MHD Flow of Thixotropic Fluid with Variable Thermal Conductivity and Thermal Radiation

Transcription

1 Article MHD Flow o Thixotropic Fluid with Variable Theral Conductivity and Theral Radiation Tasawar HAYAT Sabir Ali SHEHZAD * and Salee ASGHAR Departent o Matheatics Quaid-i-Aza University Islaabad 44 Pakistan Departent o Matheatics CIIT ChakShahzad Park Road Islaabad 44 Pakistan * Corresponding author; e-ail: ali_au7@yahoo.co Received: 8 Deceber Revised: January Accepted: January Abstract An analysis has been carried out to exaine the two-diensional and agnetohydrodynaic MHD low o thixotropic luid over a stretched surace. The theral radiation eect in the heat transer is considered when the theral conductivity is not constant. Conservation o ass oentu and energy leads to the governing partial dierential euations o the present study. The resulting euations are solved or convergent series solutions. Nuerical values o the skin-riction coeicient are presented and analyzed. Keywords: MHD low thixotropic luid theral radiation variable theral conductivity stretching surace Introduction In recent ties the dynaics o non- Newtonian luids has becoe ore and ore iportant or industrial and engineering applications. The applications o non-newtonian luids are signiicant in icro/nano luidic bioluid and heatology bacteriology building and conectionery industries cheical/petroleu engineering and ineral processing industries bubble coluns polyer solution and ood industries []. Due to diversity in behavior o the stress in the oentu euations several odels are suggested. The resulting euations or non- Newtonian luids are coplex higher order and ore nonlinear than the well known Navier-Stokes euations or viscous luids. Soe salient eatures o these luids like shear thinning/shear thickening and relaxation and retardation tie eects have already been studied in recent works [-9]. The boundary layer low over a stretching sheet with heat transer has iportance in the polyer industry. In particular non-newtonian luids are uite coon in polyer sheet extrusion ro dyes optical ibers anuacturing processes drawing o plastic ils and any other processes. In industrial processes involving such luids the uality o the inal product is closely associated with the rate o cooling. The uality o such a sheet is aected by heat transer between the sheet and the luid [-5]. Many aterials in our daily lie including drilling uds cosetic products clay suspension etc becoe less viscous with tie. To explore the rheological properties o such types o aterials a thixotropic luid odel is ore appropriate. A hysteresis inluence is noticed when the shear rate o the thixotropic luid is raped up or down. Recently Sadei et al. [6] investigated the Blasius low o thixotropic luid. They analyzed the results nuerically using the Newtonian-Kontorovitch ethod. The agnetohydrodynaic MHD low with heat transer are very iportant in agnetohydrodynaic power generators and accelerators cooling o nuclear reactors crystal growth etc. Several investigators analyzed the MHD low under various conditions. Hayat et al. [7] reported the transient low o viscoelastic luid in the presence o a agnetic ield. Magnetohydrodynaic low o viscous luid near Walailak J Sci & Tech ; : 9-4.

2 MHD Flow o Thixotropic Fluid a stagnation point was addressed by Rashidi and Erani [8]. They also considered the heat transer phenoenon and analyzed the results analytically. Turkyilazoglu [9] studied the tie-dependent MHD low o viscous luid by taking variable viscosity. MHD transient low o dusty luid was investigated by Makinde and Chinyoka []. The Navier slip condition and variable physical properties are also discussed in this study. Abbasbandy and Hayat [] reported the MHD Falkner-Skan low o viscous luids by adopting the hootopy analysis ethod. Very recently Soret and Duour the eects on MHD peristaltic low o viscous luid were exained by Hayat et al. []. They obtained the results by a perturbation techniue. All the above investigations have been carried out by considering constant theral conductivity. But it has now been proved that the theral conductivity varies linearly with teperature ro to 4 F []. Having such in ind Vyas and Rai [4] investigated the boundary layer low o viscous luid with variable theral conductivity. Furtherore the theral radiation eects are iportant in any industrial processes which involve heat transer ro nuclear uel debris underground disposal o radioactive waste aterial storage o ood stus and any others. Motivated by such acts the present work is proposed to analyze the theral radiation eects on MHD low o thixotropic luid with variable theral conductivity. The structure o this paper is as ollows. In section the atheatical odel is orulated. Section addresses the series solutions by adopting the hootopy analysis ethod HAM [5-]. Convergence analysis and discussion o the results are presented in section 4. Section 5 has the inal rearks. Basic euations Consider a Cartesian coordinate syste such that x-axis is along the stretched surace and y-axis is perpendicular to it. We consider the agnetohydrodynaic boundary layer low o thixotropic luid. A constant agnetic ield o strength B is exerted in the y-direction. The low is steady and the agnetic Reynolds nuber is taken to be sall so that an induced agnetic ield is negligible in coparison to the applied agnetic ield. The teperature o the surace T w is greater than the teperature o abient luid T. Heat transer analysis is set up in the presence o theral radiation and with variable theral conductivity. Taking into account the Rosseland s approxiation or radiative heat lux [4] ass oentu and energy conservations are sipliied as ollows: u v x y u u u 6R u u v ν x y y ρ y 4R ρ u σ B y ρ u u u u u v y y xy y u u u u u v u u v y xy y y xy u y y T T T 6σT T ρ Cp u v k x y y y k y Walailak J Sci & Tech ;

