Static wall layers in plane channel displacement flows

Abstract

Abstract We study steady and pulsating displacement flows of a Bingham fluid by a Newtonian fluid, along a plane channel. For sufficiently large yield stress a static residual wall layer can result during the displacement. The flow is parameterised by the Reynolds number (Re), the Bingham number ( B) and the viscosity ratio ( M). Perhaps intuitively, thicker layers are found with larger M and at lower Re. The residual layer is formed on the advective timescale of the displacement but drains on a slower timescale governed by M. For larger M truly stationary layers are only found for large t when the layer has thinned sufficiently to become static. Increased Re results in increased energy production locally around the finger. For large enough Re the energy production can play a significant role in yielding the fluid. As the energy production rate increases it also becomes focused around the corner or shoulder region of the front, and spreads axially along the initial part of the residual layer. This causes fluid to yield increasingly far behind the front and allows for the layer to thin. As B increases the static layer tends to decrease (see also [1,2]). At small Re the static layer thickness appears to be independent of M. At large Re the layer thickness is dependent on M and decreases asymptotically to a constant value as B → ∞. For pulsating displacement flow rates, Q( t) = 2(1 + Asin ωt) : A ∈ [0, 1) we study two ranges: ωRe ≪ 2 π and ωRe ≫ 2 π. For the viscous regime ( ωRe ≪ 2 π) a pseudo-steady 1D model predicts that the residual layer should remain static for 3(1 + Asin ωt) < MB. In practice we find that partial mobilisation of the residual layer occurs even when this inequality is satisfied, but not if MB becomes significantly larger than 3(1 + A). For ωRe ≫ 2 π we mobilise the layer for significantly larger values of MB and at smaller A, than in the viscous regime. This effect is traced to the occurrence of out-of-phase velocity fluctuations in the displacing fluid within a wall layer close to the interface.