change in vicosity consumes energy?

Look up "specific heat capacity". See: I hope this helps.

Not exactly. Some of the energy was consumed in making the oil hotter; all of that is (in theory) recoverable. I want to ignore that energy.
I'm talking about the energy that was used in changing the viscosity of the oil After all, we can imagine (& maybe there really exists) two different oils with the same specific-heat, but different Viscosity Indexii

Viscosity is describable as the resistance to the transfer of momentum between adjacent regions in a fluid.
In collisions between molecules, whether in liquids or in gases, momentum and energy are conserved and bulk thermal equilibrium is maintained. There is no storage or retrieval of energy in viscosity.
Changing the temperature of a fluid may store or retrieve energy through some other process such as a phase change or some such, but not through viscosity changes.
In general, raising the temperature will decrease the viscosity of liquids, but increase the viscosity of gases. (Interestingly, the viscosity of a mixture of gases is invariably higher than what might be expected from an "ideal" mixture relationship. I suspect the same is true of miscible liquids, for mechanistic reasons, but I have not gathered the data to demonstrate this.)
FWIW, the reciprocal of viscosity is called the fluidity.
Tom Davidson Richmond, VA

What makes you think the "energy used in changing the viscosity of the oil" is different from the energy used in heating the oil?
The chief problem with your question is that it seems to put "viscosity change" on an equal footing with a delta-E. Viscosity is a property of the physical state of the oil. In general there is a delta-E when the oil changes physical state, but no part of this is specifically assignable to "viscosity change".
There is no reason two oils with identical specific heats in some range could not undergo different changes in viscosity, even assuming all physical changes in heating were reversible: the "explanation" for this (which seems to presume it needs one), would presumably lie in the details of their molecular structures.

Dear dances_with_barka...:
Which is completely recovered when the oil cools, because its viscosity goes back up. Unless you "burn" the oil...
"indexes". Doesn't matter, you get all the heat back. All you are doing is changing the bonding between the molecules, when you consider viscosity index, and this is not a permanent change unless you alter the molecules themselves (oxidation, polmerization, scissioning).
David A. Smith

Yes...
Energy is consumed so to speak, in heating a liquid.
When a liquid cools, the energy associated with that temperature is reduced, and it goes somewhere else.
It's just the same boring old parameter for many liquids, like water for example - whose viscosity also varies with temperature. It's called the Specific Heat capacity. For water it's around 4190 joules per kilogram. degree C Oils can have half that value.
Think of viscosity as the propensity to provide internal friction on surfaces moving relative to each other in the fluid.
Brian Whatcott Altus OK