Thus our intention 80 here is to model the effect of radial magnetic field on peristaltic 81 transport of Williamson fluid in a curved channel with flexible 82 walls.. Influence of radia
Trang 15
6
7 Tasawar Hayata,b, Sadaf Nawaza,⇑ , Ahmed Alsaedib, Maimona Rafiqa
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
10
1 3 a r t i c l e i n f o
14 Article history:
15 Received 20 November 2016
16 Received in revised form 12 February 2017
17 Accepted 15 February 2017
18 Available online xxxx
19 Keywords:
20 Williamson fluid
21 Curved channel
22 Radial magnetic field
23 Complaint walls
24
2 5
a b s t r a c t
26 Peristaltic transport of Williamson fluid in a curved geometry is modeled Problem formulation is
com-27 pleted by complaint walls of channel Radial magnetic field in the analysis is taken into account
28 Resulting problem formulation is simplified using long wavelength and low Reynolds number
approxi-29 mations Series solution is obtained for small Weissenberg number Influences of different embedded
30 parameters on the axial velocity and stream function are examined As expected the velocity in curved
31 channel is not symmetric Axial velocity is noticed decreasing for Hartman number Trapped bolus
32 increases for Hartman and curvature parameters
33
Ó 2017 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://
34
creativecommons.org/licenses/by-nc-nd/4.0/)
35 36
37 Introduction
38 The mechanism of peristalsis has prime importance in
biome-39 chanics This mechanism occurs due to traveling wave along
flexi-40 ble walls of channel This travelling wave forces the contained fluid
41 to flow in the same direction even in absence of external involved
42 pressure gradient Various physiological materials are transported
43 through such mechanism Mention may be made to the transport
44 of urine from kidney to bladder, food from oesophagus to
gastro-45 intestinal tract, chyme through intestine, vasomotion of small
46 blood vessels, embro transport in the uterus, spermatic fluid in
47 the ductus effertenes of the male reproductive tract, ovum
move-48 ment in female fallopian tube etc Further peristalsis is well suited
49 to design many machines including dialysis machine, open heart
50 bypass machine, finger, roller and hose pumps, infusion pumps
51 etc There is widespread applications of peristalsis in the pump
52 industry for transport of corrosive and sterile materials A vast
53 use of peristalsis is due to the fact that the transported fluid does
54 not have direct contact with any moving part Because of all these
55 applications there are intensive attempts on peristalsis through
56 theoretical and experimental approaches Latham [1] and Shapiro
57 et al [2] were the first who presented pioneering research on
peri-58 stalsis of viscous fluids Afterwards many researchers worked on
59 peristalsis by considering both viscous and non-Newtonian fluids
60
via diverse aspects and assumptions Some literature on peristalsis
61
can be estimated through the references [3–23]
62
There is no doubt that magnetohydrodynamics (MHD) deals
63
with the motion of highly conducting fluid via a magnetic field.
64
Specifically peristalsis in presence of magnetic field has relevance
65
with processes regarding motion of conducting materials for
66
example the geothermal sources analysis, MHD power generators
67
design, hyperthermia, drug delivery, removal of arteries blockage,
68
MHD compressors, nuclear fuel debris treatment, the control of
69
underground spreading of chemical wastes and pollution Having
70
all this in mind, extensive attempts in the aforementioned quoted
71
literature have been mostly for peristalsis in presence of constant
72
applied magnetic field In other words the peristalsis subject to
73
radial magnetic field is examined scarcely Further in these
74
attempts the channel is taken straight This flow configuration is
75
not realistic since most of the geometries involved in industry
76
and physiological processes are curved Thus very recent the
77
researchers in the field analyzed the impact of curvature on
peri-78
stalsis in a channel (see [24–43] ) The objective of present
commu-79
nication is to venture further in this regime Thus our intention
80
here is to model the effect of radial magnetic field on peristaltic
81
transport of Williamson fluid in a curved channel with flexible
82
walls Note that Williamson fluid is a shear thinning viscoelastic
83
material It also represents the behavior of diverse variety of
pseu-84
doplastic liquids accurately i-e a decrease in viscosity through
85
increase in rate of shear stress is exhibited Many biophysical
86
and physiological materials posses shear-thinning behavior for
87
example blood suspension and gastro-intestinal fluids [9] Here
http://dx.doi.org/10.1016/j.rinp.2017.02.022
2211-3797/Ó 2017 Published by Elsevier B.V
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
⇑ Corresponding author
E-mail address:sadafnawaz26@gmail.com(S Nawaz)
Contents lists available at ScienceDirect
Results in Physics
j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m / r e s u l t s - i n - p h y s i c s
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 288 stream function formulation and lubrication approach are
89 employed Solution for small Weissenberg number is constructed.
