INTERNATIONAL JOURNAL OF ADVANCE RESEARCH, IJOAR.ORG ISSN

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1 ISSN International Journal of Advance Research, IJOAR.org Volume 3, Issue 6, June 015, Online: ISSN PERISTALTIC PUMPING OF COUPLE STRESS FLUID THROUGH NON - ERODIBLE POROUS LINING TUBE WALL WITH THICKNESS OF POROUS MATERIAL USING MAGNETIC FIELD N.G.Sridhar Government First Grade College, Sedam, District Gulbarga, Karnataka, India sridhar.ng1@gmail.com Abstract This paper is devoted to study the effect of thickness of porous material on the peristaltic pumping of couple stress fluid when the tube wall is provided with non- erodible porous lining with magnetic field. Long wavelength and low Reynolds number approximation is used to linearize the governing equations. The expression for axial velocity, pressure gradient and frictional force are obtained by using Beavers-Joseph Boundary conditions. The effect of various parameters on pumping characteristics is discussed with the help of graphs. Keywords: Peristaltic pumping, Magnetic field, volume flow rate, pressure rise, couple stress fluid, pumping Characteristics, Darcy s law. IJOAR 015

2 ISSN INTRODUCTION: Peristaltic pumping is now well known to physiologists to be one of the major mechanisms for fluid transport in many biological systems. In particular, Peristalsis is an important mechanism generated by the propagation of waves along the walls of a channel or tube. It occurs in the gastrointestinal, urinary, reproductive tracts and many other glandular ducts in a living body. Blood is a suspension of cells in plasma. It is a biomagnetic fluid, due to the complex integration of the intercellular protein, cell membrane and the hemoglobin, a form of iron oxide, which is present at a uniquely high concentration in the mature red cells, while its magnetic property is influenced by factors such as the state of oxygenation. A number of studies containing couple stress have been investigated by Alemayehu and Radhakrishnamacharya [1], Elshehawey and El-Sebaei [3], Raghunath Rao and Prasad Rao [11], Sohail Nadeem and Safia Akram [19] and Srivastava [0]. Couple Stress in peristaltic transport of fluids is studied by Elshehawey and Mekheimer []. The initial mathematical model of peristalsis obtained by train of sinusoidal waves in an infinitely long symmetric channel or tube has been investigated by Fung and Yahi [] and Shapiro et.al [18]. Peristaltic transport to a MHD third order fluid in a circular cylindrical tube was investigated by Hayat and Ali [5]. Hayat et al. [6] have investigated peristaltic transport of a third order fluid under the effect of a magnetic field. Effect of thickness of the porous material on the peristaltic pumping when the tube wall is provided with nonerodible porous lining has been studied by Hemadri et.al [7]. Jayarami Reddy et al. [8] have studied the Peristaltic flow of a Williamson fluid in an inclined planar channel under the effect of a magnetic field. Peristaltic transport of a Couple-stress fluid in a uniform and Non- uniform Channels is studied by Mekheimer [9]. The consideration of blood as a MHD fluid helps in controlling blood pressure and has potential for therapeutic use in the diseases of heart and blood vessels by Mekheimer [10]. Peristaltic pumping of couple stress fluid through non - erodible porous lining tube wall with thickness of porous material Rathod, Sridhar and Mahadev [1]. Rathod and Sridhar [13] have studied the peristaltic transport of couple stress fluid in uniform and non-uniform annulus through porous medium. Effect of thickness of the porous material on the peristaltic pumping of a IJOAR 015

