"Almost all interstellar objects contain inherent inhomogeneity. The inhomogeneity can manifest in many different ways, such as the uneven matter distribution in a molecular cloud, or the tangled magnetic field distribution in a Bok globule. The dynamics of interstellar objects is thus often governed by the interaction between astrophysical flows or shocks such as supernova blast waves with inhomogeneous objects. We categorize such interactions as 'heterogeneous flows' in general since many of their behaviors can be attributed to the heterogeneous nature of the underlying objects. At the computational physics group of the University of Rochester, we develop the highly sophisticated numerical tool AstroBEAR to study the physics of heterogeneous flows. One such problem is the heat conduction through interfaces between hot and cold magnetized plasmas. Through simulations, we find a simple mathematical relation for the rate of heat conduction as a function of the initial ratio of ordered to tangled field across the interface. The results can be applied to astrophysical objects such as magnetized wind blown bubbles (WBB) around evolved stars. The second problem involves the interaction between shocks and magnetized clumps. Using AstroBEAR, we consider the realistic circumstance in which the field is completely self-contained within the clumps. We find that the clump and magnetic evolution are sensitive to the fraction of magnetic field aligned with versus perpendicular to the shock normal. The relative strength of magnetic pressure and tension in the different field configurations allows us to analytically understand the different cases of postshock evolution. Interstellar heterogeneous flows can also lead to star formation. Based on the shock clump interaction model, star formation can be triggered by compression from wind or supernova driven shock waves that sweep over molecular clouds. This mechanism has been proposed as an explanation for short lived radioactive isotopes (SLRI) in the Solar System. Using AstroBEAR, we for the first time track the long term evolution of the triggering of a 1 [solar mass] cloud. We also demonstrate that through initial rotation, a circumstellar disk can be formed around such a triggering formed star. Recent progressions in the field of plasma physics, laser technology and instrumentation have led to lab platforms that are scalable to astrophysical objects. These platforms are ideal environments to study the heterogeneous flows directly. An important difference between the astrophysical objects and the lab platform is that the latter involves non-negligible resistivity. We introduce AstroBEAR simulations that investigate the magnetized shock-clump interaction problem under resistive MHD, and answer a crucial question regarding the lab design: in resistive MHD, under what conditions do the shocked behavior of a magnetized clump differ from a non-magnetized one? We find that for Rm [less-than or equal to] 100, it is impossible to distinguish the two, while for Rm [greater-than or equal to] 1000, the resistive MHD has similar morphological evolution as the ideal MHD. These numerical studies provide theoretical foundation for the study of heterogeneous flows, and give direct guidance towards observations and lab astrophysics designs"--Pages vi-vii.