Results in Physics (2016) 805–810 Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics Peristaltic flow of Sutterby fluid in a vertical channel with radiative heat transfer and compliant walls: A numerical study T Hayat a,b, Hina Zahir a,⇑, M Mustafa c, A Alsaedi b a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan b a r t i c l e i n f o Article history: Received 16 September 2016 Accepted 19 October 2016 Available online 24 October 2016 Keywords: Mixed convection Radiation effect Convective conditions Compliant walls a b s t r a c t Current study aims to investigate the impact of compliant walls on peristaltically induced flow of Sutterby fluid in a vertical channel The flow is subjected to uniform magnetic field in the transverse direction In addition heat transfer effects characterized by convective boundary conditions are considered Energy equation contains heat dissipation and radiative heat transfer effects Problem formulation is developed by negligible inertial effects and long wavelength approximation Shooting method based NDSolve of the software Mathematica has been applied to evaluate numerical results of stream function, temperature and heat transfer coefficient We found that velocity and temperature distributions in Sutterby fluid are greater than of viscous fluid Radiation parameter tends to reduce the fluid temperature inside the channel Ó 2016 The Authors 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/) Introduction Peristaltic motion has wide coverage in engineering and medical industry Peristalsis is a process in which physiological fluid (or biofluid) is propelled by means of sinusoidal waves advancing axially along the length of tube/channel In physiological world, peristalsis works as vasomotion of small blood vessels, bile in bile duct, transport of food from the mouth through oesophagus and chyme movement in the entire gastrointestinal tract Further, blood pump in heart lung machine, design of several modern medical devices, locomotion of worms and translocation of water in tall trees are also due to peristaltic principle In urinary track, peristalsis occurs due to involuntary contractions of the ureter walls which transport urine from kidneys to bladder In esophagus, peristalsis is due to the wave-like motions of smooth muscles which carry the food to stomach In view of such biomedical applications, reasonable attention has been devoted to explore the peristaltic mechanism of non-Newtonian fluids through planar channel Peristalsis through magnetohydrodynamics (MHD) has been widely addressed research topic for several years because of its occurrence in applications such as magnetic drug targeting, treatment of cancerous tissues, treatment of nuclear fuel debris, blood pumps, bleeding reduction during surgeries etc Magnet force also serves ⇑ Corresponding author E-mail address: hinazahir1@gmail.com (H Zahir) as pump for performing cardiac operations for arterial flow Some recent attempts involving peristaltic motion of Newtonian and non-Newtonian fluids under the influence of magnetic field can be found in [1–10] Buoyancy forces stemming from cooling/heating of surfaces alter the flow and thermal field and thereby heat transfer characteristics of the process The study of mixed convective flow with MHD in peristalsis has gain enormous attention due to its extensive range of applications in industry and technological fields In drying technologies, hemodialysis, chemical processing equipment, cooling of electronic devices, solar energy collectors, heat exchangers, nuclear reactors and oxygenation, the analysis of convective heat transfer in peristalsis cannot be undervalued Excellent articles related to mixed convection MHD flow have been presented by many authors [11–21] Further, thermal radiation effects in the flow are vital in applications involving plasma at high temperature, heat conduction in tissues, MHD generators, gas turbines and nuclear plants Heat transfer by means of radiation is also effective in laser surgery, cryosurgery, destruction of cancer tumors, processing of fusing of metals in an electrical