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 $\mathit{Pr}\rightarrow 0$.

Heat Transfer in Turbulent Pipe Flow for Liquids Having a Temperature-Dependent Viscosity

1978 ◽  
Vol 100 (2) ◽  
pp. 224-229 ◽  
Author(s):  
O. T. Hanna ◽  
O. C. Sandall
Keyword(s):  
Heat Transfer ◽  
Friction Factor ◽  
Pipe Flow ◽  
Turbulent Pipe Flow ◽  
Fluid Viscosity ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Analytical Approximations ◽  
Constant Viscosity ◽  
Dependent Viscosity

Analytical approximations are developed to predict the effect of a temperature-dependent viscosity on convective heat transfer through liquids in fully developed turbulent pipe flow. The analysis expresses the heat transfer coefficient ratio for variable to constant viscosity in terms of the friction factor ratio for variable to constant viscosity, Tw, Tb, and a fluid viscosity-temperature parameter β. The results are independent of any particular eddy diffusivity distribution. The formulas developed here represent an analytical approximation to the model developed by Goldmann. These approximations are in good agreement with numerical solutions of the model nonlinear differential equation. To compare the results of these calculations with experimental data, a knowledge of the effect of variable viscosity on the friction factor is required. When available correlations for the friction factor are used, the results given here are seen to agree well with experimental heat transfer coefficients over a considerable range of μw/μb.

One-dimensional viscous radiative gas with temperature dependent viscosity

2018 ◽  
Vol 38 (5) ◽  
pp. 1515-1548 ◽  
Author(s):  
Lin HE ◽  
Yongkai LIAO ◽  
Tao WANG ◽  
Huijiang ZHAO
Keyword(s):  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
One Dimensional ◽  
Dependent Viscosity

Theoretical Solutions for Combined Forced and Free Convection in Horizontal Tubes with Temperature-Dependent Viscosity

1976 ◽  
Vol 98 (3) ◽  
pp. 459-465 ◽  
Author(s):  
S. W. Hong ◽  
A. E. Bergles
Keyword(s):  
Circular Tube ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Flux Boundary Condition ◽  
Viscosity Parameter ◽  
Horizontal Tubes ◽  
Constant Viscosity ◽  
Heat Flux Boundary Condition ◽  
Dependent Viscosity ◽  
Good Agreement

A boundary layer solution is presented for fully developed laminar flow in a horizontal circular tube, assuming large Prandtl number and temperature-dependent viscosity and density. The solution is given by Nu = C1 Ra1/4, where C1 is a function of a nondimensional viscosity parameter and the heat flux boundary condition. The heat transfer predictions for large values of the viscosity parameter are 50 percent above the constant viscosity predictions. The present analysis is in good agreement with experimental data for water and ethylene glycol flowing in electrically heated tubes which approximate the boundary conditions assumed in the analysis.

scholarly journals One-dimensional compressible heat-conducting gas with temperature-dependent viscosity

2016 ◽  
Vol 26 (12) ◽  
pp. 2237-2275 ◽  
Author(s):  
Tao Wang ◽  
Huijiang Zhao
Keyword(s):  
Initial Data ◽  
Vacuum Solution ◽  
Viscosity Coefficient ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
One Dimensional ◽  
The One ◽  
Dependent Viscosity

We consider the one-dimensional compressible Navier–Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity [Formula: see text] and the heat conductivity [Formula: see text] depend on the specific volume [Formula: see text] and the temperature [Formula: see text] and are both proportional to [Formula: see text] for certain non-degenerate smooth function [Formula: see text]. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter [Formula: see text] and initial data, which imply that the initial data can be large if [Formula: see text] is sufficiently small. Such a result appears to be the first global existence result for general adiabatic exponent and large initial data when the viscosity coefficient depends on both the density and the temperature.

STATIONARY SOLUTIONS TO A QUASI-NEWTONIAN FLOW WITH VISCOUS HEATING

1995 ◽  
Vol 05 (06) ◽  
pp. 725-738 ◽  
Author(s):  
JACQUES BARANGER ◽  
ANDRO MIKELIĆ
Keyword(s):  
Negative Temperature ◽  
Stationary Flow ◽  
Finite Energy ◽  
Stationary Solutions ◽  
Viscous Heating ◽  
System Of Equations ◽  
Temperature Dependent ◽  
Newtonian Flow ◽  
Temperature Dependent Viscosity ◽  
Dependent Viscosity

System of equations describing the stationary flow of a quasi-Newtonian fluid, with temperature-dependent viscosity and with a viscous heating, is considered. Existence of at least one appropriate weak solution is proved, i.e. we get existence of at least one velocity field having finite energy and existence of a non-negative temperature field. Its regularity is a consequence of the L1-forcing term generated by the viscous heating.

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