Ultracold quantum gases provide a unique setting for studying and understanding the properties of interacting quantum systems. Here, we investigate a multi-component system of $^{87}$Rb--$^{39}$K Bose-Einstein condensates (BECs) with tunable interactions both theoretically and experimentally. Such multi-component systems can be characterized by their miscibility, where miscible components lead to a mixed ground state and immiscible components form a phase-separated state. Here we perform the first full simulation of the dynamical expansion of this system including both BECs and thermal clouds, which allows for a detailed comparison with experimental results. In particular we show that striking features emerge in time-of-flight for BECs with strong interspecies repulsion, even for systems which were separated in situ by a large gravitational sag. An analysis of the center of mass positions of the BECs after expansion yields qualitative agreement with the homogeneous criterion for phase-separation, but reveals no clear transition point between the mixed and the separated phases. Instead one can identify a transition region, for which the presence of a gravitational sag is found to be advantageous. Moreover we analyze the situation where only one component is condensed and show that the density distribution of the thermal component also show some distinct features. Our work sheds new light on the analysis of multi-component systems after time-of-flight and will guide future experiments on the detection of miscibility in these systems.
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