furnace, in the polymer processing industry controlling heat transfer and prediction of blood flow rate Inspired by the promising applications mentioned above, our main objective here is to analyze the mixed convection flow of Sutterby fluid in a planar channel having compliant wall properties Heat transfer due to viscous dissipation and radiation is analyzed The relevant equations are first modeled under adequate http://dx.doi.org/10.1016/j.rinp.2016.10.015 2211-3797/Ó 2016 The Authors 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/) 806 T Hayat et al / Results in Physics (2016) 805–810 assumptions and then solved for numerical solutions Graphical results for axial velocity, temperature and heat transfer coefficient are prepared and analyzed for broad range of embedded parameters Mathematical formulation Here peristaltic flow of incompressible Sutterby fluid in a vertical symmetric channel of half thickness d is considered The sinusoidal wave of amplitude a and wavelength k with constant speed c propagates axially across the flexible walls of channel in time t Let uðx; y; tÞ be the velocity component in the axial direction and v ðx; y; tÞ in transverse direction The wave shape is represented by the following equation: 2p y ẳ ặgx; tị ẳ ặ d ỵ a sin x ctị ; k 1ị The extra stress tensor S for Sutterby fluid is given by " S¼ #m _ l sinhÀ1 fBng Bn_ A1 ; where A; B are material constants, ð2Þ l the fluid dynamic viscosity, ( 2 2 )# mB2 @u @u @ v @v @u Sxx ẳ ỵ ỵ2 2 ỵ ; ð11Þ @y @x @y @x @x " ( ) # 2 l mB2 @u @u @ v @v @u @ v ỵ ỵ ỵ ỵ2 ; 12ị Sxy ẳ @x @y @x @y @x @y " ( 2 2 )# l mB2 @u @u @ v @v @v ỵ ỵ ỵ2 : 13ị Syy ¼ @x @y @x @y @y l " No slip conditions for this flow are: u ¼ at y ẳ ặg; Convective boundary condition for heat transport are as follows: @T ¼ À hðT À T Þ at y ¼ g; @y @T k ¼ À hT Tị at y ẳ g; @y r V ẳ 0; q 3ị dV ẳ rp ỵ div S ỵ J B ỵ qbT gT T ị; dt qcp dT ẳ kr2 T r qr ỵ L S; dt 4ị where k and ð5Þ ð6Þ Ã r are the mean absorption coefficient and Stefan- Boltzmann constant respectively In Eq (6), T we expand in Taylor series about T and then neglect square and higher powers of ðT À T Þ by assuming that temperature differences within the flow are sufficiently small Eqs (3)–(5) can be expressed in the component form as: @u @ v ỵ ẳ 0; ð7Þ @x @y @u @u @u @p @Sxx @Sxy ẳ ỵ q ỵu ỵv ỵ rB20 u þ qbT gðT À T Þ; ð8Þ @t @x @y @x @x @y @v @v @v @p @Sxx @Sxy ẳ ỵ q 9ị ỵu ỵv ỵ rB20 v ; @y @x @t @x @y @y " # @T @T @T @T @ T @ @ qr ịx qr ịy ỵu ỵv ẳk qcp ỵ @t @x @y @x2 @y2 @x @y @u @u @ v @v ỵ ỵ Sxx þ Sxy þ Syy ; ð10Þ @x @y @x @y where ð16Þ where h is the heat transfer coefficient and boundary condition for compliant wall is as under: " # @3 @3 @ s ỵ m1 þd g @x @t@x @x@t ¼ Àq @u @Sxy @Sxx ỵ ỵ ỵ qbT gT T ị rB20 u at y ẳ ặg; @t @y @x ð17Þ where s is the elastic tension in the membrane, mÃ1 the mass per unit area, d the viscous damping coefficient and r the electrical conductivity In order to non-dimensionalize the problem, we introduce the following variables w x y ct u ; xà ¼ ; yà ¼ ; tà ¼ ; uà ¼ ; cd k d k c v g T À T d p v à ¼ ; gà ¼ ; h ¼ ; pà ¼ ; c d T0 lck Rà lc à S ; k ¼ ; Sij ¼ d ij d wà ¼ where q is the density, p the pressure, B ẳ 0; 0; B0 ị the magnetic field strength, J the current density, bT the coefficient of thermal expansion, g the gravitational acceleration, T the fluid temperature, T denotes the convective temperature, cp the specific heat, k the thermal conductivity, L the velocity gradient and qr denotes the radiative heat flux given by 4rà qr ¼ À à rT ; 3k 15ị k t A1 ẳ gradVị þ ðgradVÞ the first Rivlin Erickson tensor and qffiffiffiffiffiffiffiffiffi2ffi tracA1 n_ ¼ Thus, two-dimensional flow and heat transfer of Sutterby fluid are governed by the following equations: ð14Þ ð18Þ in which the stream function in terms of velocity components is expressed below u¼ @w ; @y v ¼ Àd @w : @x ð19Þ Taking into account the assumptions of long wavelength of the peristaltic wave compared to half width of channel and negligible inertial effects, we arrive at the following equations @p @Sxy ỵ ỵ Grh H2 wy ẳ 0; @x @y 20ị @p ẳ 0; @y ỵ RnPrị 21ị @2h ỵ Brwyy Sxy ẳ 0; @y2 