90 Impacts of sundry variables on physical quantities of interest are
91 addressed in detail.
92 Modeling
93 Consider a curved channel of width 2d coiled in a circle of radius
94 R⁄and centre at O Here x -axis lies in the length of the channel and
95 r -axis normal to it The axial flow direction is along x -axis while
96 radial direction along the r -axis In the axial and radial directions
97 the components of velocity are denoted by u and v respectively An
98 incompressible electrically conducting Williamson fluid fills the
99 channel The walls shape is defined as follows:
100
r ¼ g ðx; tÞ ¼ d þ a sin 2 p
k ðx ctÞ
102
103 in which a, c and k denote the amplitude, speed and wavelength of
104 the wave respectively and t stands for time.
105 The definition of magnetic field B applied in the radial direction
107
B ¼ RB0
r þ R; 0; 0
109
110 where B0denotes the strength of applied magnetic field The
Lor-111 entz force (F = J B) in absence of electric field takes the form as
112 follows:
113
J B ¼ 0; r ðRÞ
2
B20u
ðr þ RÞ2 ; 0
115
116 Here r denotes the electrical conductivity and J(=V B) is the
117 current density of fluid Present flow is governed by the following
118 expressions:
119
@½ðr þ RÞ v
@r þ R
@u
121
122
q @ v
@t þ v @ @r v þ uR
r þ R@ v
@x
u2
r þ R
¼ @ p
@r þ
1
r þ R @
@r fðr þ R
ÞSrrg þ R
r þ R@Sxr
@x
Sxx
r þ R; ð5Þ
124
125
q @u
@t þ v @u @r þ uR
r þ R@u
@x þ
u v
r þ R
¼ R
r þ R@p
@x þ
1
ðr þ RÞ2
@
@r fðr þ RÞ
2
Srxg þ R
r þ R@Sxx
@x
r ðR
Þ2
B20u
127
128 in which relation for an extra stress tensor S for Williamson fluid is
129 [13] :
130
S ¼ ½ l1þ ð l0þ l1Þð1 C c _ Þ1A1; ð7Þ
132
133 where C denotes the time constant, l0and l1are the zero shear
134 rate and infinite shear rate viscosities and c _ and A1 are defined
135 below.
136
_
c ¼ ffiffiffiffiffiffiffi
1P
q
;
A1¼ gradV þ ðgradVÞT
;
P ¼ trðA2
1Þ:
ð8Þ
138
139 Here we have taken l1¼ 0 and C c _ < 1: Thus extra stress tensor
140 becomes
141
144
The components of extra stress tensor are
145
Srr¼ 2 l0ð1 þ C c _ Þ @ v
148
Srx¼ l0ð1 þ C c _ Þ R
r þ R@ v
@x
u
r þ Rþ @ u
@r
150 151
Sxx¼ 2 l0ð1 þ C c _ Þ v
r þ Rþ R
r þ R@u
@x
153 154
_
c ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 @ v
@r
2
þ2 v
r þRþ R
r þR@u
@x
þ R
r þR@ v
@x
u
r þRþ@ u
@r
s
: ð13Þ 156 157
The boundary conditions for the present flow considerations
158
are
159
162
R s @3
@x3þm @
3
@x@t2þd1 @2
@t@x
g
¼ 1
r þR @
@r fðr þRÞ
2
SrxgþR@Sxx
@x q ðr
þRÞ @ u
@t þ v @u @r þ uR
r þR@u
@x þ
u v
r þR
r ðR
Þ2
B20u
ðr þRÞ ; r ¼ g :
! ð15Þ
164 165
Here p represents the pressure, R⁄the curvature parameter and
166
q the density In above equation s , m and d1describe the elastic
167
tension, mass per unit area and the coefficient of viscous damping
168
respectively Further Srr, Srxand Sxxare the components of extra
169
stress tensor S.