3 ISSN Jeffry fluid with non - erodible porous lining wall is studied by Rathod and Mahadev [1]. Rathod and Sridhar [15] have studied the effects of Couple Stress fluid and an endoscope on peristaltic transport through a porous medium. Rathod and Sridhar [16] have studied the peristaltic flow of a couple stress fluid through a porous medium in an inclined channel. Ravi Kumar et.al [17] studied the unsteady peristaltic pumping in a finite length tube with permeable wall. It is now well known that blood behaves like a magneto hydrodynamic (MHD) fluid Stud et al. [1]. Subba Reddy et al. [] have studied the peristaltic transport of Williamson fluid in a channel under the effect of a magnetic field. Vijayaraj et. al [3] studied the Peristaltic pumping of a fluid of variable viscosity in a non-uniform tube with permeable wall. In view of these, we investigate the peristaltic transport of a porous material on the peristaltic pumping when the tube wall is provided with non- erodible porous line using couple stress fluid with magnetic field. The Navier stokes equations are governed by the free flow past the porous material and the flow in the permeable wall is described by Darcy s law. The axial velocity distribution, the stream function, the volume flow rate, the pressure rise and the frictional force are calculated. The effect of thickness of porous lining on the pumping characteristics is discussed with the help of graphs.. MATHEMATICAL FORMULATION: Consider the peristaltic transport of a couple stress fluids in a tube of radius a. The wall of the tube is lined with porous material of permeability k. The thickness of the porous lining is h 1. The axisymmetric flow in the porous lining is governed by Navier-Stokes equation. The flow in the porous layer is according to Darcy s law. In a cylindrical coordinate system ( r *, * z wave train is represented by ), the dimensional equation for the tube radius for an infinite R H( Z, t) a bsin ( Z ct) (.1) Where b is the wave amplitude, is the wavelength, c is the wave speed. IJOAR 015

4 ISSN Fig1. Physical Model The transformation from fixed frame to wave frame is given by R r R; z Z ct; p( z) P( Z, t); (.) Where, and are stream functions in the wave and laboratory frames respectively. We assume that the flow is inertia- free and the wavelength is infinite. In the wave frame, the equations governing the flow are * * * * * * u * u p 1 * u u * * { u w } { ( r ) } ( ( u )) ( u ) * * * * * * * (.3) r z r r r r z * * * * * * w * w p 1 * w w * * { u w } { ( r ) } ( ( w )) ( w ) * * * * * * * (.) r z z r r r z * * * u u w 0 (.5) * * * r r z 1 * Where, ( r ) * * * r r r * * * u and w are radial & axial velocities in the wave frame, is density, p is pressure, is coefficient of viscosity, is coefficient of couple stress parameter, is electric conductivity and is applied magnetic field. IJOAR 015

5 ISSN It is convenient to non-dimensionalize variables and introducing Reynolds number Re, wave number ratio as follows: * * * 1 * * z r Q h u a * * ca a h z, r, Q,, u, p p ( z ), Re,, h, a c ac c a w w c * wb k 1 1, wb, Da, u, w, l, M c a r z r r a * * * * * * * * (.6) The equations of motion (.3), (.) and (.5) becomes (dropping the stars), 3 1 u u w u p r u u Re { } ( ) r z r r r r z - ( ( u)) M ( u) (.7) w w p 1 w w 1 Re { u w } ( r ) - ( ( w)) M ( w) r z z r r r z (.8) u u w 0 r r z (.9) Where, M is the Hartmann number and a a couple-stress parameter. Using long wavelength approximation ( <<1) and dropping terms of order it follows from equation (.7) and (.8) becomes, p 0 r p 1 w 1 ( r ) ( ( w)) M ( w) z r r r (.10) (.11) The dimensionless boundary conditions are: w w w 0, 0 r r r r at r = 0 (.1a) w w w 1 wb, 0 at r r r r h (.1b) w ( w B Q) r Da at r h (.13) Da p 1 Where, Q, Z IJOAR 015

