It is a well-known fluid-mechanical phenomenon that thermocapillary forces induce surface convection on a fluid-fluid interface. This so-called Marangoni convection depends on the variation of the surface energy along the interface. In our present work we focus our attention on the evolution of a fibrous monotectic microstructure with liquid L-2 fibers. We will show, that the Marangoni convection has a strong influence on the transport of solute in front of the solidification front, despite the flow induced by density differences. The resulting flow field affects the constitutional undercooling and therefore the mean undercooling of a monotectic solidification front. In a previous paper we discussed qualitatively the influence of fluid flow on the microstructure evolution of composite monotetic growth (C. Stocker, L. Ratke, J. Crystal Growth 203 (1999) 582). We introduced an analytical model that takes the density differences of the phases and the surface convection on the L-1-L-2 surface into consideration. With this extended Jackson and Hunt theory for composite monotectic growth we derived a characteristic equation for the inter-rod distance depending on solidification velocity and temperature gradient. In this paper we develop a more accurate model. We solve numerically the diffusion equation coupled with the Navier-Stokes equation in the L-1 phase to find the minimal undercooling for a given velocity and temperature gradient. We derive a Jackson and Hunt diagram and show that the fluid flow leads to a strong dependence of the inter-rod distance on the temperature gradient opposite to eutectic solidification.