A machine for stirring a liquid food substance, such as milk or a milk-based substance, as a standalone machine or as a machine incorporated in a food processor such as a beverage maker e.g. a coffee maker, such machine comprising: - a container delimiting a cavity with a bottom and a peripheral wall for containing said liquid food substance to be stirred, the cavity extending generally along a generally upright central container axis and having a mouth for removing from the container said liquid food substance upon stirring; and - an impeller comprising an impelling member, e.g. a whisk or wipe or emulsify plate, which is configured to be rotated in the cavity against said liquid food substance about an impeller axis that is generally identical or generally parallel to the container axis, the impelling member having for instance a processing surface (21") for imparting a mechanical effect to the liquid food substance, such as for mixing the liquid food substance with another fluid, e.g. air; and optionally - a thermal conditioner for thermally conditioning the liquid food substance in the cavity, such as a heater and/or a cooler, e.g. a resistive, inductive or radiating thermal conditioner, such as a thermal conditioner configured to heat the cavity's peripheral wall and/or bottom from outside the cavity to thermally condition said liquid food substance contained in the cavity, characterised in that the impelling member is configured to be spaced above the cavity's bottom by: - a maximum stirring distance when the impelling member is rotated at a maximum stirring level at a maximum stirring speed about the impeller axis; and - a resting distance (h) when the impelling member stands still at a resting level, the maximum stirring distance being greater than the resting distance (h), for instance the maximum stirring distance exceeding the resting distance (h) by a distance difference in the range of 0.5 to 6 cm, such as 1 to 4 cm, e.g. 1.5 to 3 cm. - a container delimiting a cavity with a bottom and a peripheral wall for containing said liquid food substance to be stirred, the cavity extending generally along a generally upright central container axis and having a mouth for removing from the container said liquid food substance upon stirring; and - an impeller comprising an impelling member, e.g. a whisk or wipe or emulsify plate, which is configured to be rotated in the cavity against said liquid food substance about an impeller axis that is generally identical or generally parallel to the container axis, the impelling member having for instance a processing surface (21") for imparting a mechanical effect to the liquid food substance, such as for mixing the liquid food substance with another fluid, e.g. air; and optionally - a thermal conditioner for thermally conditioning the liquid food substance in the cavity, such as a heater and/or a cooler, e.g. a resistive, inductive or radiating thermal conditioner, such as a thermal conditioner configured to heat the cavity's peripheral wall and/or bottom from outside the cavity to thermally condition said liquid food substance contained in the cavity, characterised in that the impelling member is configured to be spaced above the cavity's bottom by: - a maximum stirring distance when the impelling member is rotated at a maximum stirring level at a maximum stirring speed about the impeller axis; and - a resting distance (h) when the impelling member stands still at a resting level, the maximum stirring distance being greater than the resting distance (h), for instance the maximum stirring distance exceeding the resting distance (h) by a distance difference in the range of 0.5 to 6 cm, such as 1 to 4 cm, e.g. 1.5 to 3 cm.