170
Considering
171
x¼x
k; r¼r
d; u¼u
c; v¼vc; t¼ct
k; g¼g
d;
Sij¼dSij
cl0; k ¼R
d; _ c¼ _ cd; p ¼d2p
ckl0; We ¼ Cc; ð16Þ 173
174
the non-dimensional problems are
175
Red d @ v
@t þ v @ @r v þ dku
r þ k
@ v
@x
u2
r þ k
¼ @ p
@r þ d
1
r þ k
@
@r fðr þ kÞSrrg þ kd
r þ k
@Sxr
@x
Sxx
r þ k
; ð17Þ
177 178
Re d @u
@t þ v @u @r þ dku
r þ k
@u
@x þ
u v
r þ k
¼ k
r þ k
@p
@x þ
1
ðr þ kÞ2
@
@r fðr þ kÞ
2
Srxg þ kd
r þ k
@Sxx
@x
k
2
180 181
Srr¼ 2ð1 þ We _ c Þ @ v
184
Srx¼ ð1 þ We _ c Þ d @ v
@x
u
r þ k þ @
u
@r
186 187
Sxx¼ 2ð1 þ We _ c Þ v
r þ k þ
k
r þ k d
@u
@x
189 190
_
c ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 @ v
@r
2
þ2 v
rþk þ
k rþk d
@u
@x
þ k rþk d
@ v
@x
u rþk þ@
u
@r
s
; ð22Þ
192
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 3u ¼ 0; at r ¼ g ¼ ð1 þ e sin 2 p ðx tÞÞ; ð23Þ 195
196
k E1@3
@x3þE2 @3
@x@t2þE3 @2
@t@x
g ¼
Reðr þkÞ d@ u
@t þ v @u @r þ ukd
rþk
@u
@x þ
u v
r þk
rþk@r@fðr þkÞ2
Srxgþdk@Sxx
@x
k2
rþkM2u ; atr ¼ g ;
!
ð24Þ 198 199
where e( = a/d) denotes the amplitude ratio, k the dimensionless
200
curvature parameter, Re( = q cd/ l0) the Reynolds number, d ð¼ d=kÞ
201
the wave number, M ¼ ffiffiffiffiffiffiffiffiffiffiffiffi
r = l0
p
B0d the Hartman number, We the
202
Williamson fluid parameter known as Weissenberg number and
203
E1ð¼ s d3=k3c l0Þ; E2ð¼ mcd3
=k3l0Þ and E3ð¼ d1d3=k2l0Þ depict the
204
non-dimensional form of elastance parameters respectively.
205
Writing the velocity components ðu; v Þ in terms of stream
func-206
tion ( w ) by [26,32,34] :
207 Fig 1 Geometry of the problem
Fig 2 u via change in M when E1= 0.003, E2= 0.003, E3= 0.01, t = 0.1, x = 0.2, e = 0.2, We = 0.01.(a)k = 5.0.(b)k? 1
Fig 3 u via change in We when E1= 0.003, E2= 0.003, E3= 0.01, t = 0.1, x = 0.2, e = 0.2, M = 0.8.(a)k = 5.0.(b)k? 1
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 4u ¼ @ w
@r ; v ¼ dk
r þ k
@w
@x :
209
210 The continuity equation is identically satisfied Using above
211 expressions in Eqs (17)(24) and then implementing the long
wave-212 length and low Reynolds number approximations we obtain
213
@p
215
216
k
r þ k
@p
@x þ
1
ðr þ kÞ2
@
@r fðr þ kÞ
2
Srxg þ k
2
ðr þ kÞ2M2@w
@r ¼ 0; ð26Þ
218
219
wr¼ 0 at r ¼ g ¼ ð1 þ e sin 2 p ðx tÞÞ; ð27Þ
221
222
k E1 @3
@x3þ E2 @3
@x@t2þ E3 @2
@t@x
g
¼ 1
r þ k
@
@r fðr þ kÞ
2
Srxg þ k
2
r þ k M
2@w
@r ; at r ¼ g ; ð28Þ
224
225
227
228
Srx¼ ð1 þ We _ c Þ 1
r þ k
@w
@r @
2
w
@r2
!
