Advertisement

Nonlinear Radiation in Bioconvective Casson Nanofluid Flow

  • I. S. Oyelakin
  • S. MondalEmail author
  • P. Sibanda
Original Paper
  • 35 Downloads

Abstract

We investigate the impact of nonlinear thermal radiation and variable transport properties on the two-dimensional flow of an electrically conducting Casson nanofluid containing gyrotactic microorganisms along a moving wedge. In some previous studies, it has been assumed that the fluid viscosity and thermal conductivity are temperature dependent. However, this study assumes that the fluid viscosity, thermal conductivity, and the nanofluid properties, are dependent on the solute concentration. Some experimental studies have shown that the viscosity and thermal conductivity of nanofluids are strongly dependent on the volume fraction of nanoparticles rather than just the temperature. The spectral local linearization method is used to solve the conservation equations. We compare our results with those in the literature, and we discuss the convergence and accuracy of the spectral local linearization method. The impact of some parameters on the skin friction, heat and microorganisms mass transport is discussed.

Keywords

Gyrotactic microorganisms Casson nanofluid Radiation–conduction interaction Spectral local linearization method 

List of Symbols

Greek Symbols

\(\alpha \)

\(\theta _w - 1\)

\(\beta \)

Casson fluid parameter

\(\eta \)

Independent similarity variable

\(\gamma \)

Velocity ratio parameter

\(\lambda \)

Wedge angle parameter

\(\mu (c)\)

Variable dynamic viscosity of the flow (\(\frac{\mathrm{kg}}{\mathrm{ms}}\))

\(\mu _{\infty }\)

Constant dynamic viscosity of the fluid (\(\frac{\mathrm{kg}}{\mathrm{ms}}\))

\(\mu _{B}\)

Plastic viscosity \(\left( \frac{\mathrm{kg}}{\mathrm{ms}}\right) \)

\(\nu _{\infty }\)

Constant kinematic viscosity of the fluid (\(\frac{\mathrm{m}^2}{\mathrm{s}}\))

\(\Omega \)

Total angle of the wedge

\(\phi (\eta )\)

Dimensionless nanoparticle volume fraction

\(\pi \)

Component of product of rate of strain

\(\pi _{c}\)

Critical value of product of rate of strain

\(\psi \)

Stream function

\(\rho _{\infty }\)

Constant density (\(\frac{\mathrm{kg}}{\mathrm{m}^3}\))

\(\sigma \)

Electrical conductivity \(\left( \frac{\mathrm{S}}{\mathrm{m}}\right) \)

\(\sigma ^*\)

Stefan–Boltzmann constant \(\left( \frac{\mathrm{W}}{\mathrm{m}^2 \mathrm{K}^4}\right) \)

\(\sigma _1\)

Microorganisms concentration difference

\(\tau \)

\(\frac{(\rho cp)_p}{(\rho cp)_f}\) Ratio of the effective heat capacity of the nanoparticle material to the fluid heat capacity

\(\tau _w\)

Local wall shear stress \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(\tau _{ij}\)

Component of the stress tensor \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(\theta (\eta )\)

Dimensionless temperature

\(\theta _w\)

Surface temperature parameter

\(\chi (\eta )\)

Dimensionless number of motile microorganism

\((\rho cp)_f\)

Heat capacity of the fluid (\(\frac{\mathrm{J}}{\mathrm{m}^3 \mathrm{K}}\))

\((\rho cp)_p\)

Effective heat capacity of the nanoparticle medium \(\left( \frac{\mathrm{J}}{\mathrm{m}^3 \mathrm{K}}\right) \)

\(B_{0}\)

Magnetic induction parameter (T)

C

Nanoparticles volume fraction (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(C_{\infty }\)

Concentration far away from the surface (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(C_w\)

Concentration of the fluid (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(Cf_x\)

Skin friction coefficient

\(D_{B,\infty }\)

Constant mass diffusivity \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_{N,\infty }\)

