Closed-form analytical solutions that describe the evolution of diffusion flames in two-dimension al laminar flowfields due to an initial line source of fuel in an infinite oxidizer environment are derived. The velocity field is assumed to be divergence free and both steady and unsteady shear and plane stagnation flows are considered in this study. The flame surface is modeled as an infinitesimally thin flame sheet at which fuel and oxidizer are consumed at an infinitely fast rate; thus, chemical kinetic effects are ignored. Validity of the solutions are discussed. The flame shapes are observed to be elliptic. The flame front moves away from the source at early times; whereas at later times, the front collapses at the location of the fuel source. An expression for the time for complete consumption of the fuel is derived in each case, and it is shown that strain fields are more conducive to combustion. Nomenclature A = aspect ratio of the elliptical front a = shear rate b = strain rate Cf = mass concentration of fuel D = mass diffusivity e = base of natural logarithm Pe = Peclet number identical to L^/Dt^f Q = mixture fraction source strength Q = fuel source strength used in Eq. (1) rm = maximum radius of reaction front