We predict the amplification of mechanical stress, force, and torque on an adherent cell due to flow within a narrow microchannel. We model this system as a semicircular bulge on a microchannel wall, with pressure-driven flow. This two-dimensional model is solved computationally by the boundary element method. Algebraic expressions are developed by using forms suggested by lubrication theory that can be used simply and accurately to predict the fluid stress, force, and torque based upon the fluid viscosity, μ, channel height, H, cell size, R, and flow rate per unit width, Q_2-d. This study shows that even for the smallest cells (γ=R/H≪1), the stress, force, and torque can be significantly greater than that predicted based on flow in a cell-free system. Increased flow resistance and fluid stress amplification occur with bigger cells (γ>0.25), because of constraints by the channel wall. In these cases we find that the shear stress amplification is proportional to Q_2-d(1−γ)⁻², and the force and torque are proportional to Q_2-d(1−γ²)^−5/2. Finally, we predict the fluid mechanical influence on three-dimensional immersed objects. These algebraic expressions have an accuracy of ∼10% for flow in channels and thus are useful for the analysis of cells in flow chambers. For cell adhesion in tubes, the approximations are accurate to ∼25% when γ>0.5. These calculations may thus be used to simply predict fluid mechanical interactions with cells in these constrained settings. Furthermore, the modeling approach may be useful in understanding more complex systems that include cell deformability and cell-cell interactions.