We observe that shape of flame of a burning candle is somewhat ellipsoidal when we burn that candle in our room at normal atmospheric pressure and room temperature.If we know the shape of an object say $f(x,y,z)$ and it is burning at room temperature at normal atmospheric pressure,then can we predict the shape of flame?
The shape of a flame is set by where light is emitted from burning material. In many cases this is a gaseous reaction between air and vapor (evaporated stearin from the candle, pyrlolysed organic compounds from a log) that only happens in a certain concentration interval. So to calculate the shape you need to model the flow of air and flammable gas. This gets complicated because (1) both flow according to the Navier-Stokes equationst with (2) density affected by local temperatures that are set by heat transport through conduction and radiation, while (3) gas is produced in a very nonlinear way from burning material dependent on temperature, composition and its structure (think of how ash on a burning log acts as an insulator). Basically you need to model hydrodynamics, thermodynamics and chemistry together - not entirely impossible in some cases, but it gets extremely compicated quickly.
For example, the candle flame is surrounded by a laminar flow of air, but a slightly larger flame will be turbulent and the mixing will become fractal. This also shows why knowing the shape of the object will not predict the flame shape well in many situations: the flame shape will be chaotic and variable even for the same object.