# Why is the temperature distribution of the non-insulated face of my domain not uniform as expected?

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While solving a dynamic 3D heat conduction problem in MATLAB using functions 'createpde' and 'solve' it is observed that the temperature distribution at the non-insulated face is non-uniform which is not expected by the corresponding setup. The setup consists of a relatively thin plate modelling the floor, where all but its top sides are temperature-insulated by means of the imposed flux. In this way, it is expected that the temperature distribution at the top plate obtains a quasi steady-state uniform temperature. Solving the transient problem in the following way,
>> dT = 15;
>> finalT = 1000*dT;
>> t = 0:dT:finalT;
>> T_full = solve(thermalModel_full,t);
>> pdeplot3D(thermalModel_full,'ColorMapData',(T_full.Temperature(:,2))-273.15);
results in a temperature distribution of the upper face of the domain which is not uniform, see the following screenshot: Why is the temperature distribution of the non-insulated face of my domain not uniform as expected?

MathWorks Support Team on 20 Jul 2021
In the latter case the problem is solved using a relatively small time step size as compared to the total time span of the simulation. Moreover, the temperature distribution is only visualized in the end of the first time step, namely,
>> pdeplot3D(thermalModel_full,'ColorMapData',(T_full.Temperature(:,2))-273.15);
Since the time discretization is so fine, the advection part of the heat conduction problem is dominant as compared to the dissipative part of the differential equation at the beginning of the simulation. The solution clearly has not reached a quasi steady-state form, thus observing a non-uniformity in the solution at the first time steps which is expected.
Waiting long enough results in having a uniform temperature distribution in the upper face of the domain. In fact at the end of the last time step, the temperature distribution looks as follows:
>> pdeplot3D(thermalModel_full,'ColorMapData',(T_full.Temperature(:,end))-273.15); Note that the temperature distribution has reached a quasi steady-state form and it is uniform in the upper face of the domain, as expected.

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