A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non‐uniform cross‐sections

The effects of non‐Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non‐uniform cross‐section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd‐B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two‐step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large‐scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non‐uniform cross‐section.