fluid dynamics - Convective boundary condition

Consider a fluid over a sheet which is placed horizontally along the x-axis. The lower face of the sheet is in contact with another fluid at temperature $T_f$ ( it is heating the sheet). The sheet is stretched and the fluid starts moving. The boundary condition for this situation (at the surface of the sheet) is $$ -k\frac{\partial T}{\partial y}= h_f(T_f-T) $$ This condition says that conduction is equal to the convection. I am not able to understand where is conduction happening and where is convection taking place, can someone help me understand this? Also, $k$ is the thermal conductivity but for which fluid: the upper or the lower?

Answer

Following from the comments...

We've established that the upper fluid is moving, suggesting that heat transfer into it is convective in nature.

We've also got that the lower fluid is being used to convectively heat the sheet. So that fluid is moving as well (convection being heat transfer by motion of a fluid). So you've got basically identical boundary conditions at the top and bottom of the sheet. Perhaps the heat transfer coefficients (HTCs) are not equal, so keep track of them separately.

The B.C. that you've got is describing the energy balance at a sheet-fluid interface. One side is conduction in the solid (sheet) the other is describing convection in the fluid. So you'd use the conductivity of the solid and the temperature gradient of the solid on the right. On the left you'd have the HTC, $h_f$, the bulk temperature of the fluid far from the surface, $T_f$, and the temperature at the interface, $T$.

Going back a bit, the boundary conditions for the top and bottom of the sheet are not exactly the same because of spatial orientation. The difference is in the negative sign in the conduction term. It indicates that a positive temperature gradient in the solid corresponds to a negative heat flow at the surface.

To work out what that means intuitively, lets consider the case you've got where the lower fluid is heating. That suggests that it's the hottest part of the system and that temperature is decreasing as we move upward in y. So, convection at the lower surface is positive ($T_{f,low} > T_{s,low}$ is positive). The temperature gradient in the solid is negative (T decreases as y increases). Lastly, the convective term in the upper fluid should be negative (T_{f,up}

For that boundary condition to be satisfied, the fluid-solid temperature difference must have the opposite sign of the solid temperature gradient (due to the negative sign in the conduction term). That's only true for the lower fluid. So this B.C. is strictly only correct for the interface between the lower fluid and the sheet.

(Note: That will hold true for any temperature distribution, not just the one described above.)

(second Note: The B.C. as you've got it is the most common way to write a convective B.C. in general. Don't be surprised if somebody casually uses it on both sides and assumes that the reader will sort out what is intended. Also, sometime people just forget....