Skip to main content
SpringerLink
Log in

Unsteady mixed convective heat transfer of two immiscible fluids confined between long vertical wavy wall and parallel flat wall

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The combined effects of thermal and mass convection of viscous incompressible and immiscible fluids through a vertical wavy wall and a smooth flat wall are analyzed. The dimensionless governing equations are perturbed into a mean part (the zeroth-order) and a perturbed part (the first-order). The first-order quantities are obtained by the perturbation series expansion for short wavelength, in which the terms of the exponential order arise. The analytical expressions for the zeroth-order, the first-order, and the total solutions are obtained. The numerical computations are presented graphically to show the salient features of the fluid flow and the heat transfer characteristics. Separate solutions are matched at the interface by using suitable matching conditions. The shear stress and the Nusselt number are also analyzed for variations of the governing parameters. It is observed that the Grashof number, the viscosity ratio, the width ratio, and the conductivity ratio promote the velocity parallel to the flow direction. A reversal effect is observed for the velocity perpendicular to the flow direction.

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

Similar content being viewed by others

Abbreviations

a :

amplitude, m

g :

acceleration due to gravity, m·s−2

Gr :

Grashof number, \((h^{(1)^3 } g\beta ^{(1)} \Delta T)/v^{(1)^2 } \)

K :

thermal conductivity, W·m−1·K−1

k :

conductivity ratio, K (2)/K (1)

m :

viscosity ratio, µ(1)/µ(2)

Nu :

Nusselt number

P :

pressure, N·m−2

ps:

static pressure, N·m−2

Pr :

Prandtl number, (C (1)p µ(1))/K (1)

r :

density ratio, ρ (2)/ρ (1)

T*, t :

dimensionless temperature

u, v :

dimensionless velocities

X, Y :

space coordinates, m

x, y :

dimensionless space coordinates

β (j) :

coefficient of thermal expansion

λ :

wavelength, m

ν (j) :

kinematic viscosity, µ(j)/ρ (j)

ρ :

density, kg·m−3

ρ 0 :

static density, kg·m−3

ω :

frequency parameter

ψ :

stream function

C (j)p :

specific heat at constant pressure, J·kg−1·K−1

U, V :

velocities along the X- and Y -directions, m·s−1

ɛ :

dimensionless amplitude parameter, a/h (1)

β :

ratio of coefficient of thermal expansion, β (2)/β (1)

λ* :

dimensionless wave number, λ/h (1)

µ:

viscosity, Kg·m−1·s−1

ωt :

periodic frequency parameter

1:

reference quantity for fluids in region-I

2:

reference quantity for fluids in region-II

References

  1. Chiu, C. P. and Chou, H. M. Transient analysis of natural convection along a vertical wavy surface in micro polar fluids. International Journal of Engineering Science, 32, 19–33 (1994)

    Article  MATH  Google Scholar 

  2. Rees, D. A. S. and Pop, I. A note on free convection along a vertical wavy surface in a porous medium. ASME Journal of Heat Transfer, 116, 505–508 (1994)

    Article  Google Scholar 

  3. Rees, D. A. S. and Pop, I. Free convection induced by a vertical wavy surface with uniform heat flux in a porous medium. ASME Journal of Heat Transfer, 117, 547–550 (1995)

    Article  Google Scholar 

  4. Yang, Y. T., Chen, C. K., and Lin, M. T. Natural convection of non-Newtonian fluids along a wavy vertical plate including the magnetic field effect. International Journal of Heat and Mass Transfer, 39, 2831–2842 (1996)

    Article  MATH  Google Scholar 

  5. Yao, L. S. Natural convection along a vertical complex wavy surface. International Journal of Heat and Mass Transfer, 49, 281–286 (2005)

    Article  Google Scholar 

  6. Yao, L. S. Natural convection along a wavy surface. ASME Journal of Heat Transfer, 105, 465–468 (1983)

    Article  Google Scholar 

  7. Yao, L. S. A note on Prandtl’s transposition theorem. ASME Journal of Heat Transfer, 110, 503–507 (1988)

    Google Scholar 

  8. Yao, L. S. and Moulic, S. G. Mixed convection along a wavy surface. ASME Journal of Heat Transfer, 111, 974–979 (1989)

    Article  Google Scholar 

  9. Vajravelu, K. and Sastri, K. S. Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. Journal of Fluid Mechanics, 80, 365–383 (1978)

    Article  MathSciNet  Google Scholar 

  10. Chen, C. K. and Wang, C. C. Transient analysis of force convection along a wavy surface in micropolar fluids. AIAA Journal of Thermophysics and Heat Transfer, 14, 340–347 (2000)

    Article  Google Scholar 

  11. Chen, C. K. and Wang, C. C. Transient force and free convection along a vertical wavy surface in micropolar fluids. International Journal of Heat and Mass Transfer, 44, 3241–3251 (2001)

    Article  MATH  Google Scholar 

  12. Cheng, C. Y. Natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. International Communications in Heat and Mass Transfer, 27, 1143–1154 (2000)

    Article  Google Scholar 

  13. Cheng, C. Y. Natural convection heat and mass transfer near a wavy cone with constant wall temperature and concentration in a porous medium. Mechanics Research Communications, 27, 613–620 (2000)