230
231
_
r þ k
@w
@r @
2
w
233
234 By cross differentiation of Eqs (25) and (26) we obtain:
235
@
@r
1
kðr þ kÞ
@
@r fðr þ kÞ
2
Srxg þ k
r þ k M
2@w
@r
237
238 Eq (25) indicates that p does not depend upon r Note that the
239 results for planar channel case are recovered when k ? 1.;
240 Solution methodology
241 It is difficult to find the closed form solution for the Eq (32)
242 Thus we intend to compute the solution for small Weissenberg
243 number (0 < We < 1) For this purpose we write
244
w ¼ w0þ Wew1þ OðWe2
246
247 The corresponding zeroth and first order systems and
248 respective stream functions are:
249
Zeroth order system and stream function
250 251
@
@r
1 kðr þ kÞ
@
@r fðr þ kÞ
2
S0rxg þ k
r þ k M
2@w0
@r
253 254
257
k E1 @3
@x3þ E2 @3
@x@t2þ E3 @2
@t@x
g
¼ 1
r þ k
@
@r fðr þ kÞ
2
S0rxg þ k
2
r þ k M
2@w0
@r ; at r ¼ g ; ð36Þ 259
260
S0rx¼ 1
r þ k
@w0
@r @
2
w0
263
w0¼ C4þ C3kr þ C3
r2
2 þ C2ðk þ rÞ1 ffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2
p
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ k2
M2
p
þ C1ðk þ rÞ1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2
p
1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ k2
M2
265
Fig 4 u via change in E1, E2, E3when t = 0.1, x = 0.2, e = 0.2, We = 0.01, M = 2.0.(a)k = 2.0.(b)k? 1
Fig 5 u via change in k when E1= 0.001, E2= 0.001, E3= 0.01, t = 0.1, x = 0.2, e = 0.2,
We = 0.01, M = 0.8
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 5266 First order system and stream function
267
268
@
@r
1
kðr þ kÞ
@
@r fðr þ kÞ
2
S1rxg þ k
r þ k M
2@w1
@r
270
271
273
274
1
r þ k
@
@r fðr þ kÞ
2
S1rxg þ k
2
r þ k M
2@w1
@r ¼ 0; at r ¼ g ; ð41Þ
276
277
S1rx¼ 1
r þ k
@w1
@r @
2
w1
@r2 þ 1
r þ k
@w0
@r @
2
w0
@r2
!2
279 280
w1¼C
2 2ð1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk2M2
p
Þþk2M2ð2þ3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2
p
Þ
ðkþrÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk 2
M 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2MIn2
p
ð8þ9k2M2Þ
þC
2ð2ð1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2
p
Þþk2M2 2þ3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2 p
ÞðkþrÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk 2 M 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2M2
p
ð8þ9k2M2Þ
þðkþrÞ 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk 2 M 2 p
B1
1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þk2
M2
1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk 2 M 2 p
B2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þk2
M2
r2
282 Fig 6.wvia change in M when E1= 0.003, E2= 0.003, E3= 0.01, t = 0, k = 5.0, e = 0.1, We = 0.03.(a)M = 5.0.(b)M = 7.0.(c)M = 9.0
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 6283 where the values for Ci,s (i = 1–4)and Bi,s(i = 1–4) are obtained with
284 the help of MATHEMATICA ( Fig 1 ).
285 Results and discussion
286 First of all the velocity is analyzed here Impacts of different
287 parameters namely the wall properties (E1, E2, E3), magnetic field
288 (M), Weissenberg number (We) and curvature parameter k on the
289
axial velocity u and stream function w have been examined.
290
Fig 2 (a) and 2(b) are plotted to see the Hartman number effect
291
on the axial velocity in the curved and straight channels
respec-292
tively It can be seen that larger magnetic field decreases axial
293
velocity in view of the resistive nature of Lorentz force It is also
294
noted that axial velocity u in case of curved channel is tilted
295
towards left Fig 3 (a) and (b) elucidate the influence of
296
Weissenberg number on the axial velocity It is noted that the axial
297
velocity shows dual behavior for Weissenberg number By
enhanc-Fig 7.wvia change in k when E1= 0.003, E2= 0.003, E3= 0.01, t = 0, M = 4.0, e = 0.1, We = 0.03.(a)k = 3.0.(b)k = 5.0.(c)k? 1
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 7298 ing Weissenberg number the velocity decreases in the region
299 ½1; 0 whereas it increases in the region ½0; 1 Fig 4 (a) and (b)
300 represent the influence of wall properties on the axial velocity.
