Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters

Abstract

The lubrication flow of a Herschel-Bulkley fluid in a symmetric long channel of varying width, 2h(x), is modeled extending the approach proposed by Fusi et al. ["Pressure-driven lubrication flow of a Bingham fluid in a channel: A novel approach," J. Non-Newtonian Fluid Mech. 221, 66-75 (2015)] for a Bingham plastic. Moreover, both the consistency index and the yield stress are assumed to be pressure-dependent. Under the lubrication approximation, the pressure at zero order depends only on x and the semi-width of the unyielded core is found to be given by σ(x) = (1 + 1/n)h(x) + C, where n is the power-law exponent and the constant C depends on the Bingham number and the consistency-index and yield-stress growth numbers. Hence, in a channel of constant width, the width of the unyielded core is also constant, despite the pressure dependence of the yield stress, and the pressure distribution is not affected by the yield-stress function. With the present model, the pressure is calculated numerically solving an integro-differential equation and then the position of the yield surface and the two velocity components are computed using analytical expressions. Some analytical solutions are also derived for channels of constant and linearly varying widths. The lubrication solutions for other geometries are calculated numerically. The implications of the pressure-dependence of the material parameters and the limitations of the method are discussed.

Lubrication solution of the flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters in an asymmetric channel

“…Below we focus on the case of a flat channel with h = 1 and derive the complete analytical solutions for various combinations of the functions f and g, which describe the dependence of k and τ 0 on pressure. As noted in Panaseti et al, 1 The pressure satisfies the following first-order integrodifferential equation…”

“…As in our previous studies, 1,8 we solve the zero-order problem. For the sake of simplicity, we will avoid introducing new symbols for the zero-order variables.…”

“…and 1 for a symmetric channel, in which case v c y = 0. The width of the unyielded core increases if the width of the channel decreases and vice versa.…”

“…In a recent study, 1 we have extended a lubrication approximation method proposed by Fusi et al 2 for solving pressure-driven flow of a Bingham-plastic in a symmetric channel, in order to solve the flow of a Herschel-Bulkley fluid with pressure-dependent consistency index k * and yield stress τ * y . Thus, we have employed the following constitutive equation…”

“…Panaseti et al 1 extended the method of Fusi et al 2 to solve the lubrication flow of a Herschel-Bulkley fluid with pressure-dependent consistency index and yield stress. For the case of a channel of constant width, they demonstrated that the width of the unyielded core is also constant, despite the pressure dependence of the yield stress, and that the pressure distribution is not affected by the yield-stress function.…”