Does Figure C Show A Constant Temperature Profile for Plane Wall Heat Transfer?

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no183
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Refer to the attached picture,
condition : assume steady state
legend : red : insulated (adiabatic)
black : no condition
Green : Temperature distribution, T(X)
I started with figure A, from heat equation, A u will get the distribution with is just a constant temperature..
For B, u will get a linear temperature gradient as shown in the figure, how bout C?
 

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mean, just a normal wall.. non-insulated
 
Then why would there be a temperature gradient in part (a)? Is there another constraint that the temperature is held at certain (different) values on the left and right sides? If so, what are the temperature constraints in part (c)?
 
ok, let's say heat is LEAVING from figure A RIGHT HAND SIDE (BLACK) so i will get the temperature distribution
While on B, heat is MOVING IN TO THE WALL from RIGHT HAND SIDE (BLACK) and i will get the temperature distribution
lastly C, what i will get if
a) heat LEAVING the wall on the right hand side
b) heat ENTERING the wall on the right hand side
 
no183 said:
While on B, heat is MOVING IN TO THE WALL from RIGHT HAND SIDE (BLACK) and i will get the temperature distribution

I don't think so, for a couple reasons. First, heat flux is always coupled to a temperature gradient (this is Fourier's Law), so the temperature profile wouldn't be flat. Second, there's nowhere for this heat to go. Therefore, there won't be any steady state condition; the temperature will just keep increasing indefinitely.