A non-affine visco-elasto-plastic constitutive equation with yield stress is proposed to simulate the response of magnetorheological fluids to driving forces. The equation yields realistic flow patterns in the pressure gradient driven fully developed tube flow which combine the effects of viscoelasticity and yield stress. The interplay of yield stress and viscoelasticity results in longitudinal velocity profiles, which can be controlled by adjusting the strength of the applied magnetic field. However, for physically realistic velocity profiles to exist the material parameters of the fluid and the steady pressure gradient must satisfy certain restrictions.
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