An analytical model for predicting the shapes of rectangular bars with variable curvatures along their lengths through a novel forming method, differential velocity sideways extrusion (DVSE), previously proposed by the authors, has been developed on the basis of the upper bound method. A new flow line function was presented to describe its deformation field. The plastic deformation zone (PDZ) was assumed to be fan-shaped, where the trajectory of the material flow within the PDZ had an elliptic shape. The proposed continuous flow line function was validated using finite element simulations. The flow patterns, extrusion pressure, curvature, and effective strain predicted by the analytical solutions agreed well with modelling results. Compared to the classical discontinuous simple shear model of channel angular extrusion (CAE) with a 90° die, the new approach was shown to predict the effective strain more closely.