PERISTALTIC PUMPING OF 4th GRADE FLUID THROUGH POROUS MEDIUM IN AN INCLINED CHANNEL WITH VARIABLE VI by IJATCA

Pradeepa, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 211-217 ISSN: 2395-3519

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PERISTALTIC PUMPING OF 4th GRADE FLUID THROUGH POROUS MEDIUM IN AN INCLINED CHANNEL WITH VARIABLE VISCOSITY UNDER THE INFLUENCE OF SLIP EFFECTS 1

Mohd.Riyazuddin, 2Y.V.K. Ravi Kumar, 3M.V.Ramana Murthy 1 Department of Mathematics, Keshav Memorial Institute of Technology, Hyderabad – 500 029, INDIA 2 Practice School Division, Birla Institute of Technology and Science,– Pilani, Rajasthan, INDIA. 3 Department of Mathematics, Mahatma Gandhi Institute of Technology, Gandipet, Hyderabad – 500 075, INDIA. 1 riyazuddin17@gmail.com, 2yvk.ravikumar@pilani.bits-pilani.ac.in, 3mv.rm50@gmail.com

Abstract: This article deals with the peristaltic pumping of 4th grade fluid through porous medium in an inclined channel with variable viscosity under the influence of slip condition. In the acquired solution expressions the long wavelength & low Reynolds number assumptions are utilized and it is noticed that we observed that pressure rise increases with the increase in

,  ,

and axial velocity decreases with increase in values of ,  ,  .

Keywords: 4th grade fluid, porous channel, slip condition, peristaltic pumping.

INTRODUCTION

Shehawes [1] considered “influence on porous boundaries on peristatic pumping through a porous passage with pulsatile Magneto fluid with a porous channel" corresponding results are obtained by Gad N.S. et al. [2]. Chaube et al. [3] considered “slip effect on peristatic pumping of micropolor fluid”. Misery et al. [4] considered “effect of a fluid with variable viscosity”. Hussain [5] studied “slip effect on the peristatic pumping of MHD fluid with variable viscosity”. Ravi Kumar [6] studied “pulsatile pumping of a viscous strafed fluid of variable viscosity”. Hayat T et al. [7] studied “Soret and Dufour effect on flow MHD fluid with variable viscosity”. Provost A [8] studied “flow of a second-order fluid in a planar passage” & corresponding results in tubes are obtained by Schwarz W.H [9]. Masuoka Stokes et al. [10] investigated “1st problem for a second grade fluid in a porous half space”. Ali N [11] studied “peristaltically induced motion of a MHD third grade fluid”. Harojun [12] studied “effect of peristatic transport of a 3rd order fluid in an asymmetric passage”. Qureshi [13] investigated the “influence of slip on the peristaltic pumping of a 3rd order fluid”. Sreenath [14] studied

“peristaltic flow of a 4th grade fluid between porous walls”. We intend to study peristaltic flow of 4th grade fluid across porous medium in an inclined channel with variable viscosity under the influence of slip situation.

II. MATHEMATICAL MODEL



dV    grad p  div S  V dt k

(1)

The law of conservation of mass is defined by

divV  0

(2)

4th grade fluid stress tensor S is

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Pradeepa, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 211-217 ISSN: 2395-3519 S   A1  1 A2   2 A12  1 A3

2 c  c2  c2 2  2 2 , 1  1 2 , , 0 d 0 d 0 d 2  c  c3  c3 3  3 2 ,   b , 1  1 3 , 2  2 3 , 0 d a a d 3 3  c  c3  c 3  3 3 ,  4  4 3 , 5  5 3 , a a a 3 3 3  c  c c 6  6 3 , 7  7 3 , 8  8 3 . a a a 2 

  2  A2 A1  A1 A2    3  tr A12 A1   1 A4  2 ( A3 A1  A1 A3 )   3 A22   4 ( A2 A12  A12 A2 )  5 (trA2 ) A2  A2   6 (trA2 ) A12 [ 7 (trA3   8tr ( A2 A1 ))] A1

(3) where

 , 1 ,  2 , 1 ,  2 , 3 ,  1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8

A being material constants and n is the Rivlin – Ericksen tensors defined dA An 1  n  An ( gradV )  ( gradV )T An , n  1 dt (4)

A1  ( gradV )  ( gradV )T

(5)

Flow of 4th grade fluid is studied in a uniform channel with porous medium of width 2d, inclined at an angle  with variable viscosity

 2  H ( X , t )  d  b Cos  X  ct      

(6)

where b : Wave amplitude  : Wave distance t : Time

(10) where  : Wave no., Re: Reynolds no.,

 : Amplitude ratio,

 2 : Porosity parameter, i  i  1, 2 

: Non-Newtonian parameters and

i  i  1, 2,3 

: Non-Newtonian parameters.

