Skip to main content
SpringerLink
Log in

A Similarity Solution for Natural Convection Flow near a Vertical Plate with Thermal Radiation

  • Original Article
  • Published:
Microgravity Science and Technology Aims and scope Submit manuscript

Abstract

This paper examined the fluid transport on boundary layer near a vertical wall with thermal radiation effect. The coupled nonlinear partial differential equations are reduced to a system of ordinary differential equations through similarity transformation. The resultant system of ordinary differential equations is integrated with maple software using RKF45 method. The effect of the active parameters such as Grashof number, thermal radiation parameter, temperature difference, Prandtl number, and local convective heat transfer parameter on the velocity profile, skin friction, temperature profile and Nusselt number are presented using graphs and tables. The outcome of the study revealed that the Prandtl number has a decreasing effect on thermal boundary layer thickness. It also revealed that the thermal radiation has an increasing effect on the Nusselt number, temperature distribution, temperature gradient and skin friction. The temperature distribution increases with increasing values of the temperature difference and local convective heat transfer parameter whereas decreases with Grashof number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

C T :

temperature difference parameter

Bi x :

local convective heat transfer parameter

R :

thermal radiation parameter

Pr:

Prandtl number

Gr x :

local Grashof number

Re:

Reynold number

θ :

(temperature)

T :

temperature (K)

U :

velocity (ms−1)

k :

thermal conductivity (W/mK)

q r :

radiative heat flux (W/m2K)

β :

thermal expansion coefficient (K−1)

References

  • Abadeh, A., Sardarabadi, M., Abedi, M., Pourramezan, M. Passandideh-Fard, M., Maghrebi, M.J.: Experimental characterization of magnetic field effects on heat transfer coefficient and pressure drop for a ferrofluid flow in a circular tube. J. Mol. Liq. 299, 12206 (2019)

  • Ali Agha, H., Bouaziz, M.N., Hanini, S.: Free convection boundary layer flow from a vertical flat plate embedded in a Darcy porous medium filled with a Nanofluid: effects of magnetic field and thermal radiation. Arab. J. Sci. Eng. 39(11), 8331–8340 (2014)

    Google Scholar 

  • Aljoufi, M.D., Ebaid, A.: Effect of a convective boundary condition on boundary layer slip flow and heat transfer over a stretching sheet in view of the exact solution. J. Theor. Appl. Mech. 46(4), 85–95 (2016)

    MathSciNet  Google Scholar 

  • Aly, E.H., Ebaid, A.: Exact analysis for the effect of heat transfer on MHD and radiation Marangoni boundary layer nanofluid flow past a surface embedded in a porous medium. J. Mol. Liq. 215, 625–639 (2016)

    Google Scholar 

  • Aydın, O., Kaya, A.: MHD mixed convective heat transfer flow about an inclined plate. Heat Mass Transf. 46(1), 129 (2009)

    Google Scholar 

  • Aziz, A.: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1064–1068 (2009)

    Google Scholar 

  • Bhattacharyya, K.: Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection. Front. Chem. Sci. Eng. 5(3), 376–384 (2011)

  • Blasius, H.: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Druck von BG Teubner (1907)

  • Blasius, H.: The Boundary Layers in Fluids with Little Friction (1950)

    Google Scholar 

  • Boetcher, S.K.S., Sparrow, E.M.: Buoyancy-induced flow in an open-ended cavity: assessment of a similarity solution and of numerical simulation models. Int. J. Heat Mass Transf. 52(15), 3850–3856 (2009)

    MATH  Google Scholar 

  • Cao, K., Baker, J.: Non-continuum effects on natural convection–radiation boundary layer flow from a heated vertical plate. Int. J. Heat Mass Transf. 90, 26–33 (2015)

  • Dixit, Desh Deepak and Arvind Pattamatta. Effect of uniform external magnetic-field on natural convection heat transfer in a cubical cavity filled with magnetic nano-dispersion. Int. J. Heat Mass Transf. 146, 118828 (2020)

  • Egorov, I.V., Palchekovskaya, N.V., Obraz, A.O., Fedorov, A.V.: Effects of injection on heat transfer and the boundary-layer instability for a hypersonic blunt body configuration. Int. J. Heat Mass Transf. 149, 119197 (2020)

