High rayleigh number convection at infinite prandtl number with weakly temperature-dependent viscosity

1996 ◽  
Vol 83 (1-2) ◽  
pp. 79-117 ◽  
Author(s):  
S. Balachandar ◽  
D. A. Yuen ◽  
D. M. Reuteler
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Dependent Viscosity

scholarly journals Analytical shock solutions at large and small Prandtl number

2013 ◽  
Vol 726 ◽  
Author(s):  
B. M. Johnson
Keyword(s):  
Prandtl Number ◽  
Maximum Error ◽  
System Of Equations ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
One Dimensional ◽  
Ideal Gases ◽  
Constant Viscosity ◽  
Dependent Viscosity ◽  
Shock Fronts

AbstractExact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-$\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small-$\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-$\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$.

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