301 We can see that with the increase in E1 and E2 the velocity
302 enhances The fact behind this is the less resistance offered to
303 the flow due to the elastance of wall which cause an enhancement
304 in the fluid velocity Decrease in the axial velocity is noticed when
305 E3increases Here damping is responsible for decrease in the axial
306 velocity Fig 5 represents the behavior of curvature parameter k on
307
the axial velocity Mixed behavior of curvature parameter on the
308
axial velocity is observed Axial velocity decreases near the lower
309
wall while it increases in rest part of channel It is also noted that
310
for very large values of curvature parameter k the curve becomes
311
symmetric Moreover in the straight channel the velocity is more
312
than in the curved channel.
313
Fig 6 (a)–(c) have been displayed for the behavior of Hartman
314
number on the stream function It can be seen that the size of
315
trapped bolus increases for larger Hartman number Fig 7 (a)–(c)
Fig 8.wvia change in We when E1= 0.003, E2= 0.003, E3= 0.01, t = 0, k = 5.0, e = 0.1, M = 7.0.(a)We = 0.01.(b)We = 0.02.(c)We = 0.03
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 8316 elucidate the impact of curvature parameter on the stream
func-317 tion Increase in the size of trapped bolus is noticed when the
cur-318 vature parameter attains larger values Effect of different values of
319 Weissenberg number on the stream function is shown through
320 Fig 8 (a)–(c) It is observed that by enhancing the values of
Weis-321 senberg parameter the size of trapped bolus decreases The effect
322 of wall properties on the stream function is examined through
323 the Fig 9 (a)–(d) We can see that by increasing the values of E1
324 and E2the size of the trapped bolus increases whereas it decreases
325 for an increase in E3.
326
Conclusions
327
Influences of wall properties and radial magnetic field on the
328
peristaltic flow of Williamson fluid in a curved channel is modeled
329
and analyzed Main points of presented analysis are listed below.
330
Axial velocity decreases with increase via Hartman number and
331
E3whereas it enhances for E1and E2.
332
Dual behavior of axial velocity is noticed for Weissenberg
num-333
ber and curvature parameter.
Fig 9.wvia change in E1, E2, E3when t = 0, k = 5.0, e = 0.1, We = 0.03, M = 7.0.(a)E1= 0.7, E2= 0.4, E3= 0.2.(b)E1= 0.9, E2= 0.4, E3= 0.2.(c)E1= 0.7, E2= 0.6, E3= 0.2.(d)E1= 0.7, E2= 0.4, E3= 0.5
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls
Trang 9334 Axial velocity in the curved channel is slanted towards left
335 when compared with the straight channel However the axial
336 velocity in straight channel is symmetric.
337 Size of trapped bolus increases for larger curvature parameter k,
338 Hartman number M, E1and E2whereas it decreases against E3.
339 Viscous fluid results are obtained when We = 0.
340
341 Uncited references
342 [44–46]
343 References
344 [1] Latham TW Fluid motion in a peristaltic pump [MS Thesis] Cambridge,
346 [2] Shapiro AH, Jafrin MY, Weinberg SL Peristaltic pumping with long
347 wavelengths at low Reynolds number J Fluid Mech 1969;37:799–825.
348 [3] Reddy MG, Reddy RV Influence of Joule heating on MHD peristaltic flow of a
349 nanofluid with compliant walls Procedia Eng 2015:1002–9.
350 [4] Asghar S, Hussain Q, Hayat T Peristaltic motion of reactive viscous fluid with
351 temperature dependent viscosity Math Comput App 2013;18:198–220.
352 [5] Asghar Z, Ali N Streamline topologies and their bifurcations for mixed
353 convective peristaltic flow AIP Adv 2015;5:097142
354 [6] Mekheimer KhS, Abd Elmaboud Y, Abdellateef AI Particulate suspension flow
355 induced by sinusoidal peristaltic waves through eccentric cylinders: thread
356 annular Int J Biomath 2013;6:1350026
357 [7] Hayat T, Abbasi FM, Ahmad B, Alsaedi A MHD mixed convection peristaltic
358 flow with variable viscosity and thermal conductivity Sains Malays
360 [8] Riaz A, Nadeem S, Ellahi R, Akbar NS The influence of wall flexibility on
361 unsteady peristaltic flow of Prandtl fluid in a three-dimensional rectangular
362 duct Appl Math Comput 2014;241:389–400.