6 ISSN = Viscosity in the free flow region, = Viscosity in the porous flow region 1 = Slip parameter 3. SOLUTIONS: Solving the equation (.10) and (.11) using boundary conditions (.1)-(.13), we obtain the velocity as X px w ( 1) 1 p ( r ( h ) ) pr Da p Da X1( M r ) X1 X1 dp Da Da Where, p, X1 Da M r, X r ( h ) ( h ) dz (3.1) Integrating the equation (3.1) and subjected to the boundary conditions 0 at r 0, we get the stream function as, 3/ 3/ 3 p (Da M r- r -Da M ArcTanh(Mr/)+ Log(-+M r )-S 1 ) Da M - -r S1 -r r S1 pda -r S1 ( - )+p Da ( - - )- ( - ) 3/ 6 Da M - p (h- ) (- ArcTanh(Mr/) (Log( -+M r )-S 1)) p r Sr Sr Sr p ( Log(-)- S 3) S3 p Da S3 pdas / 6 p(h- ) ( (Log(-)-S 3)) - ( + - ) - M 16 1 r r + ( - ) Da M - Da M ( - M ) (3.) Where, Da Da S 1=Log[ r- ], S ( h ) ( h ), S3 Log[- ] The volume flux q through each cross section in the wave frame is given by h q wrdr (3.3) 0 IJOAR 015

7 ISSN p ( h- ) 3/ 3 q [- (3Da -3 Da ( h- ) ( h- ) - Da( h- ) 8Da 8 3 ( h- ) ( )) ( ( h- )- Log(- )+Log(- + (h - ))) + M 8 ( (h- )( + ( h- ))-8 Log(- )+8 Log( (h- ) M(h- ) (h - )))+ ( ArcTanh( )+ (i +Log-Log(- )+ -M 3 3/ Log(- + (h - )))-Log(-+M (h- ) ))) + (-Da hm + 6 (- + ) M(h- ) 3/ 3/ 3 i + h +Da M - +Da M ArcTanh( ) Log- Log(- )+ Log(- + (h - ))- Log(-+M (h- ) ))]+ (h- ) (8Da + (h- ) + Da(-h+ ) ) [(h- ) - ( ( h- )- Log(- ) 8 +Log(- + (h - )))] (3.) Where, dp p dz From equation (3.), we have (h- ) (8Da + (h- ) + Da(-h+ ) ) 3 [8 q (8(h- ) - ( ( h- ) dp - Log(- ) +Log(- + (h - ))))] dz ( h- ) [- (3Da -3 Da ( h- ) ( h- ) - Da( h- ) 8 Da( h- ) 3 M ( )) ( 3/ 3 8 ( h- )- Log(- )+Log(- + (h - ))) + ( (h- )( + ( h- ))-8 Log(- )+8 Log(- + (h - )))+ 8 (h- ) M(h- ) Da ( M ArcTanh( )+ (i +Log-Log(- )+Log(- + -M 3 3/ (h - )))-Log(-+ M (h- ) ))) + (-Da hm +i + 6 (- + ) Da M(h- ) 3/ 3/ 3 M h +Da M - +Da M ArcTanh( ) + Log- 3 3 Log(- )+ Log(- + (h - ))- Log(-+M (h- ) ))] (3.5) 6 IJOAR 015

8 ISSN Following the analysis given by Shapiro et al. [18], the mean volume flow, Q over a period is obtained as Q q (1 ) (3.6) Which on using equation (3.5), yields dp (h- ) (8Da + (h- ) + Da(-h+ ) ) 3 [[8( Q (1 ) ) (8(h- ) - dz ( h- ) ( ( h- )- Log(- ) +Log(- + (h - ))))] / [- (3Da - 3 M ( h- )- Log(- )+Log(- + 8 (h - ))) + ( (h- )( + 3/ 3 3 Da ( h- ) ( h- ) - Da( h- ) 8 Da( h- ) ( )) ( 6 8 (h- ) -M M(h- ) ArcTanh( )+ (i +Log-Log(- )+Log(- + (h - )))-Log(-+M 3 (h- ) ))) + (-Da hm +i + h +Da M - (- + ) ( h- ))-8 Log(- )+8 Log(- + (h - )))+ ( 3/ 3/ 6 M(h- ) +Da M ArcTanh( ) + Log- Log(- )+ Log(- + (h - ))- Log(-+M (h- ) ))]] (3.7) 3/ THE PUMPING CHARACTERISTIC: Integrating the equation (3.7) with respect to z over one wavelength, we get the pressure rise (drop) over one cycle of the wave as 1 (h- ) (8Da + (h- ) + Da(-h+ ) ) 3 0 p [[8( Q (1 ) ) (8(h- ) - ( Da ( h- ) 3/ M ( h- )- Log(- ) +Log(- + (h - ))))] /[- (3Da -3 Da ( h- ) 3 ( h- ) - Da( h- ) 8 Da( h- ) ( 3 )) ( ( h- )- Log(- )+ M IJOAR 015