Constant microorganism diffusivity \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_B(c)\)

Variable Brownian diffusion coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_N(c)\)

Variable microorganism diffusivity coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

\(D_T\)

Thermophoretic diffusion coefficient \(\left( \frac{\mathrm{m}^2}{\mathrm{s}}\right) \)

M

Magnetic field

N

Number of motile microorganism (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(N_{\infty }\)

Concentration density of motile microorganisms far away from the surface (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

\(N_w\)

Concentration density of the motile microorganism (\(\frac{\mathrm{mol}}{\mathrm{kg}}\))

Nb

Brownian motion

\(Nn_x\)

Local density of motile microorganisms

Nt

Thermophoresis diffusion

\(Nu_x\)

Local Nusselt number

Pe

Bioconvection Péclet Number

Pr

Prandtl number

Rd

Nonlinear thermal radiation

\(Re_x\)

Reynold’s Number

S

Suction/injection

Sb

Bioconvection Lewis number

Sc

Schmidt number

\(T_{\infty }\)

Temperature far away from the surface (K)

\(T_w\)

Fluid temperature (K)

\(U_\infty \)

Constant velocity at the free stream \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(U_w\)

Constant velocity at the surface \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(W_c\)

Maximum cell swimming speed (\(\frac{\mathrm{m}}{\mathrm{s}}\))

b

Chemotaxis constant (m)

\(c_{5}\)

Dimensionless variable viscosity parameter \((c_5 = c_1C_\infty \))

\(c_{6}\)

Dimensionless variable thermal conductivity parameter \((c_6 = c_2C_\infty \))

\(c_{7}\)

Dimensionless variable mass diffusivity parameter \((c_7 = c_3C_\infty \))

\(c_{8}\)

Dimensionless variable microorganisms diffusivity parameter \((c_8 = c_4C_\infty \))

\(c_p\)

Specific heat \(\left( \frac{\mathrm{J}}{\mathrm{kg K}}\right) \)

\(e_{ij}\)

Rate of strain tensor \(\left( \frac{1}{\mathrm{s}}\right) \)

\(f(\eta )\)

Dimensionless stream function

k(c)

Variable thermal conductivity \(\left( \frac{\mathrm{W}}{\mathrm{m K}}\right) \)

\(k^*\)

Mean absorption coefficient

\(k_{\infty }\)

Constant thermal conductivity \(\left( \frac{\mathrm{W}}{\mathrm{m K}}\right) \)

m

Stream wise pressure gradient or Falkner–Skan power-law index

\(p_{y}\)

Yield stress \(\left( \frac{\mathrm{N}}{\mathrm{m}^2}\right) \)

\(q_N\)

Local density number of motile microorganisms \(\left( \frac{\mathrm{mol}}{\mathrm{kg}}\right) \)

\(q_w\)

Local surface heat flux (K)

\(u_e\)

Wedge velocity at the free stream \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(u_w\)

Stretching wedge velocity \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

uv

Velocity components in the x and y directions respectively \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

\(v_w\)

Suction/injection velocity component in the y direction \(\left( \frac{\mathrm{m}}{\mathrm{s}}\right) \)

x

Distance along the surface \(\left( \mathrm{m}\right) \)

y

Distance normal to the surface \(\left( \mathrm{m}\right) \)

