The Marangoni convective instability in a horizontal
fluid layer with the insoluble surfactant and nondeformable free
surface is investigated. The surface tension at the free surface is
linearly dependent on the temperature and concentration gradients.
At the bottom surface, the temperature conditions of uniform
temperature and uniform heat flux are considered. By linear stability
theory, the exact analytical solutions for the steady Marangoni
convection are derived and the marginal curves are plotted. The
effects of surfactant or elasticity number, Lewis number and Biot
number on the marginal Marangoni instability are assessed. The
surfactant concentration gradients and the heat transfer mechanism at
the free surface have stabilizing effects while the Lewis number
destabilizes fluid system. The fluid system with uniform temperature
condition at the bottom boundary is more stable than the fluid layer
that is subjected to uniform heat flux at the bottom boundary.