Abstract
The effect of horizontal as well as vertical temperature gradients on the stability of natural convection in a thin horizontal layer of viscous, incompressible fluid is studied on the basis of linear theory. The boundaries are taken to be rigid, perfectly thermally conducting, having prescribed temperatures and the horizontal temperature gradient is assumed to be small. It is found that for Prandtl number greater than 0.13, the critical Rayleigh number is always larger than that for the corresponding Benard problem. The preferred mode of disturbance is stationary and will be a transverse roll (having axes normal to the basic flow) or a longitudinal roll (having axes aligned in the direction of the basic flow) depending on whether the Prandtl number is less or larger than 1.7. Finally, it is shown that the instability is of thermal origin.
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Abbreviations
- d :
-
depth of the layer
- k, m :
-
wave numbers in the x and y direction
- i, j, k :
-
unit vectors
- v :
-
velocity vector
- u, v, w :
-
velocity components
- t :
-
time
- T o :
-
standard temperature
- T :
-
temperature
- U(z):
-
basic velocity field defined by (2.9)
- T(z):
-
basic temperature field defined by (2.10)
- p :
-
pressure
- D :
-
differential operator d/dz
- ▽2 :
-
Laplacian operator
- ΔT :
-
temperature difference between lower and upper plates
- P r :
-
Prandtl number ν/κ (ν = kinematic viscosity κ = thermal diffusivity)
- R :
-
Rayleigh number \(g\bar \alpha \Delta {\rm T}d^3 /\kappa \nu\) (g acceleration due to gravity \(\bar \alpha \) coefficient of volume expansion)
- \(\bar U\) :
-
defined by (3.11)
- A1, A2, A3, q1, q2, q3, D1, D2, D3 :
-
constants defined by (3.17)
- w *0 :
-
defined by (3.18)
- C1, C2, C3 :
-
constants defined by (3.19)
- G1, G2, G3 :
-
defined by (3.25)
- G 4(P r ), G 5(P r ), G 6, F 1(P r ), F 2(P r ):
-
defined by (3.43)
- \(\bar w_1 \) :
-
defined by (3.31)
- K′, K, P, P′ :
-
defined by (4.2)–(4.5)
- S :
-
defined by (4.8)
- α :
-
overall wave number
- β :
-
horizontal temperature gradient
- θ :
-
temperature
- θ *0 , \(\overline {\theta _0^* }\) :
-
defined by (3.18)
- \(\overline \Theta\) :
-
defined by (3.11)
- σ :
-
amplification factor of disturbance (≡ σ r+iσ i)
- ε, Φ, δ, \(\bar \theta _1\) :
-
defined by (3.23), (3.36), (3.41), (3.33) respectively
- +:
-
average over one wavelength in the x and y directions
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Bhattacharyya, S.P., Nadoor, S. Stability of thermal convection between non-uniformly heated plates. Appl. Sci. Res. 32, 555–570 (1976). https://doi.org/10.1007/BF00384118
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DOI: https://doi.org/10.1007/BF00384118