  • El-Hakiem, M.A.: MHD oscillatory flow on free convection--radiation through a porous medium with constant suction velocity. J. Magn. Magn. Mater. 220(2–3), 271–276 (2000)

    Google Scholar 

  • Fabregat, A., Pallarãs, J.: Heat transfer and boundary layer analyses of laminar and turbulent natural convection in a cubical cavity with differently heated opposed walls. Int. J. Heat Mass Transf. 151, 119409 (2020)

  • Fan, L.-W., Zhu, Z.-Q., Liu, M.-J.: A similarity solution to unidirectional solidification of nano-enhanced phase change materials (NePCM) considering the mushy region effect. Int. J. Heat Mass Transf. 86, 478–481 (2015)

    Google Scholar 

  • Hassan, A.R., Salawu, S.O.: Analysis of buoyancy driven flow of a reactive heat generating third grade fluid in a parallel channel having convective boundary conditions. SN Appl. Sci. 1(8), 919 (2019)

  • Hattori, T., Patterson, J.C., Lei, C.: Characterization of linear and oscillatory behaviours of radiation-induced natural convection boundary layer in response to constant and time-varying thermal forcing. Int. J. Heat Mass Transf. 87, 24–35 (2015)

    Article  Google Scholar 

  • Hossain, M.A., Takhar, H.S.: Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf. 31(4), 243–248 (1996)

    Article  Google Scholar 

  • Hossain, M.A., Alim, M.A., Rees, D.A.S.: The effect of radiation on free convection from a porous vertical plate. Int. J. Heat Mass Transf. 42(1), 181–191 (1999)

    MATH  Google Scholar 

  • Jha, B.K., Samaila, G.: Thermal radiation effect on boundary layer over a flat plate having convective surface boundary condition. SN Appl. Sci. 2(3), 381 (2020)

  • Jha, B.K., Isah, B.Y., Uwanta, I.J.: Combined effect of suction/injection on MHD free-convection flow in a vertical channel with thermal radiation. Ain Shams Eng. J. 9(4), 1069–1088 (2018)

    Google Scholar 

  • Karmakar, A., Sengupta, S.: Heat and mass transfer analysis on MHD mixed convection flow of radiative chemically heat generating fluid with viscous dissipation and thermo-diffusion effect. Front. Heat Mass Transf. 11 (2018). https://doi.org/10.5098/hmt.11.30

  • Klamka, M., Remer, M., Bobinski, T.: Beyond laminar regime – droplet interaction with air boundary layer. Int. J. Heat Mass Transf. 133, 542–547 (2019)

  • Lopez, A., Ibanez, G., Pantoja, J., Moreira, J., Lastres, O.: Entropy generation analysis of MHD nanofluid flow in a porous vertical microchannel with nonlinear thermal radiation, slip flow and convective-radiative boundary conditions. Int. J. Heat Mass Transf. 107, 982–994 (2017)

    Google Scholar 

  • Lyubimova, T., Parshakova, Y.: Onset of thermal buoyancy convection in a two-layer system with deformable Interface and fixed heat flux at the boundaries under terrestrial and microgravity conditions. Microgravity Sci. Technol. 32(3), 295–304 (2020)

    Google Scholar 

  • Magyari, E.: The moving plate thermometer. Int. J. Therm. Sci. 47(11), 1436–1441 (2008)

    Google Scholar 

  • Makinde, O.D.: Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int. Commun. Heat Mass Transf. 32(10), 1411–1419 (2005)

    Google Scholar 

  • Makinde, O.D., Olanrewaju, P.O.: Buoyancy effects on thermal boundary layer over a vertical plate with a convective surface boundary condition. J. Fluids Eng. 132(4), 44502 (2010)

  • Mukhopadhyay, S., Bhattacharyya, K., Layek, G.C.: Steady boundary layer flow and heat transfer over a porous moving plate in presence of thermal radiation. Int. J. Heat Mass Transf. 54(13), 2751–2757 (2011)

