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Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

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Abstract

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

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Abbreviations

\({\mathbf{b}}\) :

Body force (N)

\(c\) :

Positive constant

\(C_{\text{p}}\) :

Heat capacity at constant pressure (J kg−1 K−1)

\({\text{Ec}}\) :

Eckert number

\(k\) :

Thermal conductivity (W m−1 K−1)

\(k^{*}\) :

Mean absorption coefficient

\(\Pr\) :

Prandtl number

\(\Pr_{\text{eff}}\) :

Effective Prandtl number

\(p\) :

Pressure (kg m−1 s−2)

\(p_{\text{y}}\) :

Yield stress (kg m−1 s−2)

\(R\) :

Radiation parameter

\(q_{\text{r}}\) :

Radiative heat flux (W m−2)

\(T\) :

Fluid temperature (K)

\(T_{\text{w}}\) :

Wall temperature (K)

\(T_{\infty }\) :

Ambient temperature (K)

\(S\) :

Suction/injection parameter

\(u\) :

Velocity components in the \(x\) axis (m s−1)

\(v\) :

Velocity components in the \(y\) axis (m s−1)

\(\alpha\) :

Casson parameter

\(\rho\) :

Fluid density (kg m−3)

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\tau_{ij}\) :

Shear stress (kg m−1 s−2)

\(\mu_{\text{B}}\) :

Plastic dynamic viscosity (kg m−1 s−1)

\(\nu\) :

Kinematic viscosity (m2 s−1)

\(\theta\) :

Dimensionless temperature

\(\sigma\) :

Stefan–Boltzmann constant (W m−2 K−4)

\(\infty\) :

Condition at infinity

References

  1. Casson N (1959) A flow equation for pigment oil suspensions of the printing ink type. In: Mill CC (ed) Rheology of disperse systems. Pergamon Press, Oxford, pp 84–102

    Google Scholar 

  2. Dash RK, Jayaraman G, Mehta KN (2000) Shear augmented dispersion of a solute in a Casson fluid flowing in a conduit. Ann Biomed Eng 28:373–385

    Article  Google Scholar 

  3. Nagarani P, Sarojamma G, Jayaraman G (2004) Effect of boundary absorption in dispersion in Casson fluid flow in a tube. Ann Biomed Eng 5:706–719

    Article  MATH  Google Scholar 

  4. Nagarani P, Sarojamma G, Jayaraman G (2006) Exact analysis of unsteady convective diffusion in Casson fluid flow in an annulus: application to catheterized artery. Acta Mech 187:189–202

    Article  MATH  Google Scholar 

  5. Mustafa M, Hayat T, Pop I, Aziz A (2011) Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate. Heat Transf Asian Res 40:553–576

    Article  Google Scholar 

  6. Hayat T, Shehzad SA, Alsaedi A, Alhothuali MS (2012) Mixed convection stagnation point flow of Casson fluid with convective boundary conditions. Chin Phys B 29(11):114704

    Google Scholar 

  7. Mukhopadhyay S (2013) Effects of thermal radiation on Casson fluid flow and heat transfer over an unsteady stretching surface subjected to suction/blowing. Chin Phys B 22(11):114702–114707

    Article  Google Scholar 

  8. Mukhopadhyay S (2013) Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chin Phys B 22:074701–074705

    Article  Google Scholar 

  9. Mukhopadhyay S, Vajravelu K, Gorder RAV (2013) Casson fluid flow and heat transfer at an exponentially stretching permeable surface. J Appl Mech 80:054502–054509

    Article  Google Scholar 

  10. Mukhopadhyay S, Bhattacharyya K, Hayat T (2013) Exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects. Chin Phys B 22:114701–114706

    Article  Google Scholar 

  11. Rao AS, Prasad VR, Reddy NB, Beg OA (2013) Heat transfer in a Casson rheological fluid from a semi-infinite vertical plate with partial slip. Heat Transf Asian Res 2013:1–20

    Google Scholar 

  12. Khan U, Ahmed N, Khan SI, Bano S, Mohyud-Din ST (2014) Unsteady squeezing flow of a Casson fluid between parallel plates. World J Model Simul 10(4):308–319

    Google Scholar 

  13. Hussanan A, Salleh MZ, Tahar RM, Khan I (2014) Unsteady boundary layer flow and heat transfer of a Casson fluid past an oscillating vertical plate with Newtonian heating. PLoS ONE 9:1–9