S  u S u  p Re   u  v      xx  xy y  x y y  x Re   ( y )  2 (u  1)  Sin  Fr (11)

U ,V  , u , v  : Velocity components in fixed & wave

frame respectively and the relation between fixed and wave frames as

x  X ct , y Y,

u U c, v V

(7)

when we non dimensionalize eqns (3), (6) &(7)

S  A1  1 A2  2 A21  1 A3   2  A2 A1  A1 A2 

S  v v  p Re  3  u  v      2 xy y  y x  x S Re  yy   2  ( y )  2 v   Cos  y Fr

(12)

    u v      Re   u  v      2   2 2  S xy y  y x   y y   x 2

  2 v  S xx  S yy    2    ( y ) (u  1)    2   xy x   y

3  tr A21  A1  1 A4  2 ( A3 A1  A1 A3 )



3 A22  4 ( A2 A12  A12 A2 )

(13) applying the long wave distance & less Re. number approximations we obtain

5 (trA2 ) A2  A2  6 (trA2 ) A12

p S xy Re    ( y )  2 (u  1)  Sin  x y Fr

[7 (trA3  8tr ( A2 A1 ))] A1 (8)

h( x) 1   cos 2 x y y , d

S

0 c

Re 

(9)

v , c ( y)  ( y)  ,

u u , c

S  x ,

 cd , 0

v

0

Fr 

c , gd

2 

d , k

H h , d d  ,  1 c 1  , 0 d

p 0 y

S x y   ( y) where

(14) (15)

 u  u  2  y  y 

   2  3

(16)

is the Deborah number.

Eq. (15) indicates that p  p ( y ) . From equation (13) we have

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Pradeepa, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 211-217 ISSN: 2395-3519 3  u   2  u   ( y)  2     y 2  x  y  

3.1 Zero-order Equation:

p00  2u00 Re    2 (u00  1)  Sin  2 x y Fr

  v      ( y ) (u  1)    2   0 x   y 2

(17)

The boundary conditions are,

u 0 y at y  0 3   u   u u     ( y)  2      1  y  y   

u00 0 y u00  

y0

u00  1 y

(28) (29)

at y  h

(30)

(18) 3.2 First-order Equation: at

yh (19)

l  d is a slip parameter where the effect of variable viscosity on peristaltic flow

  y   ey ,   1

2 i.e.  ( y )  1   y  ( )

(20)

p01  2u01   u     2u01   2 y (u00  1)   y 00  2 x y y  y  u01 0 y0 y at u u u01   y 00   01  0 y y at y  h

(31) (32) (33)

(21) 3.3 Zero-order Solution

The volume flow rate q is Solving equation (28) we get,

q   udy

(22)

The instantaneous flow Q (X , t) is h

(23)

The time averaged volume flow rate Q over is

Q

1 T



Qdt  q  1

The pressure rise p 1

p   0



Q( X , t )   UdY   (u  1)dy  q  h 0

u00  c1 Cosh ( h) 

dp dx dx

(24)

(25)

III. SOLUTION OF THE PROBLEM PERTURBATION SOLUTION We use the below equations in order to obtain solution

u  u0   u1  ..... p  p0   p1  .....

p1  p10   p11  .....

3.4 First-order Solution Sub. zero order soln. Eq. (34) in Eq. (31) and then solving the Eq. (31), we get

u01  c3 Cosh ( h)  c4 Sinh ( h)

 G1  y   c1 y Cosh ( h) 

(26)

G1 

u0  u00   u01  ..... u1  u10   u11  ..... p0  p00   p01  .....

 1  G0  2  1 L1    L1  Cosh ( h)    Sinh ( h) p Re G0  0   2  Sin ( ) x Fr c1 



where

(27)

when we put Eq. (27) into Eqs. (17) to (25) and separate the terms of differential order in  and  we obtain

(34)

where

p01

(35)

x L2  Sinh ( h)    Cosh ( h) c L3  1  (h   ) Cosh ( h)    h Sinh ( h)  2 G h L4  1 2



c3 

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c1 L3 1   c4 L2   L4  L1  2 

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Pradeepa, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 211-217 ISSN: 2395-3519

c4 





we observe that the pressure rise increases with

c1 2

applying equations (34) & (35) with the relation

u  u0 0   u01

u

we get

1  p Re   Cosh ( h)  Cosh ( h)   2 Sin( )    1  2    x Fr L1 L1  

 Cosh ( h)  h   L3 G  L L    2 3  0 2  L3  2      2 L1 2 2 2        1 G    Sinh ( h)  0      1  y Cosh (  h )      2 L1   2         Sinh ( h)  y  3 2      

(36) The volume flow rate is h

q   u dy

decrease in values of  ,  . Fig.3 shows that the pressure rise increases with decrease in values of

  p  0 

and it is observed that pressure rise

  p  0 

increases with decrease in values of . Fig. 6 to Fig. 10 show the changes on pressure gradient due to different parameters. Fig. 7 to 8 show graphs for pressure gradient vs x. It can be noticed that there is a rise in pressure gradient with a rise in  ,  in first half wave length of the passage and the effect reverses in second half wave length of the passage. Fig. 6, Fig. 9 and Fig. 10 show graphs for pressure gradient vs x. We can notice that there is a rise in pressure gradient with lowering in

values of ,  ,  in first half wave length of the passage and the effect reverses in second half wave length of the passage. Fig. 11 to 15 show the effects of axial velocity due to various parameters.

