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For the longest time I thought an energy balance and the heat equation were identical procedures. However, recently I saw an example of a steady state, constant property, laminar flow of fluid between two flat surfaces where the top surface moves in the ##x## direction at ##V_1## and we assume fully developed hydrodynamical and thermally. The distance between plates is ##a## and the top surface is insulated with temperature ##T_o## and bottom surface has constant heat flux ##q''##.

After an energy balance I arrive at $$\frac{\partial T}{\partial x} = \frac{q''}{a V_1 \rho c}$$

But the energy equation gives (after simplification) $$\frac{\partial T}{\partial x} = \frac{\alpha}{V_1}\frac{\partial^2 T}{\partial y^2}$$

And clearly the right hand side of these two equations, while equal, are represented quite differently. Ultimately we find $$ak\frac{\partial^2 T}{\partial y^2}=q''$$ but I thought flux was equal to ##-k \nabla T=-k\partial_yT## for this problem.

Can someone please help me understand?

Thanks so much!