363 [9] Kothandapani M, Prakash J Effect of thermal radiation parameter and
364 magnetic field on the peristaltic flow of Williamson nanofluids in a tapered
365 asymmetric channel Int J Heat Mass Transfer 2015;51:234–45.
366 [10] Abbasi FM, Hayat T, Ahmad B Peristaltic flow in an asymmetric channel with
367 convective boundary conditions and Joule heating J Cent South Univ
369 [11] Hayat T, Rafiq M, Ahmad B Combined effects of rotation and thermal radiation
370 on peristaltic transport of Jeffrey fluid Int J Biomath 2015;8:21.
371 [12] Hameed M, Khan AA, Ellahi R, Raza M Study of magnetic and heat transfer on
372 the peristaltic transport of a fractional second grade fluid in a vertical tube.
373 Eng Sci Technol Int J 2015;18:496–502.
374 [13] Hayat T, Bibi S, Rafiq M, Alsaedi A, Abbasi FM Effect of inclined magnetic field
375 on peristaltic flow of Williamson fluid in an inclined channel with convective
376 conditions J Magn Magn Mater 2016;401:733–45
377 [14] Ali N, Javed T Flow of a Giesekus fluid in a planar channel due to peristalsis Z
378 Naturforsch 2013;68a:515–23
379 [15] Ellahi R, Hussain F Simultaneous effects of MHD and partial slip on peristaltic
380 flow of Jeffrey fluid in a rectangular duct J Magn Magn Mater
382 [16] Abbasi FM, Hayat T, Ahmad B, Chen GQ Peristaltic motion of non-Newtonian
383 nanofluid in an asymmetric channel Z Naturforsch 2014;69a:451–61
384 [17] Mekheimer KhS, Komy SR, Abdelsalamd SI Simultaneous effects of magnetic
385 field and space porosity on compressible Maxwell fluid transport induced by a
386 surface acoustic wave in a microchannel Chin Phys B 2013;22:124702.
387 [18] Noreen S, Hayat T, Alsaedi A, Qasim M Mixed convection heat and mass
388 transfer in the peristaltic flow with chemical reaction and inclined magnetic
389 field Indian J Phy 2013;87:889–96.
390 [19] Abd-Alla AM, Abo-Dahab SM, Kilicman A Peristaltic flow of a Jeffrey fluid
391 under the effect of radially varying magnetic field in a tube with endoscope J
392 Magn Magn Mater 2015;384:79–86.
393 [20] Ramesha K, Devakar M Magnetohydrodynamic peristaltic transport of couple
394 stress fluid through porous medium in an inclined asymmetric channel with
395 heat transfer J Magn Magn Mater 2015;394:335–48
396 [21] Sinha A, Shit GC, Ranjit NK Peristaltic transport of MHD flow and heat transfer
397
in an asymmetric channel: Effects of variable viscosity, velocity-slip and
398 temperature jump Alex Eng J 2015;54:691–704
399 [22] Mekheimer KhS, Salem AM, Zaher AZ Peristaltically induced MHD slip flow in
400
a porous medium due to a surface acoustic wavy wall J Egypt Math Soc
401 2014;22:143–51
402 [23] Gad NS Effects of Hall currents on peristaltic transport with compliant walls
403 Appl Math Comput 2014;235:546–54
404 [24] Abbasi FM, Hayat T, Alsaedi A Numerical analysis for MHD peristaltic
405 transport of Carreau-Yasuda fluid in a curved channel with Hall effects J
406 Magn Magn Mater 2015;382:104–10
407 [25] Hina S, Mustafa M, Hayat T, Alsaadi FE Peristaltic motion of third grade fluid in
408 curved channel App Math Mech-Engl Ed 2014;35:73–84
409 [26] Hayat T, Abbasi FM, Ahmad B, Alsaedi A Peristaltic transport of
Carreau-410 Yasuda fluid in a curved channel with slip effects Plos One 2014;9:e95070
411 [27] Ali N, Sajid M, Abbas Z, Javed T Non-Newtonian fluid flow induced by
412 peristaltic waves in a curved channel Eur J Mech 2010;29:387–94
413 [28] Ali N, Sajid M, Javed T, Abbas Z Heat transfer analysis of peristaltic flow in a
