Thermal Protection Tile Heat Flux (Aerospace Engineering)

An aerospace engineer designs a ceramic thermal protection tile for a hypersonic atmospheric re-entry vehicle. During peak atmospheric deceleration, intense aerodynamic friction generates non-uniform surface heating across the outer tile face.

The temperature distribution $T(x, y)$ (in degrees Celsius, $^\circ\text{C}$) across a square test section of the tile is modeled by the Gaussian heat distribution function:

where $x$ and $y$ represent spatial coordinates in centimeters ($\text{cm}$) measured from the tile center.

1. State the partial differentiation rules required to compute $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$, treating the alternate coordinate as a constant coefficient.
2. Determine the analytical expressions for the spatial temperature gradients $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$.
3. Evaluate the instantaneous temperature gradients at the specific monitoring sensor location $(x, y) = (2, 1)\text{ cm}$. State the units of each derivative and interpret the physical thermal behavior along the $x$ and $y$ directions.
4. According to Fourier's Law of Heat Conduction, conductive heat flux across the surface is given by $\vec{q} = -k \nabla T = -k \langle \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y} \rangle$, where $k > 0$ is thermal conductivity. Determine the direction of conductive heat flow at $(2, 1)$.