Notes

References

  1. 1.
    Kumari, M.: Effect of variable viscosity on non-Darcy free or mixed convection flow on a horizontal surface in a saturated porous medium. Int. Commun. Heat Mass Transf. 28(5), 723–732 (2001)CrossRefGoogle Scholar
  2. 2.
    Chin, K.E., Nazar, R., Arifin, N.M., Pop, I.: Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium. Int. Commun. Heat Mass Transf. 34(4), 464–473 (2007)CrossRefGoogle Scholar
  3. 3.
    Prasad, K.V., Pal, D., Umesh, V., Rao, N.P.: The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet. Commun. Nonlinear Sci. Numer. Simul. 15(2), 331–344 (2010)zbMATHCrossRefGoogle Scholar
  4. 4.
    Mabood, F., Khan, W., Makinde, O.: Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects. J. Eng. Thermophys. 26(4), 553–566 (2017)CrossRefGoogle Scholar
  5. 5.
    Das, S.K., Choi, S.U., Yu, W., Pradeep, T.: Nanofluids: Science and Technology. Wiley, Hoboken (2007)CrossRefGoogle Scholar
  6. 6.
    Khanafer, K., Vafai, K.: A critical synthesis of thermophysical characteristics of nanofluids. Int. J. Heat Mass Transf. 54(19–20), 4410–4428 (2011)zbMATHCrossRefGoogle Scholar
  7. 7.
    Chandrasekar, M., Suresh, S.: A review on the mechanisms of heat transport in nanofluids. Heat Transf. Eng. 30(14), 1136–1150 (2009)CrossRefGoogle Scholar
  8. 8.
    Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME Publ. Fed 231, 99–106 (1995)Google Scholar
  9. 9.
    Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128(3), 240–250 (2006)CrossRefGoogle Scholar
  10. 10.
    Kuznetsov, A.V., Nield, D.A.: Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model. Int. J. Therm. Sci. 77, 126–129 (2014)CrossRefGoogle Scholar
  11. 11.
    Khan, J.A., Mustafa, M., Hayat, T., Alsaedi, A.: Three-dimensional flow of nanofluid over a non-linearly stretching sheet: an application to solar energy. Int. J. Heat Mass Transf. 86, 158–164 (2015)CrossRefGoogle Scholar
  12. 12.
    Oyelakin, I.S., Mondal, S., Sibanda, P.: Unsteady Casson nanofluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions. Alex. Eng. J. 55(2), 1025–1035 (2016)CrossRefGoogle Scholar
  13. 13.
    Prabhakar, B., Bandari, S., Haq, R.U.: Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet. Neural Comput. Appl. 29(10), 805–814 (2018)CrossRefGoogle Scholar
  14. 14.
    Kuznetsov, A.V., Avramenko, A.A.: Effect of small particles on this stability of bioconvection in a suspension of gyrotactic microorganisms in a layer of finite depth. Int. Commun. Heat Mass Transf. 31(1), 1–10 (2004)CrossRefGoogle Scholar
  15. 15.
    Wager, H.: On the effect of gravity upon the movements and aggregation of Euglena viridis, Ehrb., and other micro-organisms. Philos. Trans. R. Soc. Lond. 201, 333–390 (1911)CrossRefGoogle Scholar
  16. 16.
    Pedley, T.J., Kessler, J.O.: Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24(1), 313–358 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kuznetsov, A.V.: Modeling bioconvection in porous media. In: Vafai, K. (ed.) Handbook of Porous Media, Chapter 16, pp. 645–686. CRC Press, Boca Raton (2005)Google Scholar
  18. 18.
    Shaw, S., Sibanda, P., Sutradhar, A., Murthy, P.: Magnetohydrodynamics and Soret effects on bioconvection in a porous medium saturated with a nanofluid containing gyrotactic microorganisms. J. Heat Transf. 136(5), 052601 (2014)CrossRefGoogle Scholar
  19. 19.
    Alsaedi, A., Khan, M.I., Farooq, M., Gull, N., Hayat, T.: Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms. Adv. Powder Technol. 28(1), 288–298 (2017)CrossRefGoogle Scholar
