Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = Stx/Cf(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number Stx and the skin friction coefficient Cf(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity fw. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f′ by the simple relationship ϑ=(f′)Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in fw. The largest of them represents a critical value, so that for fw<fw,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.