2D heat conduction through an annulus with time-dependent boundary condition

Question

So I had an old post a couple of days ago about a similar problem with Dirichlet boundary conditions for the inner and outer boundary. The link to the original post:

2D heat conduction/advection through an annulus with Dirichlet Conditions

Now I'm wondering about replacing the inner boundary with a time dependent one. Just to reiterate the setup: I want to simulate heat conduction (and advection) through an annulus and its surrounding material in 2D, as illustrated in this mesh. There is an overarching heat conduction equation that dictates heat conduction/advection in both materials:

$\[Sigma][x, y]*D[T[t, x, y], t] + ux[x, y]*D[T[t, x, y], x] -   \[Alpha][x, y]*Laplacian[T[t, x, y], {x, y}] - q[x, y]/Subscript[(Subscript[\[Rho]c, p]), f]$

In the gray region I set the advection term to 0. In the yellow region there is both conduction and advection. At the outer edge of the yellow region, I want the temperature to be at a constant 12 degrees. Originally, I set the inner boundary of the gray region to be at 45 degrees. Here is the original solution posted by Alex Trounev.

    tm = 5;
    \[CapitalOmega]1 = 
    ImplicitRegion[(x - 2.5)^2 + (y - 2.5)^2 >= 0.09 && (0 <= x <= 5) 
    && (0 <= y <= 5), {x, y}];
    \[Sigma] [x_, y_] := 
    Piecewise[ {{1, .09 <= ((x - 2.5)^2) + ((y - 2.5)^2 ) <= 0.25}, 
    {5.95396, ((x - 2.5)^2) + ((y - 2.5)^2 ) >  0.25}}]
    u [x_, y_] := 0;
    \[Alpha] [x_, y_] := 
    Piecewise[ {{2.82965, .09 <= ((x - 2.5)^2) + ((y - 2.5)^2 ) <= 
    0.25}, {5.02649, ((x - 2.5)^2) + ((y - 2.5)^2 ) >  0.25}}] 
    q [x_, y_] := Piecewise[{{0, .09 < ((x - 2.5)^2) + ((y - 2.5)^2 ) 
    < 0.25}, {1, ((x - 2.5)^2) + ((y - 2.5)^2 ) >  0.25}}]
    eq = \[Sigma] [x, y]* D[T[t, x, y] , t] - \[Alpha][x, y]*
    Laplacian[T[t, x, y], {x, y}] - q[x, y] == 0;
    Subscript[\[CapitalGamma], D] = {DirichletCondition[T[t, x, y] == 
    45,((x - 2.5)^2 + (y - 2.5)^2 ) == 0.09], 
    DirichletCondition[T[t, x, y] == 12, x == 0 || x == 5 || y == 0 
    || y == 5]};
    Temp = NDSolveValue[{eq, Subscript[\[CapitalGamma], D], T[0, x, 
    y] == 12}, T, {t, 0, tm}, {x, y} \[Element] \[CapitalOmega]1, 
    Method -> {"FiniteElement", "InterpolationOrder" -> {T -> 2}, 
    "MeshOptions" -> {"MaxCellMeasure" -> 0.001}}];
    Table[ContourPlot[Temp[t, x, y], {x, y} \[Element] \ . 
    [CapitalOmega]1, 
    Contours -> 20, ColorFunction -> "TemperatureMap", 
    PlotLegends -> Automatic, PlotRange -> All, 
    PlotLabel -> Row[{"t=", t}]], {t, .1*tm, tm, .3*tm}]

Now my question is, what if I want to set the inner boundary at a constant temperature of 12 initially, and make the temperature on the boundary change with the following equation:

D[T[t, x, y], t] + A - B(T[t, 2.81, 2.5 ] - T[t, 2.8, 2.5])==0

where A and B are constants. This equation essentially specifies the temperature on the inner boundary of the gray region, but the boundary's temperature is dependent on the solution T[t, x, y] at two points {2.81, 2.5} and {2.8, 2.5} as well as time. Note that I choose these two somewhat arbitrary points to denote the temperature difference between the temperature at the inner boundary and in the gray region immediately next to the boundary. This expression can probably be replaced by some more sophisticated expression if viable.

Do I have to solve this equation simultaneously with the previous overarching heat conduction/advection equation? If so how can I accomplish this in Mathematica?

Update:

A better way of expressing the equation that specifies the temperature on the inner boundary is probably:

D[F[t, x, y], t] + A - B(T[t, x, y ] - F[t, x, y])==0

where T[t, x, y] is defined on the gray and yellow region, excluding the inner boundary,

and F[t, x, y] is the temperature defined on the inner boundary. a.k.a ((x - 2.5)^2 + (y - 2.5)^2 ) == 0.09

has parameters:

fluidTempOp = D[F[t, x, y], t] + Subscript[u, z]*zTempGrad-  (2*Subscript[\[Alpha],p] (T[t, x, y ]- F[t, x, y]))/(Subscript[r, i]* Subscript[(Subscript[\[Rho]c, p]), fl])

where

    Subscript[u, z] = 0.65;
    Subscript[r, i] = 5*10^(-2);
    Subscript[(Subscript[\[Rho]c, p]), fl] =4.12262*10^6;
    zTempGrad = 0.1;
    Subscript[\[Alpha], p ] =2.82965*10^-7;