q

 Sinh( h) 1  p Re   Sinh ( h)   2 Sin( )    h   2  x Fr  L  L1  1 

 Sinh( h)  h   L3 G  L L     2 3  0 2  L3  2      2  L1   2 2 2        2      G02  1 h Sinh( h)  Cosh (4 h)  h 2  14     2    2  L1  

(37) 1

p   The pressure rise is where

,  ,  for a stable y value. In Fig. 15 it is noticed that velocity rises with rise in  for stable y value. In Fig. 12 it is noticed that there is a fall in velocity with rise in the porosity parameter  . 4.1 Trapping: Fig. 16 show the stream line patterns and trapping for

various values of the  and here we noticed that the

dp dx dx

size of the bolus decreases with rise in  . Fig. 17 show the stream line patterns and trapping for various

  L2     h   L3     2  2  2 3    Sinh (  h )        L1  G0  L2      L      2  3       2        3 L1 dp  q  Sinh( h)     h Sinh ( h)  G0  1   2 dx Sinh( h)  h  L1   L1 2  L1         Cosh (  h )    4        2   h  1  2 4  2         2

Fig. 11, 13 and 14 show velocity vs y. It is noticed that there is a fall in velocity with a rise in values of

values of  , here we observed that the size of bolus

falls with rise in  . Fig. 18 show the stream line patterns and trapping for various values of  , here we observed that the size of bolus increases with rise in  . Fig. 19 show the stream line patterns and trapping for various values of  , here we noticed that the size of the bolus increases with increase of  .

Re Sin( ) Fr

(38)

IV.

RESULTS AND DISCUSSION

Fig. 1 to 5 show that the changes in pressure rise due to different parameters. In Fig. 1 to Fig. 5, it is noticed that pressure rise increases with the increase in

 ,  . In Fig.2 and Fig.4

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Pradeepa, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 211-217 ISSN: 2395-3519 [2]. Afif N.A.S., and Gad N.S., Interaction of peristaltic flow with pulsatile Magneto fluid through a porous medium, Acta Mechanica, 149, (2001), 229-237. [3]. Chaube M.K., Pandey S.k., and Tripathi D., Slip Effect on Peristatic transport of Micropolor fluid, Applied Mathematical Sciences, 4, (2010), 2105-2117. [4]. Misery A.M.E.I., Hakeem A.E.I., Naby A.E.I., Nagar A.H.E.I., Effects of a fluid with variable viscosity and an endoscope on peristaltic motion, J. Phys. Soc. J Pn. 72, (2003), 89-93. (b) [5]. Ali N., Hussain Q., Hayat T., Asghar S., Slip effects on the peristatic transport of MHD fluid with variable viscosity, Phy. Lett. A372, (2008) , 1477-1489. [6]. Ravi Kumar Y.V.K., Avinash K., Rao JA., and Sreenadh S., Pulsatile flow of a viscous stratified fluid of variable viscosity between permeable beds, Journal of Porous Media, (2011). [7]. Abbasi F. M., Hayat T., Alsaedi A., Ahmed B., Soret and Dufour Effects on Peristaltic Transport of MHD Fluid with variable viscosity, Appl. Math. Inf. Sci., 8 (1), (2014), (d) 211-219. [8]. Siddiqui A.M., Provost A., Schwarz W.H., Peristaltic pumping of a second-order fluid in a planar channel, Rheol Acta 30, (1991), 249. [9]. Siddiqui A.M., Schwarz W.H., Peristaltic flow of a second-order fluid in tubes, J. Non-Newtonian fluid Mech. 53(257), (1994). [10]. Tan W.C. and Masuoka Stokes T., First problem for a second grade fluid in a porous half space with heated boundary, Int. J. Non-linear Mech., Vol.40, (2005), 512-522. [11]. Hayat T., Ali N., Peristaltically induced motion of a MHD third grade fluid in a deformable tube, Physica, A 370, (2006), 225. [12]. Harojun M.H., Effect of Deborah number and phase difference on peristatic transport of a third-order fluid in an (b) asymmetric channel, commun. Nonlinear Sci. Number. Simulat., 12(1464), (2007). [13]. Hayat T., Qureshi M.U., Ali N., The influence of slip on the peristaltic motion of a third order fluid in an asymmetric channel, Phy. Lett. A 372, (2008), 2653-2664. [14]. Sreenadh S., Prabakaran P.H., Kavith A., Saravan R., Reddy R.H., Peristaltic transport of a fourth grade fluid between porous walls with suction and injection, International Journal of pure and applied Mathematics, 86(2), (2013).

(d)

References [1]. Shehawes E.F.EI., and Husseny S.Z.A., Effects on porous boundaries on peristaltic transport through a porous medium, Acta Mechanica, 143, (2000), 165-177.

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