# The onset of natural convection and heat transfer correlation in horizontal fluid layer heated uniformly from below

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## Abstract

The critical condition of the onset of buoyancy-driven convective motion of uniformly heated horizontal fluid layer was analysed by the propagation theory which transforms the disturbance quantities similarly. The dimensionless critical time,*τ* _{ c }, is obtained as a function of the Rayleigh number and the Prandtl number. Based on the stability criteria and the boundary-layer instability model, a new heat transfer correlation which can cover whole range of Rayleigh number was derived. Our theoretical results predict the experimental results quite reasonably.

## Key words

Buoyancy Effect Stability Analysis Heat Transfer Correlation Propagation Theory## Nomenclature

- a
Horizontal wave number [−]

- d
Fluid depth [m]

- g
Gravitational acceleration [m/s

^{2}]- k
Thermal conductivity [J/mK]

- Nu
Nusselt number (=q

_{w}/kΔ*T*) [−]- P
Pressure [Pa]

- Pr
Prandtl number (=ν/α) [−]

- Ra
Rayleigh number (=gβΔ

*T*d^{3}/αν) [−]- Ra
_{q} Rayleigh number based on the heat flux (=gβq

_{w}d^{4}/kαν) [−]- q
_{w} Wall heat flux [J/m

^{2}]- T
Temperature [K]

- t
Time [s]

- \(\overrightarrow U \)
Velocity vector [m/s]

- w
Dimensionless vertical velocity [−]

- X, Y, Z
Space in Cartesian coordinate [m]

## Greeks

- α
Thermal diffusivity [m

^{2}/s]- β
Thermal expansion coefficient [1/K]

- ΔT
Temperature difference [K]

- ζ
Similarity variable [−]

- θ
Dimensionless temperature [−]

- μ
Viscosity [Pa s]

- ν
Kinematic viscosity [m

^{2}/s]- ρ
Density [kg/m

^{3}]- τ
Dimensionless time [−]

## Subscripts

- 0
Basic quantity

- 1
Disturbed quantity

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