Gap/clearance flows (also known as tip gap flows) affect the hydrodynamic forces, flow structure, and cavitation in both turbomachines and hydrofoils. Computational Fluid Dynamics (CFD) solvers permit high-fidelity, viscous simulations of gap-flow, but the computational expense often prohibits their use for fully exploring design spaces. Conventional potential flow solvers, on the other hand, cannot capture the viscosity-dominated gap-flow dynamics. In the present study, a physics-based gap-flow model is presented to capture the critical effects of gap-flow using general potential flow solvers. This is accomplished by re-casting a lift-retention model as a corrected boundary condition. A simple lifting-line formulation is used to demonstrate the applicability of the gap-flow model. Results from the lifting-line analysis, modified by the gap-flow model, are compared with experimental measurements and high-fidelity CFD simulations over a range of gap sizes for two confined-wing arrangements with different geometries and flow conditions. The modified lifting-line analysis improves significantly upon the circulation, downwash, and drag predictions, compared to a standard lifting-line formulation without the gap-flow model. A viscosity-corrected expression for tip-vortex strength is proposed. Using the viscous correction, qualitatively-correct trends are predicted for vortex strength and tip vortex minimum pressure coefficients across a range of foil aspect-ratios, gap sizes, and angles of attack.