3 MHD Flow o Thixotropic Fluid where uv are the velocity coponents parallel to the x- and y-axes. R and R are the constants v the dynaic viscosity o the luid ρ the density o luid σ* the electrical conductivity T the teperature k the variable theral conductivity C p the speciic heat σ the Stean-Boltzann constant and k* the ean absorption coeicient. Es. - have to be solved subject to the boundary conditions α u u cx v v T T x T Dx at y w w u T T as y 4 in which c is the stretching rate. We introduce the ollowing change o variables c T T u cx v cν y ν T T w 5 where prie denotes the dierentiation with respect to and we consider T w x T Dx α at and variable theral conductivity k k [ ε] k is the luid ree strea conductivity and ε is given by kw k. ε 6 k The incopressibility condition is autoatically satisied by E. 5 and Es. -4 becoe 4 K x K x M iv 7 4 ε ε N Pr[ α ] 8 S at as. 6R c x Here the non-newtonian paraeters are Hartan nuber the radiation paraeter. K x and K x M σ B / ρc ρν S v / νc the suction paraeter The diensionless or o the skin riction coeicient is Re 4 4R c x ρν ρcpν Pr k the Prandtl nuber and 9 the N 4σT kk K / 6[ ]. / x C Walailak J Sci & Tech ;

4 MHD Flow o Thixotropic Fluid Series solutions The hootopic solutions or and in a set o base unctions k { exp n k n } can be expressed as n k k k a a n exp n n k k b k exp n n k where a and b are the coeicients. The appropriate initial approxiations and auxiliary linear n k n operators or the considered probles are S exp exp 4 L L 5 subject to the ollowing properties L 6 C Ce Ce L C4e C e. 5 In which C i i 5 denote the arbitrary constants and the zeroth order deoration probles are expressible in the or L ˆ ; N ˆ ; 7 L ˆ ; N ˆ ; ˆ ; 8 ˆ ˆ ˆ ; S ; ; ˆ; and ˆ ;. Here shows ebedding paraeter operators N and N are given by and 9 the non-zero auxiliary paraeters and the nonlinear Walailak J Sci & Tech ;

5 MHD Flow o Thixotropic Fluid Walailak J Sci & Tech ; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ] ˆ [ 4 4 ˆ ˆ ˆ ˆ 4 ˆ ˆ ˆ ˆ x K M x K N. ˆ ˆ Pr ˆ ˆ Pr ˆ ˆ ˆ ˆ 4 ] ˆ ˆ [ α ε ε N N For and one has ; ˆ and ; ˆ ; ˆ and ; ˆ and when increases ro to then and vary ro to and to. Eploying the Taylor's series expansions we have ;! 4. ;! 5 Clearly the convergence o series E. 7 and E. 8 is closely associated with h and. h The values o h and h are chosen such that the series E. 7 and E. 8 converge at. Hence 6. 7 I we denote the special solutions and then the general solutions and are e C e C C e C e C 9

