Energy flows inside the wire both for AC and DC
Not necessarily. Take a superconductive wire at DC Inside the superconductor, electric field is zero everywhere (electric field potential is the same at any point inside the superconductor).
In other words: E = 0 inside such wire.
Electromagnetic energy flow is described by Poynting vector:
S = E x H = 0 (if E is zero vector, then its cross product is also zero vector no matter H field strength).
So no, at DC electromagnetic energy does not flow inside a superconductive wire.
If you replace superconductor with a wire that has resistance, then there will be an electric field inside the wire along its axis (equal to wire’s voltage drop divided by the length of the wire).
There is also a magnetic filed inside the wire when it conducts DC current. It can be quantified by Maxwell’s 4th equation. Such magnetic field circulates around wire’s axis. It’s strength is zero at the axis, and increases to a maximum value at wire surface.
Therefore EM energy flow inside such wire will be non zero. However the direction of the flow inside the wire will be strictly perpendicular to the wire (remember direction of E, and the fact that cross product of two vectors is perpendicular to the vectors being multiplied). The total flow at surface (surface integral S over surface of the wire) is equal to Joule heating of the wire.
Remaining EM energy flows outside the wire. Using Poynting vector formula and some math, it can be shown that the energy flown outside the wire + Joule heating is all the energy that flows out of battery. From conservation of energy principal, we can declare that there no other EM energy flows.