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Arterial endothelial cell (EC) responsiveness to flow is essential for normal vascular function and plays a role in the development of atherosclerosis. EC flow responses may involve sensing of the mechanical stimulus at the cell surface with subsequent transmission via cytoskeleton to various intracellular transduction sites. We model the deformation of coupled networks of cell-surface flow sensors and intracellular structures in response to steady and oscillatory flow. The various structures are represented as viscoelastic materials with standard linear solid behavior. The analysis reveals that flow induces an instantaneous deformation in all structures followed by creeping to the asymptotic response. For simple sensor-cytoskeleton-nucleus networks, the results show that, consistent with the experimentally observed temporal chronology of EC flow responses, the flow sensor attains its peak deformation faster than intracellular structures and the nucleus deforms more rapidly than cytoskeletal elements. The results have also revealed that a 1-Hz oscillatory flow induces significantly smaller deformations than steady flow. This may provide insight into why a number of EC responses induced by steady flow are not induced by oscillatory flow.