Solutions of Herschel-Bulkley flows

Two mixed boundary value problems for longitudinal shear flows of a Herschel-Bulkley fluid are formulated and solved analytically. The rheological model encompasses as limiting cases the Bingham plastic, power law and Newtonian fluid models. There is also an interpretation in terms of nonlinear filtration and nonlinear elastic problems.

The constitutive equations are, in general, nonlinear, and the boundary value problems considered here are not solvable using similarity solutions or related methods. Nevertheless, progress can be made when the specific problems considered here are formulated in a hodograph plane. A Legendre transformation is used to interchange the roles of dependent and independent variables thereby obtaining a linear problem in this hodograph plane. Hypergeometric integral transforms are then used to obtain exact solutions in this hodograph plane. These exact solutions are examined to determine both near field behaviour at singular points and positions of plug regions in the physical domain. Asymptotic methods for large and small values of a parameter are used as a check on the velocity field near the singular points. For small values of this parameter a matching procedure is used to find the asymptotic form of the plug regions. This provides an estimate of the region occupied by the plug region and acts as a check on the exact solutions.

A reciprocal theorem is used to provide an independent check upon the results which are used to determine the fields near singular points. Separable solutions which correspond to flows in the neighbourhood of a wedge apex are also briefly considered.

Quarterly Journal of Mechanics and Applied Mathematics, 48, pp.343-374, 1995

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