414 curved channel Int J Heat Mass Transfer 2010;53:3319–25
415 [29] Hina S, Mustafa M, Hayat T, Naif Alotaibi D On peristaltic motion of
416 pseudoplastic fluid in a curved channel with heat/mass transfer and wall
417 properties Appl Math Comput 2015;263:378–91
418 [30] Sato H, Kawai T, Fujita T, Okabe M Two dimensional peristaltic flow in curved
419 channels Trans Jpn Soc Mech Eng B 2000;66:679–85
420 [31] Ali N, Sajid M, Hayat T Long wavelength flow analysis in a curved channel Z
421 Naturforsch A 2010;65:191–6
422 [32] Hayat T, Javed M, Hendi AA Peristaltic transport of viscous fluid in a curved
423 channel with complaint walls J Heat Mass Transfer 2011;54:1615–21
424 [33] Hayat T, Hina S, Hendi AA, Asghar S Effect of wall properties on the peristaltic
425 flow of a third grade fluid in a curved channel with heat and mass transfer Int J
426 Heat Mass Transfer 2011;54:5126–36
427 [34] Hina S, Hayat T, Mustafa M, Alsaedi A Peristaltic transport of pseudoplastic
428 fluid in a curved channel with wall properties and slip conditions Int J
429 Biomath 2014;7:16
430 [35] Hayat T, Tanveer A, Alsaadi F, Alotaibi ND Homogeneous-hetrogeneous
431 reaction effects in peristalsis through curved channel geometry AIP Adv
432 2015;5:067172
433 [36] Hayat T, Farooq S, Alsaedi A, MHD peristaltic flow in a curved channel with
434 convective condition J Mech doi:http://dx.doi.org/10.1017/jmech.2016.76
435 [37] Hayat T, Tanveer A, Alsaadi F, Mousa G Impact of radial magnetic field on
436 peristalsis in curved channel with convective boundary conditions J Magn
437 Magn Mater 2016;403:47–59
438 [38] Hina S, Mustafa M, Hayat T, Alsaedi A Peristaltic transport of Powell-Eyring
439 fluid in a curved channel with heat/mass transfer and wall properties Int J
440 Heat Mass Transfer 2016;101:156–65
441 [39] Ali N, Javid K, Sajid M, Zaman A, Hayat T Numerical simulations of Oldroyd
8-442 constant fluid flow and heat transfer in a curved channel Int J Heat Mass
443 Transfer 2016;94:500–8
444 [40] Tanveer A, Hayat T, Alsaadi F, Alsaedi A Mixed convection peristaltic flow of
445 Eyring-Powell nanofluid in a curved channel with compliant walls Comput
446 Biol Med 2017;82:71–9
447 [41] Hayat T, Quratulain A, Alsaedi M Rafiq, Ahmad B Joule heating and thermal
448 radiation effects on peristalsis in curved configuration Results Phys
449 2016;6:1088–95
450 [42] Hayat T, Farooq S, Ahmad B, Alsaedi A Homogeneous-heterogeneous reactions
451 and heat source/sink effects in MHD peristaltic flow of micropolar fluid with
452 Newtonian heating in a curved channel J Mol Liq 2016;223:469–88
453 [43] Hayat T, Farooq S, Alsaedi A, Ahmad B Numerical study for Soret and Dufour
454 effects on mixed convective peristalsis of Oldroyd 8-constants fluid Int J
455 Therm Sci 2017;112:68–81
456 [44] Turkyilmazoglu M Magnetic field and slip effects on the flow and heat transfer
457
of stagnation point Jeffrey fluid over deformable surfaces ZNA
458 2016;71:549–56
459 [45] Turkyilmazoglu M Determination of the correct range of physical parameters
460
in the approximate analytical solutions of nonlinear equations using the
461 Adomian decomposition method Mediterranean J Math 2016;13:4019–37
462 [46] Turkyilmazoglu M Three dimensional MHD flow and heat transfer over a
463 stretching/shrinking surface in a viscoelastic fluid with various physical
464 effects Int J Heat Mass Transfer 2014;78:150–5
465
Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls