Viscoplastic flow development in a channel with slip along one wall

The flow development of a Herschel-Bulkley fluid in a horizontal channel is considered assuming that slip occurs only on the upper wall due to slip heterogeneities. Hence, the velocity profile is allowed to be asymmetric as was the case in recent experiments on softy glassy suspensions [13]. A power-law slip equation is employed, which generalizes the Navier-slip law. The one-dimensional fully-developed solutions are derived and the different flow regimes are identified. The two-dimensional development flow is solved numerically using finite elements along with the Papanastasiou regularization for the constitutive equation. Due to the asymmetry and the viscoplastic character of the flow, the classical definition of the development length is not applicable. The global and upper-wall development lengths are thus considered and the combined effects of slip and the Bingham number are investigated. Numerical results are presented for two values of the power-law exponent, i.e. n = 1 (Bingham plastic) and n = 1/2 (Herchel-Bulkley fluid). It is demonstrated that the global development length increases with the Bingham number and that flow development is slower near the no-slip wall. The global development length increases with slip exhibiting two plateaus and an intermediate rapid increase zone and doubles in the limit of infinite slip.