For oil, we can generally expect a relatively high Prandtl number (Pr >> 1) meaning that momentum diffusion should occur significantly faster than thermal diffusion. You may find useful to check this somewhat related RG discussion, even if not specifically addressing the transport properties of oil: https://www.researchgate.net/post/What_is_the_relationship_between_specific_heat_capacity_and_viscosity_cSt_Is_there_any_reference_document

The Thermal Conductivity of oil doesn't actually vary very much I have attached a link to reference by J.C, Jones. A search for "Thermal Conductivity" will find about 80 occurrences. Page 105 has a correlation that might be useful to you.

https://kosalmath.files.wordpress.com/2010/08/hydrocarbons.pdf

I hope this helps.

Article Diffusion, Thermal Conductivity, and Viscous Flow of Liquids

Hi,

There is no relationsship between these two thermodynamic properties. High viscosity oils can have the same thermal conductivity as low viscosity oils if the substance class is the same or nearly the same. The same is true for oils with "high" thermal conductivity (polar ones) and oils with low thermal conductivity for viscosities. The can have the same viscosities from 0.x to xx.xxx at room temperature but other thermal conductivity. This depends only on the substances.

Some dimensionless numbers like Prandtl number combine some thermodynamic properties to forecast or compare observations in real circumstances. The properties in such numbers are mostly not connected by physical meaning like temperature dependence of thermophysical properties.

This comment is only to keep things in mind and driving the discussion in the right direction.

Best Regards

Steffen.