Figure 1:
A “timeline” focussed on the topics covered in this review, including chemists, engineers, mathematicians, philosophers, and physicists who have contributed to the development of nonrelativistic fluids, their relativistic counterparts, multifluid versions of both, and exotic phenomena such as superfluidity. 

Figure 2:
A schematic illustration of two possible versions of parallel transport. In the first case (a) a vector is transported along great circles on the sphere locally maintaining the same angle with the path. If the contour is closed, the final orientation of the vector will differ from the original one. In case (b) the sphere is considered to be embedded in a threedimensional Euclidean space, and the vector on the sphere results from projection. In this case, the vector returns to the original orientation for a closed contour. 

Figure 3:
A schematic illustration of the Lie derivative. The coordinate system is dragged along with the flow, and one can imagine an observer “taking derivatives” as he/she moves with the flow (see the discussion in the text). 

Figure 4:
The projections at point P of a vector onto the worldline defined by and into the perpendicular hypersurface (obtained from the action of ). 

Figure 5:
An object with a characteristic size is modeled as a fluid that contains fluid elements. From inside the object we magnify a generic fluid element of characteristic size . In order for the fluid model to work we require and . 

Figure 6:
A local, geometrical view of the Euler equation as an integrability condition of the vorticity for a singleconstituent perfect fluid. 

Figure 7:
The pushforward from “fluidparticle” points in the threedimensional matter space labelled by the coordinates to fluidelement worldlines in spacetime. Here, the pushforward of the “” () fluidparticle to, say, an initial point on a worldline in spacetime can be taken as where is the spatial position of the intersection of the worldline with the time slice. 

Figure 8:
The pushforward from a point in the constituent’s threedimensional matter space (on the left) to the corresponding “fluidparticle” worldline in spacetime (on the right). The points in matter space are labelled by the coordinates , and the constituent index . There exist as many matter spaces as there are dynamically independent fluids, which for this case means two. 
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