9 ISSN Log(- + (h - ))) + ( (h- )( + ( h- ))-8 Log(- ) (h- ) M(h- ) Log(- + (h - )))+ ( ArcTanh( )+ (i +Log-Log(- )+ -M 3 3/ (-Da hm +i + + ) Log(- + Da M (h - )))-Log(-+M (h- ) ))) + 6 (- Da 3/ 3/ 3 3 M h +Da M - +Da M ArcTanh( ) + Log- Log(- )+ 3 Log(- + (h - ))- Log(-+M (h- ) ))]] M(h- ) dz (.1) The pressure rise required to produce zero average flow rate is denoted by p0 and is given by 1 (h- ) (8Da + (h- ) + Da(-h+ ) ) 3 [[8( Q (1 ) ) (8(h- ) - 0 p ( h- ) ( ( h- )- Log(- ) +Log(- + (h - ))))] /[- (3Da -3 3/ 3 3 Da ( h- ) ( h- ) - Da( h- ) 8 Da( h- ) ( )) ( M 8 ( h- )- Log(- )+Log(- + (h - )))+ ( (h- )( (h- ) ( h- ))-8 Log(- )+8 Log(- + (h - )))+ ( -M M(h- ) ArcTanh( )+ (i +Log-Log(- )+Log(- + (h - )))-Log(-+M 3 (h- ) )))+ (-Da hm +i + h +Da M - 6 (- + ) 3/ 3/ M(h- ) +Da M ArcTanh( ) + Log- Log(- )+ Log(- + (h - ))- Log(-+M (h- ) ))]] dz (.) 3/ IJOAR 015

10 ISSN The dimensionless frictional force F at the wall across one wavelength in the tube is given by (h- ) (8Da + (h- ) + Da(-h+ ) ) ( h- ) [-8( Q-(1- ) - ) (8(h- ) - F [ ( h- ) ( ( h- )- Log(- ) +Log(- + (h - ))))] ] dz Da Da h h Da h Da 3/ 3 [- (3-3 ( - ) ( - ) - ( - ) 8 3 ( h- ) ( )) ( ( h- )- Log(- )+Log(- + M 8 (h - ))) + ( (h- )( + ( h- ))-8 Log(- ) (h- ) M(h- ) Log(- + (h - )))+ ( ArcTanh( )+ -M (i +Log-Log(- )+Log(- + (h - )))-Log(-+M (h- ) ))) (- + ) Da 3/ 3/ (-Da hm +i + h +Da M - M(h- ) 3/ 3 3 M +Da M ArcTanh( ) + Log- Log(- )+ 3 Log(- + (h - ))- Log(-+M (h- ) ))] (.3) 5. RESULTS AND DISCUSSIONS: In order to see the effect of various pertinent parameters such as the thickness of porous lining ( ), amplitude ratio ( ), ratios of viscosities in the free flow region and porous region ( ), Darcy number (Da), slip parameter ( ), couple stress parameter ( ) and Magnetic parameter (M) on pumping characteristics have plotted in Figs The variation of pressure rise p with average flow rate Q for different values of Da with = 0.7, = 0.01, = 0., = 0.6, = 0.5, M = 1 is presented in Fig.. It is observed that, in pumping region the time averaged flow rate Q decreases with increasing Darcy number. Further, it is noted that smaller Darcy number, larger pressure IJOAR 015

