# From the nano- to the macroscale: bridging scales for the moving contact line problem

#### Benjamin D. Goddard, (Edinburgh), Andreas Nold (Imperial), Nikos Savva (Cardiff), David N. Sibley (Loughborough) and Peter Yatsyshin (Imperial). Part of the applied mathematics seminar series.

The moving contact line problem occurs when modelling one fluid replacing another as it moves along a solid surface, a situation widespread throughout industry and nature. Classically, the no-slip boundary condition at the solid substrate, a zero-thickness interface between the fluids, and motion at the three-phase contact line are incompatible - leading to the well-known shear-stress singularity. At the heart of the problem is its multiscale nature: a nanoscale region close to the solid boundary where the continuum hypothesis breaks down, must be resolved before effective macroscale parameters such as contact line friction and slip, often adopted to alleviate the singularity [1], can be obtained.

In this talk we will review very recent progress made by our group, considering the problem and related physics from the nano- to macroscopic length scales. In particular, to capture nanoscale properties very close to the contact line and to establish a link to the macroscale behaviour, we employ elements from the statistical mechanics of classical fluids, namely density-functional theory (DFT) [2-4], in combination with extended Navier-Stokes-like equations. Using simple models for viscosity and no slip at the wall, we compare our computations with the Molecular Kinetic Theory model for the dynamic vs. equilibrium contact angle behaviour, by extracting the contact line friction, depending on the imposed temperature of the fluid [4]. A key fluid property captured by DFT is the fluid layering at the wall-fluid interface, which has a large effect on the shearing properties of a fluid.

[1] J. Fluid Mech. 764, 445-462 (2015)

[2] J. Phys.: Condens. Matter 25, 035101 (2013)

[3] Math. Model. Nat. Phenom. 10, 111-125 (2015)

[4] Phys. Fluids 26, 072001 (2014)

[5] A. Nold, PhD Thesis, Imperial College London (2016)

Joint work with Benjamin D. Goddard (Edinburgh), Andreas Nold (Imperial), Nikos Savva (Cardiff), David N. Sibley (Loughborough) and Peter Yatsyshin (Imperial)

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