Skip to main content
SpringerLink
Log in

Thermal instability of a rotating curved plate subjected to an applied magnetic field

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The effect of a magnetic field on thermal instability in mixed convection flow on a heated rotating convex surface is studied in this paper. The onset position characterized by the Goertler number G δ depends on the Grashof number, the rotational number, the Prandtl number, the magnetic field parameter, and the wave number. The buoyancy force, the centrifugal force, the Lorentz force, and the Coriolis force are found to significantly affect the flow structure and heat transfer of the flow. Negative rotation (clockwise) destabilizes the boundary layer flow on a convex surface. However, the Lorentz force stabilizes the flow. Numerical data in this study show the same order of magnitude like experimental data.

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′:

Dimensional wave number, a′ = 2π/λ (1/m)

(a′δ)mod :

Modified wave number, defined in Eq. (28)

a :

Dimensionless wave number, a = a′R/Re 1/2

B 0 :

The uniform magnetic field strength (Wb/m 2)

C f :

Local friction factor

F :

Velocity, pressure or temperature function

f :

The reduced stream function, ψ(ν XU ∞)−1/2

g :

Gravity acceleration (m/s2)

Gr :

The Grashof number, \({Gr=\frac{g\beta (T_w -T_\infty )R^{3}}{\nu^{2}}}\)

Gr X :

Local Grashof number, \({Gr_X =\frac{g\beta (T_w -T_\infty )X^{3}}{\nu^{2}}}\)

G :

The Goertler number based on the radius of curvature of the convex surface, G = 2R Re 1/2

G L :

The Goertler number based on the characteristic length of the flat plate, \({G=2L{Re}_L^{1/2}}\)

G X :

Local Goertler number, \({G_X =2X{Re}_X^{1/2} /R=G\times x^{3/2}}\)

G δ :

Local Goertler number based on the boundary layer thickness, defined in Eq. (26)

h :

Local heat transfer coefficient (W/m 2 K)

Ha :

The Hartmann number, \({Ha^{2}=\frac{\sigma B_0^2 R^{2}}{\rho \nu}}\)

L :

The characteristic length of the flat plate

M :

The magnetic field parameter, \({M=\frac{\sigma B_0^2 R}{\rho U_\infty}=\frac{Ha^{2}}{{Re}}}\)

p′, p:

Dimensional and dimensionless pressure, \({{p}'=\rho U_\infty^2 p/{Re}}\) (N/m2)

Pr :

The Prandtl number, ν/α

Nu X :

Local Nusselt number, hX/k

R :

Radius of curvature (m)

Ro :

Rotational number, Ω R/U ∞

Re :

The Reynolds number, U ∞ R/ν

Re L :

The Reynolds number based on the characteristic length of the flat plate, U ∞ L/ν

Re X :

Local Reynolds number, U ∞ X/ν

T :

Temperature (K)

t′, t:

Dimensional and dimensionless perturbation temperature, t′ = (T w − T ∞)t

t°:

Initial constant perturbation temperature at x = 0

U, V, W :

Dimensional velocity components (m/s)

u, v, w:

Dimensionless perturbation velocity components

u′, v′, w′:

Perturbation velocity components (m/s)

X, Y, Z :

Curvilinear coordinates (m)

x, y, z:

Dimensionless curvilinear coordinates

α :

Thermal diffusivity of fluid (m2/s)

β :

The coefficient of thermal expansion (1/K)

δ :

Boundary layer thickness (m)

η :

The similarity variable, \({YRe_X^{1/2} /X}\)

θ b :

Dimensionless basic temperature, (T − T ∞)/(T w − T ∞)

λ :

The wavelength in Z-direction (m)

σ :

Electrical conductivity, (mho/m)

τ :

The elapsed time (s)

τ w :

Local wall shear stress

ν :

Kinematic viscosity of the fluid (m2/s)

ξ :

Vorticity function in X-direction (1/s)

ψ :

Stream function (m2/s)

Ω:

Angular speed ( rad/s)

*:

Onset position

b :

Basic flow quantity

p :

Perturbation quantity

w :

Surface condition

X :

Local coordinate

∞:

Free stream condition

References

  1. Bejan A.: Heat Transfer. 1st edn. Wiley, New York (1993)

    Google Scholar 

  2. Chen C.T., Lin M.H.: Effect of rotation on Goertler vortices in the boundary layer flow on a curved surface. Int. J. Numer. Methods Fluids 40, 1327–1346 (2002)