    Article  Google Scholar 

  14. Hussanan A, Salleh MZ, Khan I, Tahar RM (2016) Unsteady heat transfer flow of a Casson fluid with Newtonian heating and thermal radiation. Jurnal Teknologi 78(4):1–7

    Google Scholar 

  15. Khan I, Shah NA, Vieru D (2016) Unsteady flow of generalized Casson fluid with fractional derivative due to an infinite plate. Eur Phys J Plus 131:1–12

    Article  Google Scholar 

  16. Mohyud-Din ST, Khan U, Ahmed N, Bin-Mohsin B (2015) Heat and mass transfer analysis for MHD flow of nanofluid inconvergent/divergent channels with stretchable walls using Buongiorno’s model. Neural Comput Appl. doi:10.1007/s00521-016-2289-5

    Google Scholar 

  17. Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647

    Article  Google Scholar 

  18. Grubka LJ, Bobba KM (1985) Heat transfer characteristics of a continuous, stretching surface with variable temperature. J Heat Transf 107:248–250

    Article  Google Scholar 

  19. Chen CK, Char MI (1988) Heat transfer of a continuous, stretching surface with suction or blowing. J Math Anal Appl 135:568–580

    Article  MathSciNet  MATH  Google Scholar 

  20. Ali ME (1994) Heat transfer characteristics of a continuous stretching surface. Warme-und Stoffubertragung 29(4):227–234

    Article  Google Scholar 

  21. Sharma R (2012) Effect of viscous dissipation and heat source on unsteady boundary layer flow and heat transfer past a stretching surface embedded in a porous medium using element free Galerkin method. Appl Math Comput 219:976–987

    MathSciNet  MATH  Google Scholar 

  22. Rashidi MM, Momoniat E, Rostami S (2012) Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving stretching surface by homotopy analysis method with two auxiliary parameters. J Appl Math 2012:1–19

    Article  MathSciNet  MATH  Google Scholar 

  23. Soid SK, Kechil SA, Shah AHM, Halim NZISNSA (2012) Hydromagnetic boundary layer flow over stretching surface with thermal radiation. World Appl Sci J 17:33–38

    Google Scholar 

  24. Nadeem S, Hussain ST, Lee C (2013) Flow of a Williamson fluid over a stretching sheet. Braz J Chem Eng 30(03):619–625

    Article  Google Scholar 

  25. Qasim M, Khan I, Shafie S (2013) Heat transfer in a micropolar fluid over a stretching sheet with Newtonian heating. PLoS ONE 8:e59393

    Article  Google Scholar 

  26. Rashidi MM, Ali M, Rostami B, Rostami P, Xie GN (2015) Heat and mass transfer for MHD visco-elastic fluid flow over a vertical stretching sheet with considering Soret and Dufour effects. Math Probl Eng 2015:1–12

    Article  Google Scholar 

  27. Freidoonimehr N, Rashidi MM, Mahmud S (2015) Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid. Int J Therm Sci 87:136–145

    Article  Google Scholar 

  28. Mohamed MKA, Salleh MZ, Ishak A, Pop I (2015) Stagnation point flow and heat transfer over a stretching/shrinking sheet in a viscoelastic fluid with convective boundary condition and partial slip velocity. Eur Phys J Plus 130:1–9

    Article  Google Scholar 

  29. Abolbashari MH, Freidoonimehr N, Nazari F, Rashidi MM (2015) Analytical modeling of entropy generation for Casson nano-fluid flow induced by a stretching surface. Adv Powder Technol 26:542–552

    Article  Google Scholar 

  30. Khalili S, Tamim H, Khalili A, Rashidi MM (2015) Unsteady convective heat and mass transfer in pseudoplastic nanofluid over a stretching wall. Adv Powder Technol 26:1319–1326

    Article  Google Scholar 

  31. Mohyud-Din ST, Khan U, Ahmed N, Hassan SM (2015) Magnetohydrodynamic flow and heat transfer of nanofluids in stretchable convergent/divergent channels. Appl Sci 5:1639–1664

    Article  Google Scholar 

  32. Alinejad J, Samarbakhsh S (2012) Viscous flow over nonlinearly stretching sheet with effects of viscous dissipation. J Appl Math 2012:1–10