  20. 20.
    Mukhopadhyay, S.: 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 (2013)CrossRefGoogle Scholar
  21. 21.
    Malik, M.Y., Naseer, M., Nadeem, S., Rehman, A.: The boundary layer flow of Casson nanofluid over a vertical exponentially stretching cylinder. Appl. Nanosci. 4(7), 869–873 (2014)CrossRefGoogle Scholar
  22. 22.
    Nadeem, S., Haq, R.U., Akbar, N.S.: MHD three-dimensional boundary layer flow of Casson nanofluid past a linearly stretching sheet with convective boundary condition. Inst. Electr. Electron. Eng. Trans. Nanotechnol. 13(1), 109–115 (2014)Google Scholar
  23. 23.
    Abolbashari, M.H., Freidoonimehr, N., Nazari, F., Rashidi, M.M.: Analytical modeling of entropy generation for Casson nano-fluid flow induced by a stretching surface. Adv. Powder Technol. 26(2), 542–552 (2015)CrossRefGoogle Scholar
  24. 24.
    Oyelakin, I.S., Mondal, S., Sibanda, P.: Unsteady MHD three-dimensional Casson nanofluid flow over a porous linear stretching sheet with slip condition. Front. Heat Mass Transf. 8, 1–9 (2017)Google Scholar
  25. 25.
    Pal, D.: Hall current and MHD effects on heat transfer over an unsteady stretching permeable surface with thermal radiation. Comput. Math. Appl. 66(7), 1161–1180 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ibrahim, F.S., Hady, F.M.: Mixed convection-radiation interaction in boundary-layer flow over horizontal surfaces. Astrophys. Space Sci. 168(2), 263–276 (1990)zbMATHCrossRefGoogle Scholar
  27. 27.
    Hossain, M.A., Alim, M.A.: Natural convection–radiation interaction on boundary layer flow along a thin vertical cylinder. Heat Mass Transf. 32(6), 515–520 (1997)CrossRefGoogle Scholar
  28. 28.
    Mustafa, M., Mushtaq, A., Hayat, T., Ahmad, B.: Nonlinear radiation heat transfer effects in the natural convective boundary layer flow of nanofluid past a vertical plate: a numerical study. PLoS ONE 9(9), e103946 (2014)CrossRefGoogle Scholar
  29. 29.
    Devi, S.P., Raj, J.W.S.: Nonlinear radiation effects on hydromagnetic boundary layer flow and heat transfer over a shrinking surface. J. Appl. Fluid Mech. 8(3), 613–621 (2015)CrossRefGoogle Scholar
  30. 30.
    Ghadikolaei, S., Hosseinzadeh, K., Hatami, M., Ganji, D., Armin, M.: Investigation for squeezing flow of ethylene glycol (\({\rm C}_2{\rm H}_{6}{\rm O}_2\)) carbon nanotubes (CNTs) in rotating stretching channel with nonlinear thermal radiation. J. Mol. Liq. 263, 10–21 (2018)CrossRefGoogle Scholar
  31. 31.
    Ghadikolaei, S., Hosseinzadeh, K., Ganji, D.: MHD raviative boundary layer analysis of micropolar dusty fluid with graphene oxide (Go)-engine oil nanoparticles in a porous medium over a stretching sheet with joule heating effect. Powder Technol. 338, 425–437 (2018)CrossRefGoogle Scholar
  32. 32.
    Ghadikolaei, S., Hosseinzadeh, K., Ganji, D.: Numerical study on magnetohydrodynic CNTs-water nanofluids as a micropolar dusty fluid influenced by non-linear thermal radiation and Joule heating effect. Powder Technol. 340, 389–399 (2018)CrossRefGoogle Scholar
  33. 33.
    Shojaei, A., Amiri, A.J., Ardahaie, S.S., Hosseinzadeh, K., Ganji, D.: Hydrothermal analysis of Non-Newtonian second grade fluid flow on radiative stretching cylinder with Soret and Dufour effects. Case Stud. Therm. Eng. 13, 100384 (2019)CrossRefGoogle Scholar
  34. 34.
    Prakash, J., Raju, C.S.K., Sandeep, N.: Dual solutions for heat and mass transfer in MHD Jeffrey fluid in the presence of homogeneous–heterogeneous reactions. Front. Heat Mass Transf. 7(1), 1–8 (2016)Google Scholar
