We study a convection problem in a container with a surface open to the air and heated by a long wire placed
at the bottom. Coupled buoyancy and thermocapillarity effects are taken into account. A basic convective state
appears as soon as a temperature gradient with horizontal component different from zero is applied. It consists
of two big rolls that fill the convective cell and are parallel to the heater. A numerical solution allows us to
determine this basic state. A linear stability analysis on this solution is carried out. For different values of the
applied temperature gradient the basic rolls undergo a stationary bifurcation. The thresholds depend on the
fluid properties, on the geometry of the heater, and on the heat exchange on the free surface. This confirms the
results obtained in recent experiments.