# Can heat travel up an airflow

I know heat travel from hot to cold places and natural airflow (e.g. wind) travel from high pressure to low pressure. But, if a forced airflow is blowing toward a hotter place, would the heat travel upstream and warm up the source of the airflow?

If you are talking about (a) heat conduction and (b) heat convection by fluid flow in the opposite direction, then this can definitely be modeled to help answer your question. Consider heat conduction in the positive x-direction combined with fluid flow in the negative x-direction with speed V. Let's focus on the region from $x = 0$ to $x =\infty$, and let the temperature at x = 0 be maintained at $T_1$, while the temperature at large x is equal to $T_0$. The heat transport equation for this situation with combined conduction and convection is described by: $$-\rho CV \frac{dT}{dx}=k\frac{d^2T}{dx^2}$$where the left hand side represents the convection back toward x = 0, and the right hand side represents conduction. $\rho$ is the fluid density, C is the fluid heat capacity, and k is the fluid thermal conductivity. The solution to this equation, subject to the prescribed boundary conditions is given by: $$T=(T_1-T_0)e^{-\frac{Vx}{\alpha}}+T_0\tag{1}$$where $\alpha$ is the thermal diffusivity: $$\alpha=\frac{k}{\rho C}$$According to Eqn. 1, the length scale for the temperature variation near x = 0 is on the order $\alpha/V$. The thermal diffusivity of water at room temperature is $0.143 \times 10^{-6}\ \frac{m^2}{s}$, so even for a small fluid velocity of 0.1 m/s, the temperature disturbance at x = 0 would only propagate upstream a few microns in x. Even for air, which has a thermal diffusivity at room temperature and 1 atm a little over 100 times that of water, the disturbance would still only propagate upstream on the order of a few tenths of a millimeter.