    Article  MATH  Google Scholar 

  14. Malashetty, M. S., Umavathi, J. C., and Leela, V. Magnetoconvective flow and heat transfer between a vertical wavy wall and a parallel flat wall. International Journal of Applied Mechanics and Engineering, 6(2), 437–456 (2001)

    MATH  Google Scholar 

  15. Fasogbon, P. F. Analytical studies of heat and mass transfer by free convection in a two-dimensional irregular channel. International Journal of Applied Mechanics and Engineering, 6(4), 17–37 (2010)

    Google Scholar 

  16. Batchelor, G. K. An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge (1967)

    MATH  Google Scholar 

  17. Ishii, M. Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris (1975)

    MATH  Google Scholar 

  18. Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. Journal of Applied Physics, 25, 1400–1406 (1954)

    Article  Google Scholar 

  19. Loharsbi, J. and Sahai, V. Magnetohydrodynamic heat transfer in two-phase flow between parallel plates. Applied Scientific Research, 45, 53–66 (1988)

    Article  Google Scholar 

  20. Akai, M., Inoue, A., and Aokis, S. The prediction of stratified two-phase flow with two-equation model of turbulence. International Journal of Multiphase Flow, 7, 21–29 (1981)

    Article  MATH  Google Scholar 

  21. Jones, W. P. and Launder, B. E. The calculation of low Reynolds number phenomena with a two-equation model of turbulence. International Journal of Heat and Mass Transfer, 16, 1119–1130 (1992)

    Article  Google Scholar 

  22. Packham, B. A. and Shail, R. Stratified laminar flow of two immiscible fluids. Proceedings of Cambridge Philosophical Society, 69, 443–448 (1971)

    Article  MATH  Google Scholar 

  23. Malashetty, M. S. and Leela, V. Magnetohydrodynamic heat transfer in two-phase flow. International Journal of Engineering Science, 30, 371–377 (1992)

    Article  MATH  Google Scholar 

  24. Malashetty, M. S., Umavathi, J. C., and Prathap-Kumar, J. Convective flow and heat transfer in an inclined composite porous medium. Journal of Porous Media, 15, 15–22 (2001)

    Google Scholar 

  25. Malashetty, M. S., Umavathi, J. C., and Prathap-Kumar, J. Two fluid flow and heat transfer in an inclined composite porous and fluid layer. Heat and Mass Transfer, 40, 871–876 (2004)

    Article  Google Scholar 

  26. Umavathi, J. C., Chamkha, A. J., Mateen, A., and Al-Mudhaf, A. Unsteady two fluid flow and heat transfer in a horizontal channel. Heat and Mass Transfer, 42, 81–90 (2005)

    Article  Google Scholar 

  27. Umavathi, J. C., Chamkha, A. J., Mateen, A., and Prathap-Kumar, J. Unsteady magnetohydrodynamic two-fluid flow and heat transfer in a horizontal channel. International Journal of Heat and Technology, 26, 121–133 (2008)

    Google Scholar 

  28. Umavathi, J. C., Manjula, M. H., Pop, I., and Liu, I. C. Flow and heat transfer of couple stress viscous fluids in a vertical channel. International Journal of Applied Mechanics and Engineering, 12, 537–555 (2007)

    Google Scholar 

  29. Prathap-Kumar, J., Umavathi, J. C., and Biradar, B. M. Mixed convective flow of immiscible viscous fluids in a vertical channel. Heat Transfer — Asian Research, 40, 1–25 (2011)

    Article  Google Scholar 

  30. Hossain, M. Z. and Sadrul-Islam, A. K. M. Numerical investigation of unsteady flow and heat transfer in wavy channels. The 15th Australasian Fluid Mechanics Conference, Sydney, 13–17 (2004)

  31. Deka, R. K. and Neog, B. C. Unsteady natural convection flow past an accelerated vertical plate in a thermally stratified fluid. Theoretical and Applied Mechematics, 36(4), 261–274 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rostami, J. Unsteady natural convection in an enclosure with vertical wavy walls. Heat and Mass Transfer, 44, 1079–1087 (2008)

    Article  Google Scholar 

  33. Soundalgekar, V. M., Deka, R. K., and Das, U. N. Transient free convection flow of a viscous incompressible fluid past an infinite vertical plate with periodic heat flux. Indian Journal of Engineering and Material Sciences, 10, 390–396 (2003)

    Google Scholar 

  34. Ostrach, S. An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force, NACA-report-1111, Washington (1952)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Karnataka, 585106, India

    J. C. Umavathi & M. Shekar

  2. Department of Civil Engineering, National Chi Nan University, Nantou, 54561, Taiwan, China

    I. C. Liu

Authors
  1. J. C. Umavathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. C. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Shekar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. C. Umavathi.

Additional information

Project supported by the Major Research Projects of University Grants Commission of India (No. F.No. 37-178 (2009))

Rights and permissions

Reprints and permissions

About this article

Cite this article

Umavathi, J.C., Liu, I.C. & Shekar, M. Unsteady mixed convective heat transfer of two immiscible fluids confined between long vertical wavy wall and parallel flat wall. Appl. Math. Mech.-Engl. Ed. 33, 931–950 (2012). https://doi.org/10.1007/s10483-012-1596-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-012-1596-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation