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A new hybrid d fini te-volume rrethod for the nurrerical simulation of steady flow of 01 droyd-B fluid is presented. Our study has two purposes: one is to devel op a new hybrid finite vol ume rrethod to simulate the flow of Oldroyd-B fluid and the other is to test the rrethod in an attempt to simulate the planar 4 ·i contraction flows of such fluid. The main features of this method are that the technique of Chorirrs pseudo-corrpressibility is used to solve the steady incompressible continuity equati on and morrentum equations and the streamli ne-upwi nding approach is used to discretize the consti tutive equations. The method is appl ied to a sudden-contr acti ons vi scoel asti c-flow problem for a range of Weissenberg numbers. NJmerical experiments suggest that the results of the present nurrerical method be in good qualitative agreerrent with exi sting experi rrental resul ts.