    Article  MATH  Google Scholar 

  3. Hwang G.J., Lin M.H.: Estimation of the onset of longitudinal vortices in a laminar boundary layer heated from below. ASME J. Heat Transf. 117, 835–842 (1995)

    Article  Google Scholar 

  4. Jean, J.F.: Visualization of Vortices in a Rotating Concave-Rectangular Passage. Ms. Thesis, National Tsing-Hwa University, Taiwan, ROC (1997)

  5. Lee S.L.: Weighting function scheme and its application on multidimensional conservation equations. Int. J. Heat Mass Transf. 32, 2065–2073 (1982)

    Google Scholar 

  6. Lin, C.H.: Visualization of Vortex Instability in Radically Rotating Rectangular Channel. Master thesis, National Tsing-Hwa University, Taiwan (1996)

  7. Lin M.H., Hwang G.J.: Numerical study of formation of longitudinal vortices in a laminar boundary layer with rotation. J. Chin. Soc. Mech. Eng. 2, 341–349 (2000)

    Google Scholar 

  8. Lin M.H., Chen C.T.: Study on the formation of Goertler vortices in natural convection flow over a rotating concave surface. Appl. Math. Comput. 169, 778–796 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lin M.H., Chen C.T.: Study of thermal instability in natural convection flow over a rotating convex surface. J. Enhan. Heat Transf. 12, 259–272 (2005)

    Article  Google Scholar 

  10. Lin M.H., Chen C.T.: Study of thermal instability in mixed convection flow over a rotating convex surface. J. Chin. Soc. Mech. Eng. 17, 343–351 (2006)

    Google Scholar 

  11. Lin M.H.: Thermal instability in natural convection flow over a rotating convex surface subject to external magnetic field. Appl. Math. Comput. 198, 511–525 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lin M.H.: Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating laminar boundary layer. Appl. Math. Model. 32, 547–561 (2008)

    Article  MATH  Google Scholar 

  13. Lin M.H., Chen C.T.: Magnetic effect on the formation of longitudinal vortices in natural convection flow over a horizontal plate. J. Aeronaut. Astron. Aviat. 39, 145–152 (2007)

    Google Scholar 

  14. Matsson O.J.E., Alfredsson P.H.: The effect of spanwise system rotation on dean vortices. J. Fluid Mech. 274, 243–265 (1994)

    Article  Google Scholar 

  15. Matsson O.J.E.: Experiments on streamwise vortices in curved wall jet flow. Phys. Fluids 7, 2978–2988 (1995)

    Article  Google Scholar 

  16. Matubara, M., Masuda, S.: Three-Dimensional Instability in Rotating Boundary Layer. FED-Vol. 114, ASME, Boundary layer stability and transition to turbulence, pp. 103–107 (1991)

  17. Mutabazi I., Normand C., Wesfreid J.E.: Gap size effects on centrifugally and rotationally driven instabilities. Phys. Rev. A 4, 1199–1205 (1992)

    MATH  Google Scholar 

  18. Selmi A., Nandakumar K., Finley W.H.: A bifurcation study of viscous flow through a rotating concave duct. J. Fluid Mech. 262, 353–375 (1994)

    Article  MATH  Google Scholar 

  19. Wang L., Cheng K.C.: Flows in concave channel with a low negative rotation speed. Phys. Rev. E 51, 1155–1161 (1995)

    Article  Google Scholar 

  20. Wang L., Cheng K.C.: Flow transitions and combined free and forced convective heat transfer in rotating concave channels: the case of positive rotation. Phys. Fluids 8, 1553–1573 (1996)

    Article  MATH  Google Scholar 

  21. Wang L.: The effect of negative spanwise rotation on dean vortices. ASME J. Fluids Eng. 119, 718–721 (1997)

    Article  Google Scholar 

  22. Wang L.: Buoyancy-force–driven transitions in flow structures and their effects on heat transfer in a rotating concave channel. Int. J. Heat Mass Transf. 40, 223–235 (1997)

    Article  MATH  Google Scholar 

  23. White, F.M.: Viscous Fluid Flow, 2nd edn., p. 378. McGraw-Hill, New York (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Engineering Management, Ta-Hwa Institute of Technology, Hsinchu, 307, Taiwan, Republic of China

    Chin-Tai Chen

Authors
  1. Chin-Tai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chin-Tai Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, CT. Thermal instability of a rotating curved plate subjected to an applied magnetic field. Acta Mech 214, 343–356 (2010). https://doi.org/10.1007/s00707-010-0289-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-010-0289-6

Keywords

Search

Navigation