Transient behavior and steady-state rheology of dense frictional suspensions in pressure-driven channel flow

Abstract Results from particle-resolved Direct numerical simulations are presented for dense suspensions of frictional non-colloidal spheres in viscous pressure-driven channel flow. The bulk solid volume fraction varies between $$\phi _b=0.2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.2</mml:mn> </mml:mrow> </mml:math> and 0.6, and the Coulomb friction coefficient is either $$\mu _c = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> or 0.5. The main objectives are to unravel the influence of (1) $$\phi _b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> and $$\mu _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> on the flow development time and of (2) heterogeneous shear on the steady-state suspension rheology. Starting from an initially homogeneous distribution, the particles show shear-induced migration toward the core until equilibrium is reached. The flow development time decays exponentially with increasing $$\phi _b/\Phi _R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$\Phi _R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:math> is a friction-dependent reference bulk concentration beyond which particle contacts cause a rapid increase in the particle stress. The steady-state rheology is studied by means of the ‘viscous’ and ‘frictional’ rheology frameworks. Excluding the central core and wall regions, the data for the local relative suspension viscosity collapse onto a single curve as function of the normalized local concentration $${\bar{\phi }}/\phi _m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$\phi _m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> is the friction-dependent maximum flowable packing fraction. The frictional rheology shows ‘subyielding’ at low viscous number $$I_v$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:math> in the core region, where the macroscopic friction coefficient $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> drops below the minimal value found for homogeneous shear flows. A modified frictional rheology model is presented that captures subyielding. Finally, a model is presented for $${\bar{\phi }}/\phi _{mp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>mp</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> as function of $$I_v$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:math> , where $$\phi _{mp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ<