Hydrodynamic boundary conditions

The no-slip boundary condition (NSBC) states that there is no relative motion at the fluid-solid interface. It is one of the two pillars in hydrodynamics (the other one being the Navier-Stokes equation). Over the years, NSBC has demonstrated broad applicability, in spite of the fact that there has been no first-principles justification. However, starting about fifty years ago there has been a particularly persistent problem associated with the application of NSBC to the moving contact line in two-phase immiscible flows. The contact line is defined to be the intersection of the fluid-fluid interface with the solid wall. In 1974, it was shown by Dussan and Davis that the application of NSBC leads to a stress singularity for the moving contact line (MCL) problem. This was followed by molecular dynamic (MD) simulations in 1988 showing total slip, not no-slip, for the MCL. And away from the MCL MD simulations have also shown partial slip in the single-phase flow regime. Thus the MCL problem means that the usual continuum hydrodynamics can not accurately model fluids flow on the micro/nano scale. This was sometimes referred to as an example of the “breakdown of continuum.”

By applying Onsager’s principle of minimum energy dissipation, which underlies almost all linear response phenomena in dissipative systems, it was shown that the hydrodynamic boundary conditions can be derived consistently with the equations of motion. And the continuum system that results can quantitatively predict hydrodynamic behavior at the molecular scale, in excellent agreement with MD simulations. In particular, the near-total slip of the MCL is reproduced, and away from the MCL, the partial slip decays in a power-law fashion. Implications of our results are the following. (1) Hydrodynamic boundary conditions should be consistent with the general principle that underlies all the linear response phenomena in dissipative systems. (2) Near-total slip of MCL is present in all flow speed, independent of the shear rate. It is basically a linear response phenomenon. (3) Slip at the fluid-solid interface is characterized by a length scale, usually denoted the slip length. The ratio of slip length to solid particle size (e.g., in a colloid) determines the importance of slipping in the continuum hydrodynamic behavior. Thus NSBC is an excellent approximation for macroscopic bodies much larger than the slip length, but breaks down for the hydrodynamics of micro/nano particles. (4) The slip length (or the slip coefficient in general) is a statistical mechanic parameter just like the viscosity. Its magnitude depends on solid-fluid molecular interactions and the interfacial geometric constraints.


Key publications:

  1. A Scaling Approach to the Derivation of Hydrodynamic Boundary Conditions“, T. Qian, C. Qiu and Ping Sheng, J. Fluid Mech. 611, 333-364 (2008).
  2. Moving Contact Line on Chemically Patterned Surfaces“, X. P. Wang, T. Qian and Ping Sheng, J. Fluid Mech. 605, 59-78 (2008).
  3. A Variational Approach to Moving Contact Line Hydrodynamics“, T. Qian, X.P. Wang and Ping Sheng, J. Fluid Mech. 564, 333-360 (2006).
  4. Power-Law Slip Profile of the Moving Contact Line in Two-Phase Immiscible Flows“, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. Lett. 93, 094501-094504 (2004).
  5. Molecular Scale Contact Line Hydrodynamics of Immiscible Flows“, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. E68, 016306 (2003).