# Nanoscale hydrodynamics near solids

- Duque Zumajo, Diego

- Pep Español Director

Defence university: UNED. Universidad Nacional de Educación a Distancia

Fecha de defensa: 12 July 2019

- Rafael Delgado Buscalioni Chair
- Miguel Angel Rubio Alvarez Secretary
- Eduardo Santiago Sanz García Committee member

Type: Thesis

## Abstract

This dissertation studies the behaviour of a fluid in contact with a solid in the nanoscale. Theoretical formulations of non-local continuum and discrete hydrodynamics are presented in which the interaction of the solid with the fluid appears explicitly in terms of extended forces on the fluid, confined to the vicinity of the solid object. The discrete theory is validated through MD simulations, where we encounter the plateau problem in the determination of the transport coefficients. We offer a method that solves the problem and allows us to evaluate the transport coefficients unambigously. In the course of the MD investigation we find that the Markovian assumption implicit in the theoretical derivations is not satisfied near the wall when the hydrodynamics is resolved at molecular scales. However, for sufficiently large bins in which the discrete hydrodynamic variables are defined the behaviour is fully Markovian. The final outcome of the present dissertation is the derivation of the slip boundary condition from the microscopically formulated discrete hydrodynamic theory. The slip length and the position of the wall are defined through Green-Kubo formulas and seen to coincide with the original proposal of Bocquet and Barrat. We test the validity of the slip boundary condition thus obtained in a particularly challenging flow, an initial plug flow that is discontinuous near the wall at initial stages of the flow. We observe that the slip boundary condition is violated at the initial stages of the flow and we explain the reasons for this failure. More specifically,we derive, using the Theory of Coarse-Graining (ToCG), the equations of motion of a fluid in contact with a solid sphere of large dimensions compared to molecular scales. We use the Kawasaki-Gunton projection operator technique which leads to a set of nonlinear equations for the relevant variables of the system. We assume the Markovian approximation in order to obtain a set of memoryless equations. We address the well-known plateau problem that appears in the expressions of the transport coefficients present in the equations of motion of the fluid in contact with a solid. We solve this problem and we obtain an alternative expression for the transport coefficients without the plateau problem. In order to validate the theory and measure the transport coefficients with Molecular Dynamics (MD) simulations, we derive the discrete equations of motion for a set of discrete hydrodynamic variables. These variables are defined in terms of finite element basis functions based on regular bins constructed by dividing the domain of the fluid with equiespaced parallel planes. The discrete set of hydrodynamic equations are shown to be identical to a Petrov-Galerkin discretization of the continuum equations. The only assumption made in order to derive the equations of motion of the fluid is the Markovian hypothesis. In order to validate it, we use Mori theory which let us obtain the time evolution of the correlation of the relevant variables of the system. The equations are linear, implying an exponential behaviour of the correlation. We take deviations from exponential decay as an indication of the violation of the Markovian hypothesis. As a first step to address the problem of Markovianity we study the simpler case of a fluid in periodic boundary conditions (PBC) by performing MD simulations. We monitor the relevant variales of the system and compute their correlations. We realize that is necessary to move to the reciprocal space (i.e. the Fourier space for unconfined fluids) in order to validate the Markovian hypothesis. Only for an exponential decay of the modes of the matrix of correlations we can expect the hypothesis to be valid. Once the methodology is well established we address the more complicated case of a confined fluid between two solid slabs. We show that if the size of the bin is of the order of the molecular length the Markovian hypothesis fails for modes near the walls. Nevertheless, for large bins the hypothesis is satisfied but the nonlocal effect of layering is lost. Finally, we derive the slip boundary condition from the theory proposed, by considering mechanical balance within a slab of fluid near the wall, and assuming that the flow field inside this slab is linear. This allows to infer the slip length and the position of the wall where the boundary condition is to be satisfied in microscopic terms, through Green-Kubo expressions. The microscopic expression obtained coincides with the original proposal by Bocquet and Barrat. We demonstrate that the friction coefficient is an intrinsic property of the surface of the solid as it does not depend on the width of the channel.