    MATH  Google Scholar 

  • Mythreye, A., Pramod, J.P., Balamurugan, K.S.: Chemical reaction on unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. Procedia Eng. 127, 613–620 (2015)

    Google Scholar 

  • Oh, G., Noh, K.M., Park, H., Choi, J.-I.: Extended synthetic eddy method to generate inflow data for turbulent thermal boundary layer. Int. J. Heat Mass Transf. 134, 1261–1267 (2019)

    Google Scholar 

  • Raptis, A.: Radiation and free convection flow through a porous medium. Int. Commun. Heat Mass Transf. 25(2), 289–295 (1998)

    Google Scholar 

  • Rashad, A.M.: Perturbation analysis of radiative effect on free convection flows in porous medium in the presence of pressure work and viscous dissipation. Commun. Nonlinear Sci. Numer. Simul. 14(1), 140–153 (2009)

    Google Scholar 

  • Sattar, M.D.A., Kalim, H.: Unsteady free-convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. J. Math. Phys. Sci. 30(1), 25–37 (1996)

    MATH  Google Scholar 

  • Sharypov, O.V., Kuibin, P.A.: 2D Flow Structure in a Thin Liquid Layer Under Thermal Wave Propagation. Microgravity Sci. Technol. 21(1), 321 (2009)

  • Sheikholeslami, M., Chamkha, A.J.: Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field. Numer. Heat Transf. Part A Appl. 69(10), 1186–1200 (2016)

    Google Scholar 

  • Taghite, M., Barake, B., Rahmattulla, A., Lanchon-Ducauquis, H.: Evaluation of the thermal boundary layer in the plate of a heat exchanger and the error estimate. Comput. Methods Appl. Mech. Eng. 178(1), 141–152 (1999)

    MATH  Google Scholar 

  • Voller, V.R.: A similarity solution for solidification of an under-cooled binary alloy. Int. J. Heat Mass Transf. 49(11), 1981–1985 (2006)

    MATH  Google Scholar 

  • Wang, Z.C., Tang, D.W., Hu, X.G.: Similarity solutions for flows and heat transfer in microchannels between two parallel plates. Int. J. Heat Mass Transf. 54(11), 2349–2354 (2011)

    MATH  Google Scholar 

  • Weyl, H.: On the differential equations of the simplest boundary-layer problems. Ann. Math. 43(2), 381–407 (1942)

    MathSciNet  MATH  Google Scholar 

  • Wu, Y., Zhang, W., Zou, Z., Chen, J.: Effects of heat transfer on separated boundary layer behavior under adverse pressure gradients. Int. J. Heat Mass Transf. 142, 118348 (2019)

    Google Scholar 

  • Xu, G., Liu, Q., Qin, J., Zhu, Z.-Q.: Numerical study of Thermocapillary-buoyancy convection of volatile liquid layer in an enclosed cavity. Microgravity Sci. Technol. 32(3), 305–319 (2020)

    Google Scholar 

  • Yao, S., Fang, T., Zhong, Y.: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 16(2), 752–760 (2011)

    MATH  Google Scholar 

  • Zhang, Y., Tan, H., Li, Y., Shan, S., Liu, Y.: A modified heat transfer correlation for flow boiling in small channels based on the boundary layer theory. Int. J. Heat Mass Transf. 132, 107–117 (2019)

    Google Scholar 

  • Zhou, Y., Hu, X., Li, T., Zhang, D., Zhou, G.: Similarity type of general solution for one-dimensional heat conduction in the cylindrical coordinate. Int. J. Heat Mass Transf. 119, 542–550 (2018)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ahmadu Bello University Zaria, Kaduna, Nigeria

    Basant K. Jha

  2. Department of Mathematics, Air Force Institute of Technology Kaduna, Kaduna, Nigeria

    Gabriel Samaila

Authors
  1. Basant K. Jha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriel Samaila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Samaila.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jha, B.K., Samaila, G. A Similarity Solution for Natural Convection Flow near a Vertical Plate with Thermal Radiation. Microgravity Sci. Technol. 32, 1031–1038 (2020). https://doi.org/10.1007/s12217-020-09830-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12217-020-09830-y

Keywords

Search

Navigation