos. 3 4, P , Bucharest, 008

2 500 S. A. Odejide, Y. A. S. Aregbesola, O. K. Makinde lies along the center of the tube, r is the distance measured radially and is the azimuthal angle. Let u and be the elocity components in the directions of z and r increasing respectiely. Assume r a0 (1 t), where is a constant of dimension [T 1 ] which characterizes unsteadiness in the flow field, a 0 is the characteristic radius of the tube at time t = 0 as shown in the figure below. Fig. 1 Geometry of the problem. For axisymmetric unsteady iscous incompressible flow, the goerning equations are as follows: Continuity equation ( r) r u 0 r z Naier-Stokes equations u 1 p u u u u t z r z 1 p u ( u ) t z r r r Energy equation c T p u T T K T t z r (1) () (3) (4) where 1, p is the pressure, the density, the kinematic r r r z iscosity of the fluid, T is the temperature, K the coefficient of thermal

3 3 Fluid flow and heat transfer in a collapsible tube 501 conductiity, the coefficient of iscosity and c p is the specific heat capacity at constant pressure. The appropriate boundary conditions are: Reqularity of solution along z- axis i.e. u 0 0 T 0 on r 0 (5) r r The axial and normal elocities at the wall are prescribed as: Tz 0 u0 a T on r a( t) (6) t a0 1t Introducing the stream function and orticity as follows: u 1 and 1 (7) r r r z u z r r z r r r r Eliminating pressure p from () and (3) by using (7) and (8), we obtain 1 t r rz r z r Also, using (7) in (4), we hae T T 1 K T t r rz cp The boundary conditions become Tz 0 0 a da T r a( t) r z dt a 1t T 0 r 0 r r r z r Introducing the following transformations: r 0 zg( ) 0 T 3 0 a0 1t 0 a zf( ) T z( ) a 1 t a 1t Substituting (13) into equations (9) (1), we hae d 1 d( G) RG df F d G dg 3G G d 1dF d d d d d d d 0 (8) (9) (10) (11) (1) (13) (14)

4 50 S. A. Odejide, Y. A. S. Aregbesola, O. K. Makinde 4 d d PR df F d d d d d d d d 1 d( G) 0 F 0 d 0 on 0 d d d df 0 F 1 1 on 1 d (15) (16) (17) a 0 ca p 0 where R is the Reynolds number and PR is the product of K Prandtl number and Reynolds number (i.e. Peclet number). We now sole equations (14) (17) by assuming a power series expansion, for small Reynolds number, of the form i i and i i i i i0 i0 i0 (18) F FR G G R R Substituting equation (18) into equations (14) (17) and collecting the coefficients of like powers of R, we hae the following: zeroth order and higher order equations as in appendix A1. We hae written a MAPLE program that calculates successiely the coefficients of the solution series. In outline, it consists of the following segments: 1. Declaration of arrays for the solution series coefficients e.g. F = array(0 10), G array(0 10), array(010).. Input the leading order term and their deriaties i.e. F 0, G 0, Using MAPLE loop procedure to sole equations (3) (6) (in Appendix A1) for the higher order terms i.e. F n, G n and n, n = 1,, 3,. Details of the MAPLE program can be found in Appendix A. 3. GRAPHICAL RESULTS AND CONCLUSION In our numerical calculation, we hae used different Prandtl numbers such as air (P = 0.71), water (P = 7.1) etc. Fig. and Fig. 3 show that the fluid temperature increases with an increase in fluid Prandtl number with maximum alue at the center. In Fig. 4, we fixed the Prandtl number and arying the Reynolds number. An increase in Reynolds number leads to an increase in the fluid temperature with maximum magnitude at the pipe center and minimum at the wall. Fig. 5 and Fig. 6 show the rate of heat transfer across the wall. For R < 0 (wall expansion), as the Prandtl number increases the rate of heat transfer

5 5 Fluid flow and heat transfer in a collapsible tube 503 increases. Also, for R > 0 (wall contraction), the rate of heat transfer decreases with increases in the Prandtl number. Fig. 7 shows the fluid elocity profile which is parabolic in nature and increases as Reynolds number increases. From our analysis, it is important to note that wall expansion is represented by negatie alues of flow Reynolds number (R < 0) while wall contraction is Fig. s. for R = 1. Fig. 3 s. for R = 1. Fig. 4 s. for P = 7.1. Fig. 5 Nu s R. Fig. 6 Nu s R. Fig. 7 u s.

