Abstract
The thermal stability of a vibrationally nonequilibrium molecular gas with a fixed vibrational energy flowing in a circular tube with a constant surface temperature was theoretically analyzed. The analysis determined the boundaries for the existence of the vibrationally nonequilibrium gas state. It was shown that the upper bound for the existence of a vibrationally nonequilibrium state of a molecular gas increases linearly with increasing Reynolds number over the investigated range, 0 ≤ Re ≤ 550. It also increases when the length of the tube relative to its radius is decreased, and when the nondimensional quantity, \({\delta ={\frac{\rho\varepsilon_{\rm eq}({\rm T}_{\rm S}){\rm r}_{0}^{2}}{\lambda\tau({\rm T}_{\rm S}){\rm T}_{\rm S}}}}\), is decreased. In this nondimensional quantity, ρ is the density of the gas, T_{S} is the tube surface temperature, τ is the gas vibrational relaxation time, ε _{eq} is the gas vibrational energy at T_{S}, and λ is the gas coefficient for heat conduction. To obtain a large storage of excess vibrational energy, low values of δ must be used.
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Abbreviations
 ε :

Excess gas vibrational energy
 ε _{eq} :

Equilibrium gas vibrational energy
 ε _{c} :

Critical excess gas vibrational energy
 T:

Gas translational temperature
 T_{S} :

Tube surface temperature
 T_{max} :

Maximum gas translational temperature along tube axis
 T_{v} :

Gas vibrational temperature
 τ :

Gas vibrationalrelaxation time
 λ:

Gas heatconduction coefficient for equilibrium degrees of freedom
 C_{V} :

Gas specificheat capacity for equilibrium degrees of freedom
 ρ :

Gas density
 U:

Gas velocity
 P:

Gas pressure
 μ :

Coefficient of gas viscosity
 r_{0} :

Tube radius
 L:

Tube length
 \({\gamma=\frac{{\rm L}}{{\rm r}_{\rm 0}}}\) :

Ratio of tube length to tube radius
 ε_{eq}(T_{S}):

Equilibrium gas vibrational energy at tube surface temperature T_{s}
 \({\beta=\frac{\varepsilon_{\rm c}}{\varepsilon_{\rm eq} \left({{\rm T}_{\rm s}}\right)}}\) :

Nondimensional critical vibrational energy
 τ(T_{s}):

Gas vibrational relaxation time at tube surface temperature T_{s}
 \({\delta=\frac{\rho\varepsilon_{\rm eq}\left({\rm T}_{\rm S}\right){\rm r}_{\rm 0}^{\rm 2}}{\lambda\tau\left({\rm T}_{\rm S}\right){\rm T}_{\rm S}}}\) :

Nondimensional quantity containing gas parameters
 \({\phi_{\rm f}}\) :

Heat flux out of the gas by flow
 \({\phi_{\rm s}}\) :

Heat flux out of the gas by conduction
 \({\phi=\phi_{\rm f}/\phi_{\rm s}}\) :

Nondimensional heat flux
 Re:

Reynolds number
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Younis, S.M. Analysis of the Thermal Stability of a Vibrationally NonEquilibrium Molecular Gas with Fixed Vibrational Energy Flowing in a Circular Tube. Arab J Sci Eng 36, 137–143 (2011). https://doi.org/10.1007/s1336901000166
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Keywords
 Molecular energy transfer
 Heat transfer