The influence of elasticity and inertia for steady flow of a thin viscoelastic fluid jet is examined theoretically. The fluid is assumed to emerge from a vertical channel and driven by a pressure gradient and/or gravity. The boundary-layer equations are generalized for a viscoelastic thin film obeying the Oldroyd-b constitutive model. Special emphasis is placed on the initial stages of jet development. The formulation and simulation are carried out for two-dimensional jet flow in order to better understand the intricate wave and flow structures for a viscoelastic jet. In contrast to the commonly used depth-averaging solution method, the strong nonlinearities are preserved in the present formulation as the viscoelastic boundary-layer equations are solved by expanding the flow field in terms of orthonormal shape functions. It is found that for a steady viscoelastic jet, a reduction in inertia or a rise in elasticity leads to the emergence of surface waviness and excessive normal stress, which leads to the formation of sharp gradients in the velocity and shear stress. These gradients can be sufficiently substantial to cause a discontinuity or shock in the flow. During transition, the surface profiles adhere earlier to the shape of the final steady state instead of a traveling wave, the transition between the two states takes the form of a standing wave, which grows essentially in amplitude only.

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