11 ISSN rise. Fig.3. depicts the variation of on pumping characteristics with = 0.7, Da = 0.0, = 0.01, = 0.6, = 0.5, M = 1. It is seen that, an increase in increases the pressure rise p against which the pumping works. In addition, it is noted that the flux Q increases with increase of.the variation of pressure rise p with average flow rate Q for different values of slip parameter with = 0.7, = 0.01, = 0., Da = 0.0, = 0.5, M = 1 is presented in Fig.. It is observed that, an increase in the value of slip parameter, decreases the pressure rise p.the effect of amplitude ratio on the pumping performance is shown in Fig.5 with Da = 0.0, = 0.01, = 0., = 0.6, = 0.5, M = 1, it is noted that, larger the amplitude ratio, greater the pressure rise against which the pump works.i.e., of pressure rise p increases with increase in. Fig.6. shows the variation p with average flow rate Q for different values of with = 0.7, Da = 0.0, = 0., = 0.6, = 0.5, M =. It is found that, larger the thickness of porous lining, greater the pressure rise against which the pump works. Fig.7. shows the relation between the pressure rise p and averaged flux Q for different value of with = 0.7, = 0.01, = 0., Da = 0.0, = 0.6, M = 1. It is found that in the entire pumping region the volumetric flow rate increases with the increase in couple stress parameter. Fig.8. shows the relation between the pressure rise p and averaged flux Q for different value of M with = 0.7, = 0.01, = 0., Da = 0.0, = 0.6, = 0.5. It is observed that, an increase in the value of Magnetic parameter M, decreases the pressure rise p. The variation of friction force F with averaged flow rate Q under the influence of all emerging parameters such as Da,,,,,,M. It is observed that the effect of all the parameters on friction force are opposite to the effects on pressure with the averaged flow rate is shown in figs IJOAR 015

12 ISSN Fig.. Effect of Da on p when =0.7, =0., =0.6, =0.01, =0.5,M=1 Fig.3. Effect of µ on p when =0.7,Da=0.0, =0.6, =0.01, =0.5,M=1 Fig.. Effect of on p when =0.7, Da=0.0, =0., =0.01, =0.5,M=1 IJOAR 015

13 ISSN Fig.5. Effect of on p when Da=0.0, =0., =0.6, =0.01, =0.5,M=1 Fig.6. Effect of on p when =0.7,Da=0.0, =0., =0.6, =0.5,M= Fig.7. Effect of on p when =0.7,Da=0.0, =0., =0.6, =0.01,M=1 IJOAR 015

14 ISSN Fig.8. Effect of M on p when =0.7,Da=0.0, =0., =0.6, =0.01, =0.5 Fig.9. Effect of on F when Da=0.0, =0., =0.6, =0.01, =0.5,M=1 Fig.10. Effect of Da on F when =0.7, =0., =0.6, =0.01, =0.5,M=1 IJOAR 015

15 ISSN Fig.11. Effect of on F when =0.7,Da=0.0, =0., =0.6, =0.,M=1 Fig.1. Effect of on F when =.7,Da=0.0, =0., =0.01, =0.5,M=1 Fig.13. Effect of µ on F when =0.7,Da=0.0, =0.6, =0.01, =0.5,M=1 IJOAR 015

16 ISSN Fig.1. Effect of on F when =.7,Da=0.0, =0., =0.6, =0.01, M=1 Fig.15. Effect of M on F when =0.7,Da=0.0, =0., =0.6, =0.01, = CONCLUSION: In this analysis peristaltic transport of a porous material on the peristaltic pumping when the tube wall is provided with non- erodible porous line using couple stress fluid with magnetic field has been studied. We conclude the following observations: Pressure with average flow rate Q Pressure decreases with increase in Da, & M and Pressure increases with increasing in,, &. Q Friction force with average flow rate It is observed that the effects of the parameters on friction force are opposite to the effects on pressure with the averaged flow rate. IJOAR 015

