  Current theories on planetary formation establish that giant planet formation
should be contextual to their quick migration towards the central star due to
the protoplanets-disc interactions on a timescale of the order of $10^5$ years,
for objects of nearly 10 terrestrial masses. Such a timescale should be smaller
by an order of magnitude than that of gas accretion onto the protoplanet during
the hierarchical growing-up of protoplanets by collisions with other minor
objects. These arguments have recently been analysed using N-body and/or
fluid-dynamics codes or a mixing of them. In this work, inviscid 2D simulations
are performed, using the SPH method, to study the migration of one protoplanet,
to evaluate the effectiveness of the accretion disc in the protoplanet dragging
towards the central star, as a function of the mass of the planet itself, of
disc tangential kinematics. To this purpose, the SPH scheme is considered
suitable to study the roles of turbulence, kinematic and boundary conditions,
due to its intrinsic advective turbulence, especially in 2D and in 3D codes.
Simulations are performed both in disc sub-Keplerian and in Keplerian kinematic
conditions as a parameter study of protoplanetary migration if moderate and
consistent deviations from Keplerian Kinematics occur. Our results show
migration times of a few orbital periods for Earth-like planets in
sub-Keplerian conditions, while for Jupiter-like planets estimates give that
about $10^4$ orbital periods are needed to half the orbital size. Timescales of
planet migration are strongly dependent on the relative position of the planet
with respect to the shock region near the centrifugal barrier of the disc flow.