6 MHD Flow o Thixotropic Fluid Convergence analysis and discussion We recall that the auxiliary paraeters and are useul in controlling and adjusting the convergence o series solutions. We draw the curves at 9 th order o approxiation or the eaningul values o and. It is noticed through Figures and that the adissible values o and are.7. 5 and Thus our series solutions converge in the whole region o or. 5 and. 7. To see the behaviors o dierent eerging paraeters we have plotted Figures -4 or the velocity ield and teperature proile. Variations o K K M and S on the velocity ield are depicted in the Figures -6. Both the non-newtonian paraeters K and K have siilar eects on the velocity ield in a ualitative sense. By increasing K and K both the velocity and boundary layer thickness increase. It is noticed here that K is negative and K is positive. That is why both paraeters have sae behaviors. I we take the value o K to be negative then it has opposite behavior as copared to K. Eects o the Hartan nuber can be seen in Figure 5. The Hartan nuber opposes the low because the applied agnetic ield noral to the low direction induces the drag in ters o Lorentz orce due to which the luid velocity and boundary layer thickness decreases. Figure 6 deonstrates the variation o S on the velocity. Suction is an agent that leads to a decrease in luid low due to which the luid velocity decreases by increasing S. Figures 7-4 represent the eects o dierent paraeters on the teperature proile. Figures 7 and 8 illustrate that teperature and theral boundary layer thickness are decreasing unctions o K and K. Through coparative study o Figures 4 7 and 8 we ound that the variation in velocity is ore signiicant than the variation in teperature. Hartan nuber increases the teperature and theral boundary layer thickness Figure 9. The eects o the Hartan nuber on velocity and teperature are uite opposite. Figure illustrates that suction decreases the teperature proile. Fro Figures 6 and we observed that the velocity proile disappears uickly when copared with the teperature. In Figure we observed that the Prandtl nuber decreases the theral boundary layer thickness. In act an increase in the Prandtl nuber increases the theral diusitivity and thus there is a decrease in the teperature proile. By increasing α the teperature proile decreases but by increasing ε both the teperature and theral boundary layer thickness increase Figure. In addition an increase in theral radiation paraeter increases the teperature proile and theral boundary layer thickness. So teperature and theral boundary layer thickness are decreasing unctions o N Figure. Table is provided to see how uch approxiations are reuired to ind a convergent series solution. Fro this table we see that 5 th order coputations are enough or velocity and th order coputations are suicient or teperature. It is observed that less coputations are reuired or velocity when copared to teperature. Table shows the nuerical values o skin-riction coeicient or dierent values o K K M and S. The values o skin-riction coeicient increase by increasing S and M and decrease by increasing K and K. 4 Walailak J Sci & Tech ;

7 MHD Flow o Thixotropic Fluid Figure curve or the unction. Figure curve or the unction. Figure Inluence o K on. Walailak J Sci & Tech ; 5

8 MHD Flow o Thixotropic Fluid Figure 4 Inluence o K on. Figure 5 Inluence o M on. Figure 6 Inluence o S on. 6 Walailak J Sci & Tech ;

9 MHD Flow o Thixotropic Fluid Figure 7 Inluence o K on. Figure 8 Inluence o K on. Figure 9 Inluence o M on. Walailak J Sci & Tech ; 7

10 MHD Flow o Thixotropic Fluid Figure Inluence o S on. Figure Inluence o Pr on. Figure Inluence o α on. 8 Walailak J Sci & Tech ;

11 MHD Flow o Thixotropic Fluid Figure Inluence o ε on. Figure 4 Inluence o N on. Walailak J Sci & Tech ; 9

12 MHD Flow o Thixotropic Fluid Table Convergence o series solutions or dierent order o approxiations when K₁. K₂. S.5 M.6 ε. α.8 Pr. N. ħ.5 and ħ.8. Order o approxiations Table Nuerical values o skin-riction coeicient or dierent values o K₁ K₂ M and S. S M K₁ K₂ -Re / x C Closing rearks MHD low o thixotropic luid over a stretched surace with theral radiation is studied. Further heat transer is considered in the presence o variable theral conductivity. The ain results o the conducted study are:. The non-newtonian paraeters K and K have uite opposite eects on the velocity and teperature proiles.. The eects o M and S on the velocity ield are siilar in a ualitative sense.. An increase in S decreases the teperature and theral boundary layer thickness. 4. Variations in α and ε on the teperature are uite opposite. 5. The behaviors o S and M on the skin riction coeicient are uite opposite to that o K and K. Reerences [] A Nejat A Jalali and M Sharbatdar. A Newton-Krylov inite volue algorith or the power-law non-newtonian luid low using pseudo-copressibility techniue. J. Non-Newtonian Fluid Mech. ; [] H Qi and M Xu. Unsteady low o viscoelastic luid with ractional Maxwell odel in a channel. Mech. Research Coun. 7; 4 -. [] S Wang and WC Tan. Stability analysis o soret-driven double-diusive convection o Maxwell luid in a porous ediu. Int. J. Heat Fluid Flow ; [4] M Khan SH Ali C Fetecau and Haitao Qi. Decay o potential vortex or a viscoelastic luid with ractional Maxwell odel. Appl. Math. Modelling 9; 56-. [5] M Jail and C Fetecau. Soe exact solutions 4 Walailak J Sci & Tech ;