6 504 S. A. Odejide, Y. A. S. Aregbesola, O. K. Makinde 6 represented by positie alues of flow Reynolds number (R > 0). It is interesting to note that the fluid temperature increases transersely with maximum alue along the channel centerline for fluid with Prandtl number greater than or equal to one whereas the opposite is obsered i.e. a transerse decrease in temperature, for fluid with Prandtl number less than one (Figs. 3). Furthermore, it is obsered that the fluid temperature increases with increasing positie alues of flow Reynolds number (R > 0) as illustrated in Fig. (4). In Figs. (5-6), the wall heat flux is illustrated for both positie and negatie alues of flow Reynolds number. It is noteworthy that for R > 0, the rate of heat transfer across the wall decreases with increasing alues of Prandtl number (P) whereas for R < 0, the wall heat flux increases with increasing alues of Prandtl number. Finally, we obsered that the fluid elocity increases with increasing positie alues of flow Reynolds number. APPENDIX A1: Zeroth Order d 1 d( G0) 1 d( F0) 0 G d 0 0 d d d d d d( 0 ) 0 d d d 1 df ( 0) d 0 0 F0 0 0 on 0 d d d df0 d 0 F 1 1 on (19) (0) (1) () Higher Order (n 1): n1 n i n i 1 d n i 1 n1 Fi i0 d 1 d( G ) G df G dg 3Gn1 d d d d d G n d 1 df ( n ) d d (3) n1 n n i 1 n i 1 n1 P i Fi n1 i 0 d d df d d d d d d (4) d d 1 df ( n) d n 0 Fn 0 0 on 0 d d d (5)

7 7 Fluid flow and heat transfer in a collapsible tube 505 dfn d 0 F 0 0 on 1 n n (6) A: The MAPLE procedure to sole the equations (3) (6) # Declare the arrays to store the computed results F array(0 10), G array(0 10), array(0 10), F array(010), Garray(0 10), array(010). # Input the zero order solution F[0], G[0] and [0]. F[0] ( 4) G[0] 8 [0] 1 F[0] diff( F[0] ) G[0] diff( G[0] ) [0] diff( [0] ) # Compute the higher order terms, n > 0, for n from 1 by 1 to 10 do A1 normal(1sum( G[ i] F[ n i1] F[ i] ( G[ n i1] G[ ni1]) i0n1)) A normal( G[ n1] 3 G[ n1]) A R( A1 A) A1 0 A 0 g11 normal( (int( A ) K)) A 0 g1 normal(int( g11 ) ) g11 0 f11 normal( (int( g1 ) M)) f1 normal(int( f11 )) B normal( P Rsum(( [ i] F[ n i1] Fi [ ] [ n i1]) [ n1] [ n1] i 0n 1)) t11 normal((int( B ) )) B 0 t1 normal(int( t11 ) L) 1 K normal(sole( f11 0 K)) M normal(sole( f1 0 M)) L normal(sole( t1 0 L)) f11 0 F[ n] normal( f1) f1 0 Gn [ ] normal( g1) [ n] normal( t1) g1 0 F[ n] normal(diff( Fn [ ] )) G[ n] normal(diff( Gn [ ] )) [ n] normal(diff( [ n] )) KKMMLLprint( F[ n]) print( Gn [ ]) print( [ n]) od quit() Some of the stream-function and orticity are then gien as follows: 4 R R 6 4 F( ) ( ) ( 1) ( 10) ( 1) ( R ) ( 1) ( ) OR 4 R R G( ) 8 ( 1 7) ( (7)

8 506 S. A. Odejide, Y. A. S. Aregbesola, O. K. Makinde 8 3 R 71) ( ) OR ( 4) The temperature is gien by PR ( ) 1 PR ( 1)( 4) ( 1) ( 1) ( 13) PR 6 4 P( ) ( 1)[4( 1) ( ) P( )) 9 P ( )] O( R4 ) The wall heat transfer rate, Nu d at 1 is gien by d Nu 3 PR PR R 163 R R R R P R R R R R P4R R R R P R R R P R R. (8) (9) (30) REFERENCES 1. M. Heil, Stokes Flow in Collapsible Tubes-Computational and Experiment, Journal of Fluid Mechanics, 353, (1997).. O. D. Makinde, Extending the Utility of Perturbation Series in Problems of Laminar Flow in a Porous Pipe and Dierging Channel, Jour. of Austral. Math. Soc. Ser. B 41, (1999). 3. O. D. Makinde, Heat and Mass Transfer in a Pipe with Moing Surface: Effect of Viscosity Variation and Energy Dissipation, Quaestiones Mathematicae. Vol. 4, (001). 4. O. D. Makinde, Collapsible Tube Flow: A Mathematical Model, Rom. Journ. Phys., Vol. 50, Nos. 5 6, (005). 5. O. D. Makinde, Y. A. S. Aregbesola, S. A. Odejide, Wall Drien Steady Flow and Heat Transfer in a Porous Tube, Kragujeac J. Math., (006).

Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe

Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe T S L Radhika**, M B Srinivas, T Raja Rani*, A. Karthik BITS Pilani- Hyderabad campus, Hyderabad, Telangana, India. *MTC, Muscat,