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This paper proves that there are no zero-net wall-transpiration control strategies that can sustain net heat flux below the laminar level in an incompressible channel flow with constant-temperature walls. The result represents a fundamental limit on the performance of a controlled nonlinear system as measured by a linear cost function over a broad class of admissible initial conditions and control inputs, not a zero-sum tradeoff in the frequency domain or time domain. Both buoyancy effects (via the Boussinesq approximation) and viscous heating effects are accounted for, and phenomenological justification for the result is also given. The boundedness of solutions of the two-way coupled Navier-Stokes/energy equations (when both buoyancy and viscous heating are accounted for) is also discussed, and a new proof of existence under an appropriate small-data assumption is provided.