22ị Sxy ẳ f1 bw2yy gwyy ; 23ị B20 r where Gr ẳ qblT gT is the Grashof number, H2 ¼ l the Hartman c à lcp 16r number, Rn ¼ 3kk à lcp T the radiation parameter, Pr ¼ k the Prandtl 2 c the Sutterby number, Br ¼ EPr the Brinkman number and b ¼ mB 6d2 fluid parameter The boundary conditions (14)–(17) in view of variables (18) become wy ¼ at y ẳ ặg; 24ị T Hayat et al / Results in Physics (2016) 805810 @h ỵ Bih ¼ at y ¼ g; @y ð25Þ @h À Bih ẳ at y ẳ g; @y 26ị 807 " # @3 @3 @2 @ E1 ỵ E2 þ E3 g ¼ Sxy @y @x @x@t @x@t ỵ Grh H2 wy at y ẳ ặg In above equations E1 ¼ À ks3dlc ; E2 ¼ m1 cd3 27ị and E3 ẳ dlkd2 are the non-dimensional elasticity and damping parameters, Bi ¼ hd the k Biot number and ¼ da the amplitude ratio parameter Through Eqs (20) and (21) one obtains the following: @ @Sxr ỵ Grh H2 wy ẳ 0: @y @y k3 l Fig Variation in velocity u for wall parameters E1 ; E2 ; E3 when ¼ 0:2; t ¼ 0:01; x ¼ 0:2; H ¼ 0:1; Br ¼ 0:5; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5, Gr ¼ 4, and Rn ¼ ð28Þ Numerical results and discussion The governing nonlinear Eqs (22) and (28) subject to the conditions (24)–(27) have been solved for the numerical solutions by built in function NDSolve of Mathematica This routine is based on the standard shooting method with fourth order Runge–Kutta integration technique Velocity distribution The results presented in Figs 1–4 reveal variation in velocity profile uðyÞ with the change in Grashof number Gr, compliant wall parameters E1 ; E2 , and E3 , Hartman number H and Sutterby fluid parameter b As the Grashof number Gr enlarges, the velocity distribution increases inside the channel (see Fig 1) Physically, an increase in Gr implies larger buoyancy forces which accelerates the flow in axial direction Fig shows that axial velocity uðyÞ is an increasing function of tension and mass characterizing parameters E1 and E2 whereas velocity decays when damping parameter E3 is increased It is understandable since fluid experiences larger resistance to flow in case of stronger wall damping Moreover, larger values of E1 implies a reduction in wall tension due to which velocity increases In Fig 3, the behavior of magnetic field effect on axial velocity uðyÞ is portrayed Hartman number H is defined as the ratio of magnetic force to viscous force As H enlarges, the magnetic force normal to the flow direction dominates the viscous effect Consequently, velocity in the axial direction decreases upon increasing the Hartman number H Fig shows the influence of Fig Variation in velocity u for Hartman number H when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2; Br ¼ 0:9; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and Rn ¼ Fig Variation in velocity u for Sutterby fluid parameter b when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Br ¼ 0:8; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and Rn ¼ Sutterby fluid parameter on the velocity profile uðyÞ Flow appears to accelerate along the axial direction when bigger values of b are employed Temperature distribution Fig Variation in velocity u for Grashof number Gr when E1 ¼ 0:04, E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2; H ¼ 0:1; Br ¼ 0:5; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5, and Rn ¼ Figs 5–11 are sketched to observe the physical effects of embedded parameters on temperature profile hðyÞ From Fig it is observed that temperature his inversely proportional to the radiation parameter Rn Temperature is found to be maximum at the centre of planar channel Fig indicates that temperature 808 T Hayat et al / Results in Physics (2016) 805–810 Fig Variation in temperature h for Radiation Rn when E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, Br ¼ 0:9; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and H ¼ 0:1 Fig Variation in temperature h for wall parameters E1 ; E2 ; E3 when ¼ 0:2; t ¼ 0:01; x ¼ 0:2; Br ¼ 0:7; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4; H ¼ 0:1 and Rn ¼ Fig Variation in temperature h for Sutterby fluid parameter b when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2; Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and