17 ISSN ACKNOWLEDGEMENT: The author is highly grateful to the reviewers for careful reading of the manuscript and for valuable suggestions. The present work is part of the Minor Research Project (plan) [Grant No.: MRP(S)-035/13-1/KAGU056/UGC-SWRO, Date: 8-Mar-1] sponsored by the UNIVERSITY GRANTS COMMISSION (UGC), Bangalore, Karnataka, India. REFERENCES: [1] Alemayehu and Radhakrishnamacharya.G, Dispersion of a Solute in Peristaltic Motion of a Couple stress fluid in the presence of Magnetic Field, Word Academy of Science, Engg. and Tech., 75(011), [] Elsehawey EF and El-Sebaei W, Couple-stress in Peristaltic transport of a Magneto- Fluid, Physica Scripta, 6(001), [3] Elsehawey EF and Mekheime Kh S, Couple-stresses in Peristaltic transport of fluids, J. of Phys. D: appl. Phys., 7(199), [] Fung YC and Yih CS, Peristaltic transport. J. of Appl. Mech., Trans. ASME, 5(1968), [5] Hayat T and Ali N, Physica A: Statistical Mechanics and its Applications, 370(006), [6] Hayat, T., Afsar, A., Khan, M. and Asghar, S. Computers and Mathematics with Applications, 53(007), [7] Hemadri Reddy R, Kavitha A, Sreenadh S, Hariprakashan P, Effect of thickness of the porous material on the peristaltic pumping when the tube wall is provide with Non-erodible porous lining, Adv. App. Sc. Research, (011), [8] Jayarami Reddy B, Subba Reddy M.V, Nadhamuni Reddy C and Yogeswar Reddy P, Peristaltic flow of a Williamson fluid in an inclined planar channel under the effect of a magnetic field, Adv. Appl. Sci. Res., 01, 3 (1):5-61. [9] Mekheimer Kh S, Peristaltic transport of a Couple-stress fluid in a uniform and Non- uniform Channels, Biorheology, 39(00), [10] Mekheimer, KH.S. Appl. Math. Comput, 153 (00), [11] Raghunath Rao T & Prasad Rao D.R.V, Peristaltic flow of a couple stress fluids through a porous medium in a channel at low Reynolds number, Int. J. of Appl. Mech., 8(3)(01), [1] Rathod V. P, Sridhar N. G and Mahadev M. Peristaltic pumping of couple stress fluid through non - erodible porous lining tube wall with thickness of porous material, Advances in Applied Science Research, 01, 3 (): [13] Rathod V. P, Sridhar N. G, Peristaltic transport of couple stress fluid in uniform and non-uniform annulus through porous medium, international journal of mathematical archive, 3() (01) [1] Rathod V.P and Mahadev M., Effect of thickness of the porous material on the peristaltic pumping of a Jeffry fluid with non - erodible porous lining wall, Int. j. of Mathematical Archive, (10), Oct IJOAR 015

18 ISSN [15] Rathod V.P and Sridhar N.G, Effects of Couple Stress fluid and an Endoscope on Peristaltic Transport through a Porous Medium, International J. of Math. Sci. & Engg. Appls., 9(1) (015) [16] Rathod V.P and Sridhar N.G, Peristaltic flow of a couple stress fluid through a porous medium in an inclined channel, Journal of Chemical, Biological and Physical Sciences, 5() (015) [17] Ravikumar S, Prabhakar Rao G, and Siva Prasad R, Peristaltic flow of a Couple stress fluid in a flexible channel under an oscillatory flux, Int. J. of Appl. Math and Mech., 6(13) (010), [18] Shapiro A.H., Jaffrin M.Y, and Weinberg S.L, Peristaltic pumping Long Wave Length at Low Reynolds Number J.Fluid Mech., 37 (1969) [19] Sohail Nadeem and Safia Akram, Peristaltic flow of a Couple stress fluids under the effect of induced magnetic field in an asymmetric channel, Arch Appl. Mech., 81(011), [0] Srivastava LM, Peristaltic transport of a couple stress fluids, Rheol. Acta, 5(1986), [1] Stud, V.K., Sekhon, G.S. and Mishra, R.K. Bull. Math. Biol. 39(1977), [] Subba Reddy, M. V., Jayarami Reddy, B. and Prasanth Reddy, D. International Journal of Fluid Mechanics,3(1)(011), [3] Vijayaraj K, Krishnaiah G, Ravikumar M.M., Peristaltic pumping of a fluid of variable in a non-uniform tube with permeable, J. Theo. App. Inform. Science, 009, ****** IJOAR 015