13 MHD Flow o Thixotropic Fluid or rotating lows o a generalized Burgers' luid in cylindrical doains. J. Non- Newtonian Fluid Mech. ; [6] M Jail and C Fetecau. Helical lows o Maxwell luid between coaxial cylinders with given shear stresses on the boundary. Nonlinear Analysis: Real World Appl. ; 4-. [7] T Hayat SA Shehzad M Qasi and S Obaidat. Steady low o Maxwell luid with convective boundary conditions. Z. Naturorsch A ; [8] A Ahad and S Asghar. Flow o a second grade luid over a sheet stretching with arbitrary velocities subject to a transverse agnetic ield. Appl. Math. Lett. ; [9] T Hayat SA Shehzad M Qasi and S Obaidat. Theral radiation eects on the ixed convection stagnation-point low in a Jeery luid. Z. Naturorsch A ; [] K Bhattacharyya S Mukhopadhyay and GC Layek. Slip eects on an unsteady boundary layer stagnation-point low and heat transer towards a stretching sheet. Chin. Phy. Lett. ; [] K Bhattacharyya and GC Layek. Eects o suction/blowing on steady boundary layer stagnation-point low and heat transer towards a shrinking sheet with theral radiation. Int. J. Heat Mass Transer ; [] S Nadee S Zaheer and T Fang. Eects o theral radiation on the boundary layer low o a Jeery luid over an exponentially stretching surace. Nuer. Algor. ; [] T Hayat M Qasi and Z Abbas. Radiation and ass transer eects on the agnetohydrodynaic unsteady low induced by a stretching sheet. Z Naturorsch A ; [4] S Yao T Fang and J Zhang. Heat transer o a generalized stretching/shrinking wall proble with convective boundary conditions. Co. Nonlinear Sci. Nu. Siu. ; [5] T Hayat SA Shehzad M Qasi and S Obaidat. Radiative low o Jeery luid in a porous ediu with power law heat lux and heat source. Nuclear Eng. Design ; [6] S Sadei N Khabazi and K Sadeghy. Blasius low o thixotropic luids: A nuerical study. Co. Nonlinear Sci. Nu. Siu. ; [7] T Hayat C Fetecaua and M Sajid. Analytic solution or MHD Transient rotating low o a second grade luid in a porous space. Nonlinear Analysis: Real World Appl. 8; [8] MM Rashidi and E Erani. A new analytical study o MHD stagnation-point low in porous edia with heat transer. Coput. Fluids ; [9 M Turkyilazoglu. Unsteady MHD low with variable viscosity: Applications o spectral schee. Int. J. Theral Sci. ; [] OD Makinde and T Chinyoka. MHD transient lows and heat transer o dusty luid in a channel with variable physical properties and Navier slip condition. Coput. Math. Appl. ; [] S Abbasbandy and T Hayat. Solution o the MHD Falkner-Skan low by hootopy analysis ethod. Coun. Nonlinear Sci. Nuer. Siulat. 9; [] T Hayat FM Abbasi and S Obaidat. Peristaltic otion with Soret and Duour eects. Magnetohydrodynaics ; [] WM Kay. Convective heat and ass transer Mc-Graw Hill New York 996. [4] P Vyas and A Rai. Radiative low with variable theral conductivity over a nonisotheral stretching sheet in a porous ediu. Int. J. Contep. Math. Sciences ; [5] SJ Liao. Beyond perturbation: Introduction to hootopy analysis ethod. Chapan and Hall CRC Press Boca Raton. [6] H Vosughi E Shivanian and S Abbasbandy. A new analytical techniue to solve Volterra's integral euations. Math. Methods Appl. Sci. ; [7] MM Rashidi and SAM Pour. Analytic approxiate solutions or unsteady boundary-layer low and heat transer due to a stretching sheet by hootopy analysis ethod. Nonlinear Anal. Model. Contr. ; [8] B Yao. Approxiate analytical solution to Walailak J Sci & Tech ; 4

14 MHD Flow o Thixotropic Fluid the Falkner-Skan wedge low with the pereable wall o unior suction. Coun. Nonlinear Sci. Nuer. Siulat. 9; 4-6. [9] MM Rashidi SAM Pour and S Abbasbandy. Analytic approxiate solutions or heat transer o a icropolar luid through a porous ediu with radiation. Coun. Nonlinear Sci. Nuer. Siulat. ; [] T Hayat SA Shehzad and M Qasi. Mixed convection low o a icropolar luid with radiation and cheical reaction. Int. J. Nu. Methods Fluids ; Walailak J Sci & Tech ;