Laminar flow of a Harschel-Bulkley fluid in channels of finite width

Laminar flow of a viscoplastic fluid in a shallow and wide channel is examined under the long-wave approximation. The fluid is described by the three-parameter Herschel-Bulkley constitutive equation. The complete set of equations governing the flow is presented, generalizing earlier results for a Bingham fluid. The paper then focuses on steady uniform flow: for different geometries of the channel (polynomials like triangular and parabolic, trapezoidal, and rectangular), the velocity distribution and the total discharge are derived analytically as functions of the fluid properties and of the channel cross-section. Results show the existence of dead zones close to bed and banks of the channel; discharges through any given cross-section are higher for dilatant than for pseudo-plastic fluids.