I need proper explanation of linear thermal radiation and nonlinear thermal radiation. And how to differentiate whether a particular fluid flow is subjected to linear or nonlinear thermal radiation by observing the governing equations?
But@dudley i have noticed in many research articles that the concerned flow being subjected to either linear/non-linear thermal radiation
That may be so but it just means that the authors are using the terms incorrectly. Perhaps they have created a boundary condition with linearly varying temperature and are considering radiative heat transfer in their calculations, but this doesn't mean that it's "linear radiation". Radiative heat transfer is not often considered in a flow. Are you trying to model the Space-X rocket or something like that? Many years ago when the Saturn V engine was first developed, Lockheed promised to provide a numerical model of the engines to NASA. Launch was approaching and no model. William R. Waldrop said he would do it and he did... in FORTRAN on punch cards. He wanted to get a PhD and they weren't interested in the Aerospace department, so he got into the Chemical Engineering department at LSU and turned the Saturn V rocket engine code into a sediment transport code because there was funding to model the Mississippi River as it flowed out into the Gulf of Mexico. Then he converted it into a reservoir model. Not that's some accomplishment!
In simplified heat transfer calculations the radiative convective coefficient is defined as: hr=sigma x [(T1^2-T2^2) x (T1+T2)] x emissivity, W/m^2/K. This is simply: sigma x (T1^4-T2^4)/(T1-T2) x emissivity. This, linearised form can be used in some complex cavities where radiation is small compared with convection, rather than the highly non-linear T^4 format: the value of hr is then added to the convective coefficient. It is a useful approximation, often used in engineering calculations, but must be used with due care and attention.
Jack Broughton the first relation should be hr=sigma x {(T1^2+T2^2) x (T1+T2)] x emissivity
Madiha Takreem: you are quite correct: a typo I'm afraid. If the temperature differences in the zone are small it is a good simplification.