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Rayleigh-B\'enard convection (RBC) and Taylor-Couette Flow (TCF) are twoparadigmatic fluid dynamical systems frequently discussed together because oftheir many similarities despite their different geometries and forcing. Oftenthese analogies require approximations, but in the limit of large radii whereTCF becomes rotating plane Couette flow (RPC) exact relations can beestablished. When the flows are restricted to two spatial degrees of freedomthere is an exact specification that maps the three velocity components in RPCto the two velocity components and one temperature field in RBC. Using this, wededuce several relations between both flows: (i) The Rayleigh number $Ra$ inconvection and the Reynolds $Re$ and rotation $R_\Omega$ number in RPC flow arerelated by $Ra= Re^2 R_\Omega (1-R_\Omega)$. (ii) Heat and angular momentumtransport differ by $(1-R_\Omega)$, explaining why angular momentum transportis not symmetric around $R_\Omega=1/2$ even though the relation between $Ra$and $R_\Omega$ has this symmetry. This relationship leads to a predicted valueof $R_\Omega$ that maximizes the angular momentum transport that agreesremarkably well with existing numerical simulations of the full 3D system.(iii) One variable in both flows satisfy a maximum principle i.e., the fields'extrema occur at the walls. Accordingly, backflow events in shear flow\emph{cannot} occur in this two-dimensional setting. (iv) For free slipboundary conditions on the axial and radial velocity components, previousrigorous analysis for RBC implies that the azimuthal momentum transport in RPCis bounded from above by $Re^{5/6}$ with a scaling exponent smaller than theanticipated $Re^1$.

Exact relations between Rayleigh-B\'enard and rotating plane Couette flow in 2D