  In this paper we address the theory of Type Ia supernovae from the moment of
carbon runaway up to several hours after the explosion. We have concentrated on
the boiling-pot model: a deflagration characterized by the (nearly-)
simultaneous ignition of a number of bubbles that pervade the core of the white
dwarf. Thermal fluctuations larger than >1% of the background temperature
(7x10^8 K) on lengthscales of < 1m could be the seeds of the bubbles.
Variations of the homogeneity of the temperature perturbations can lead to two
alternative configurations at carbon runaway: if the thermal gradient is small,
all the bubbles grow to a common characteristic size related to the value of
the thermal gradient, but if the thermal gradient is large enough, the size
spectrum of the bubbles extends over several orders of magnitude. The explosion
phase has been studied with the aid of a smoothed particle hydrodynamics code
suited to simulate thermonuclear supernovae. In spite of important procedural
differences and different physical assumptions, our results converge with the
most recent calculations of 3D deflagrations in white dwarfs carried out in
supernova studies by different groups. For large initial numbers of bubbles
(>3-4 per octant), the explosion produces about 0.45 solar masses of 56Ni, and
the kinetic energy of the ejecta is 0.45x10^{51} ergs. However, all
three-dimensional deflagration models share three main drawbacks: 1) the scarce
synthesis of intermediate-mass elements, 2) the loss of chemical stratification
of the ejecta due to mixing by Rayleigh-Taylor instabilities during the first
second of the explosion, and 3) the presence of big clumps of 56Ni at the
photosphere at the time of maximum brightness.