H ¼ 0:1 Fig Variation in temperature h for Hartman number H when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and b ¼ 0:02 Fig Variation in temperature h for Brinkman number Br when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and b ¼ 0:02 Fig 10 Variation in temperature h for Prandtl number Pr when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Br ¼ 0:9; Gr ¼ 4, and b ¼ 0:02 Fig 11 Variation in temperature h for Biot number Bi when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Pr ¼ 1:5; Br ¼ 0:9; Gr ¼ 4, and b ¼ 0:02 henlarges by increasing tension and mass characterizing parameters while damping parameter cools down the fluid inside the channel Fig reveals that as Sutterby fluid parameter increases, temperature distribution increases It means that temperature in Sutterby fluid is larger than that in viscous fluid In Fig 8, it is clear that magnetic force has important role in reducing fluid temperature within the channel Fig preserves the behavior of Brinkman number Br on temperature Brinkman number Br represents the intensity of dissipation effects Since temperature at the wall is controlled convectively, therefore heat transfer is only considered due to heat dissipation effects Due to this reason, we expect temperature to be higher in case of larger Brinkman number, as also observed in Fig Fig 10 shows that temperature hhas direct T Hayat et al / Results in Physics (2016) 805–810 809 relationship with Prandtl number Pr Fig 11 indicates that temperature decreases when strength of convective heating is enhanced Heat transfer coefficient In Figs 12–16, we plot heat transfer coefficient ZðxÞ for different values of embedded parameters Interestingly, ZðxÞ has oscillatory profile across channel walls Fig 12 indicates that buoyancy forces reduce the heat transfer rate from the channel walls The effect of compliant wall parameters E1 ; E2 and E3 on ZðxÞ appears to be similar to that on the temperature profile (see Fig 13) It can be concluded through Fig 14 that radiative heat transfer opposes the heat flow from the channel walls Moreover this effect is preserved when Hartman number is varied from H ¼ to H ¼ 0:9 Fig 15 Variation in heat transfer coefficient Z for Hartman number H when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and b ¼ 0:02 Fig 12 Variation in heat transfer coefficient for Grashof number Gr when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:02; t ¼ 0:01; H ¼ 0:1; Br ¼ 0:9; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5, and Rn ¼ Fig 16 Variation in heat transfer coefficient Z for Sutterby fluid parameter when E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01; ¼ 0:2; t ¼ 0:01; H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and b ¼ 0:02 Conclusions In this attempt, we presented numerical solutions for peristaltically driven flow of Sutterby fluid in planar channel Radiative heat transfer and wall properties effects are preserved in the mathematical model The key observations of this study are listed below: Fig 13 Variation in heat transfer coefficient Z for wall parameters E1 ; E2 ; E3 when ¼ 0:02; t ¼ 0:01; Br ¼ 0:3; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4; H ¼ 0:1 and Rn ¼ Axial velocity is directly proportional to both Sutterby fluid parameter b and Grashof number Gr Magnetic force acting normal to the flow resists the fluid motion due to which velocity decreases in the axial direction Magnetic field effect tends to reduce fluid temperature while heat transfer coefficient enlarges when stronger magnetic force is employed Velocity and temperature distributions increase when either wall tension reduces or wall mass per unit area enlarges The influence of wall damping parameter E3 on the solutions is qualitatively opposite to that of E1 and E2 Temperature rises and heat flux from the wall diminishes when convective heating at the walls is intensified Role of radiation parameter Rn is to reduce temperature and wall heat flux References Fig 14 Variation in heat transfer coefficient Z for Radiation Rn when E1 ¼ 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