  35. 35.
    Ghadikolaei, S., Hosseinzadeh, K., Ganji, D., Jafari, B.: Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet. Case Stud. Therm. Eng. 12, 176–187 (2018)CrossRefGoogle Scholar
  36. 36.
    Amirsom, N.A., Uddin, M.J., Ismail, A.I.: Three dimensional stagnation point flow of bionanofluid with variable transport properties. Alex. Eng. J. 55(3), 1983–1993 (2016)CrossRefGoogle Scholar
  37. 37.
    Noghrehabadi, A., Behseresht, A.: Flow and heat transfer affected by variable properties of nanofluids in natural-convection over a vertical cone in porous media. Comput. Fluids 88, 313–325 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Raju, C.S.K., Hoque, M.M., Sivasankar, T.: Radiative flow of Casson fluid over a moving wedge filled with gyrotactic microorganisms. Adv. Powder Technol. 28(2), 575–583 (2017)CrossRefGoogle Scholar
  39. 39.
    Motsa, S.S.: A new spectral local linearization method for nonlinear boundary layer flow problems. J. Appl. Math. 2013, Article ID 423628, p. 15 (2013). https://doi.org/10.1155/2013/423628 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Nadeem, S., Mehmood, R., Motsa, S.: Numerical investigation on MHD oblique flow of Walter’s B type nano fluid over a convective surface. Int. J. Therm. Sci. 92, 162–172 (2015)CrossRefGoogle Scholar
  41. 41.
    Sithole, H.M., Mondal, S., Sibanda, P., Motsa, S.S.: An unsteady MHD Maxwell nanofluid flow with convective boundary conditions using spectral local linearization method. Open Phys. 15(1), 637–646 (2017)CrossRefGoogle Scholar
  42. 42.
    Shateyi, S, Marewo, G.: Numerical analysis of MHD stagnation point flow of casson fluid, heat and mass transfer over a stretching sheet. In: Balicki, J. (ed.) Proceedings of the 7th International Conference on Finite Differences, Finite Elements, Finite Volumes, Boundary Elements (F-and-B ’14), Adv. Appl. Pure Math. pp. 128–132, WSEAS, Poland (2014) Google Scholar
  43. 43.
    Sayehvand, H.-O., Parsa, A.B.: A new numerical method for investigation of thermophoresis and Brownian motion effects on MHD nanofluid flow and heat transfer between parallel plates partially filled with a porous medium. Results Phys. 7, 1595–1607 (2017)CrossRefGoogle Scholar
  44. 44.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  45. 45.
    Rajagopal, K.R., Gupta, A., Na, T.-Y.: A note on the Falkner–Skan flows of a non-Newtonian fluid. Int. J. Non-Linear Mech. 18(4), 313–320 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kuo, B.: Application of the differential transformation method to the solutions of Falkner–Skan wedge flow. Acta Mech. 164(3–4), 161–174 (2003)zbMATHCrossRefGoogle Scholar
  47. 47.
    Khan, M., Azam, M., Munir, A.: On unsteady Falkner–Skan flow of MHD Carreau nanofluid past a static/moving wedge with convective surface condition. J. Mol. Liq. 230, 48–58 (2017)CrossRefGoogle Scholar
  48. 48.
    White, F.M.: Viscous Fluid Flow, vol. 2. McGraw-Hill, New York (1991)Google Scholar
  49. 49.
    Mustafa, M., Mushtaq, A., Hayat, T., Alsaedi, A.: Numerical study of MHD viscoelastic fluid flow with binary chemical reaction and Arrhenius activation energy. Int. J. Chem. React. Eng. 15(1), 627–633 (2017) Google Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer ScienceUniversity of KwaZulu-NatalPietermaritzburgSouth Africa
  2. 2.Department of MathematicsAmity University, KolkataNewtownIndia

Personalised recommendations