The similarity solutions for mixed convection boundary-layer flow when the wall heat flux is prescribed are analysed in detail in terms of a buoyancy parameter \(\alpha\) and m the exponent of the free stream flow. It is shown that for \(\alpha >0\) the solution approaches the free convection limit, and for \(\alpha <0\), there is a range of \(\alpha\), \(\alpha_ s<\alpha <0\), over which dual solutions exist. The nature of the bifurcation at \(\alpha =\alpha_ s\) and how the lower branch of solutions behaves as \(\alpha \to 0^-\) are also considered. It is established that the solution becomes singular as \(m\to 1/5\) and the nature of this singularity is also discussed, where it is shown that two separate cases have to be treated, namely when \(\alpha\) is of O(1) and when \(\alpha\) is small. Finally it is shown that for m large the solution approaches that corresponding to exponential forms for the free stream and prescribed wall heat flux. Taken all together this information enables a complete description of how the solution behaves over all possible ranges of the parameters \(\alpha\) and m to be deduced.