Implicit methods for partial differential equations using unstructured meshes allow for an efficient solution strategy for many real-world problems (e.g., simulation-based virtual surgical planning). Scalable solvers employing these methods not only enable solution of extremely-large practical problems but also lead to dramatic compression in time-to-solution. We present a parallelization paradigm and associated procedures that enable our implicit, unstructured flow-solver to achieve strong scalability. We consider fluid-flow examples in two application areas to show the effectiveness of our procedures that yield near-perfect strong-scaling on various (including near-petascale) systems. The first area includes a double-throat nozzle (DTN) whereas the second considers a patient-specific abdominal aortic aneurysm (AAA) model. We present excellent strong-scaling on three cases ranging from relatively small to large; a DTN model with O(10<sup>6</sup>) elements up to 8,192 cores (9 core-doublings), an AAA model with O(10<sup>8</sup>) elements up to 32,768 cores (6 core-doublings) and O(10<sup>9</sup>) elements up to 163,840 cores.