    Article  MathSciNet  MATH  Google Scholar 

  33. Nandeppanavar MM, Siddalingappa MN (2013) Effect of viscous dissipation and thermal radiation on heat transfer over a non-linearly stretching sheet through porous medium. Int J Appl Mech Eng 18:461–474

    Google Scholar 

  34. Qasim M, Noreen S (2014) Heat transfer in the boundary layer flow of a Casson fluid over a permeable shrinking sheet with viscous dissipation. Eur Phys J Plus 129:1–8

    Article  Google Scholar 

  35. Khan SI, Khan U, Ahmed N, Jan SU, Waheed A, Mohyud-Din ST (2015) Effects of viscous dissipation and convective boundary conditions on Blasius and Sakiadis problems for Casson fluid. Natl Acad Sci Lett 38:247–250

    Article  MathSciNet  Google Scholar 

  36. Reddy MG, Padma P, Shankar B (2015) Effects of viscous dissipation and heat source on unsteady MHD flow over a stretching sheet. Ain Shams Eng J 6:1195–1205

    Article  Google Scholar 

  37. Mohyud-Din ST, Zaidi ZA, Khan U, Ahmed N (2015) On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates. Aerosp Sci Technol 46:514–522

    Article  Google Scholar 

  38. Khan U, Ahmed N, Mohyud-Din ST (2015) Heat transfer effects on carbon nanotubes suspended nanofluid flow in a channel with non-parallel walls under the effect of velocity slip boundary condition: a numerical study. Neural Comput Appl. doi:10.1007/s00521-015-2035-4

    Google Scholar 

  39. Herbert OJ, Prandtl L (2004) Prandtl’s essentials of fluid mechanics. Springer, New York

    Google Scholar 

  40. Hussanan A, Ismail Z, Khan I, Hussein AG, Sharidan S (2014) Unsteady boundary layer MHD free convection flow in a porous medium with constant mass diffusion and Newtonian heating. Eur Phys J Plus 129:1–16

    Article  Google Scholar 

  41. Ahmed N, Khan U, Mohyud-Din ST, Bin-Mohsin B (2016) A finite element investigation of the flow of a Newtonian fluid in dilating and squeezing porous channel under the influence of nonlinear thermal radiation. Neural Comput Appl. doi:10.1007/s00521-016-2463-9

    Google Scholar 

  42. Chen CH (1998) Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transf 33:471–476

    Article  Google Scholar 

  43. Khader MM, Megahed AM (2014) Differential transformation method for the flow and heat transfer due to a permeable stretching surface embedded in a porous medium with a second order slip and viscous dissipation. J Heat Transf 136:0726021–0726027

    Article  Google Scholar 

  44. Mustafa M, Ahmad R, Hayat T, Alsaedi A (2016) Rotating flow of viscoelastic fluid with nonlinear thermal radiation: a numerical study. Neural Comput Appl. doi:10.1007/s00521-016-2462-x

    Google Scholar 

  45. Hussanan A, Anwar MI, Farhad A, Khan I, Sharidan S (2014) Natural convection flow past an oscillating plate with Newtonian heating. Heat Transf Res 45:119–137

    Article  Google Scholar 

  46. Khan U, Ahmed N, Mohyud-Din ST, Bin-Mohsin B (2015) Nonlinear radiation effects on MHD flow of nanofluid over a nonlinearly stretching/shrinking wedge. Neural Comput Appl. doi:10.1007/s00521-016-2187-x

    Google Scholar 

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Acknowledgements

The authors would like to acknowledge the support received from Universiti Malaysia Pahang, Malaysia, through vote numbers RDU121302 and RDU140111.

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Authors and Affiliations

  1. Applied and Industrial Mathematics Research Group, Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300, Kuantan, Pahang, Malaysia

    Abid Hussanan & Mohd Zuki Salleh

  2. College of Engineering, Majmaah University, P.O. Box 66, Majmaah, Saudi Arabia

    Ilyas Khan

  3. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310, Skudai, Malaysia

    Sharidan Shafie

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  1. Abid Hussanan
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Correspondence to Abid Hussanan.

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Hussanan, A., Salleh, M.Z., Khan, I. et al. Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation. Neural Comput & Applic 29, 1507–1515 (2018). https://doi.org/10.1007/s00521-016-2674-0

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