Session M11: Viscous Flows

Viscous Added Mass of a Moving Solid Object in a Closed Liquid-Filled Container

A moving solid object in a closed liquid-filled container is shown to have a viscous added mass in the quasi-steady Stokes limit. The viscous added mass is similar to the added mass for potential flow. The added-mass force is the product of the viscous added mass and the object's acceleration and is analogous to but distinct from the drag force, which is the product of the drag coefficient and the object's velocity. Both the drag coefficient and the viscous added mass can be computed directly from the quasi-steady Stokes solution for the liquid velocity field. The viscous added mass arises from the fact that the object's acceleration changes the kinetic energy of the liquid as well as the object. If the object fills most of the container's cross section, the viscous added mass is much larger than the object's mass and thus is dynamically significant. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

The effect of the torsion on the axial flow in helical pipes

The effect of the torsion on the flow in helical pipes remains difficult to predict and slightly controversial. Also in the simple case of a circular cross section and laminar flow it is difficult to have a clear idea of the influence of the torsion. In particular as regards the axial flow we register a second order effect of the torsion on the flow rate but a strong effect on the asymmetry of the velocity profile produced by the curvature. In the presentation a new bulk indicator of the rotation of the axial flow induced by the torsion is defined. This indicator is applied to the accurate calculations of Gammack and Hydon, (JFM 433, 357-382, (2001)), and an estimate of the bulk rotation angle of the axial flow is produced in the limiting case of low curvature and high values of the torsion. In this case we register a strong and direct dependence of the bulk rotation of the axial flow with the torsion number.   

Flow in and geometry of microstructured optical fibres

Microstructured optical fibres (MOFs) have revolutionised optical fibre technology, promising a virtually limitless range of fibre designs for a wide range of applications. Extrusion of a preform and drawing to form a fibre is a promising fabrication process for mass production. However, understanding of the flow during fabrication and its effect on the complex air-solid structure in the MOF cross section is lacking, and this impedes MOF development. We propose a modelling methodology suitable for complex structure, and focus on flow in the cross section during preform extrusion. Excellent qualitative agreement of model results and experiment is shown and areas for model improvement are identified.   

The moving contact line problem. Is there a solution?

The moving contact line problem occurs when one fluid replaces another as it moves along a solid surface, a situation arising in a vast range of applications. The apparent paradox of motion of the fluid-fluid interface, yet static fluid velocity at the solid satisfying the no-slip boundary condition, has been known for a number of decades, with a wealth of publications suggesting methods to resolve it since. Here we consider the behaviour of a solid-liquid-gas system near the three-phase contact line using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. We first obtain a wetting boundary condition on the solid that allows us to consider the motion without any additional physics. Careful examination then of the asymptotic behaviour as the contact line is approached is shown to resolve the moving contact line problem. Various features of the model are scrutinised alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall. But these are not necessary to resolve the moving contact line problem.   

Using Simple Flows to Tie Knots in Flexible Fibers

Flexible fibers, such as DNA and other polymer chains, have sometimes been found to contain knotted regions. While such fibers are not strict, closed knots, they exhibit similar characteristics; the formation of these ``open knots'' and the effects they have on material properties are the subject of current research. We investigate the possibility that simple flows can generate open knots in sufficiently long and flexible elastic fibers. Using the HYDROMULTIPOLE algorithm, which solves the multipole expansion of Stokes equations, we use numerical simulations to study the time evolution of a bead-spring model fiber in a shear flow. The model is described by the dimensionless parameters $k$, $A$, and $l_0$, which account for the elastic forces, bending forces, and equilibrium distance between beads, respectively. We find that for certain systems, the characteristic tumbling motion of a fiber in shear flow will result in the formation of $3_1$ and $5_1$ knots, as identified by their Alexander polynomial knot invariants. Investigation of the knotting mechanism is ongoing.   

Inlet Jet Interaction in Horizontal Pipe Flow

Laminar incompressible flow (Re $<$ 1000) inside a horizontal channel with multiple cross-flow inlets was studied numerically. First, two cross-flow inlets were used to observe the flow interference phenomenon between the inlets. This concept was extended to axisymmetric pipe flow with five cross-flow inlets. Three basic flow regimes - trickle flow, partially blocked flow and fully blocked flow - were identified with respect to the blocking of upstream inlets by the downstream ones. The effects of inlet pressure and different inlet sizes on the flow regimes under steady state condition were studied. A hydrostatic model of fluid reservoirs draining into the channel was constructed using a linear function for pressure at the inlet boundaries to study the dynamic behavior of the inlets. Three different time scales related to the depletion of the reservoirs were identified. The dynamic behavior of two cross-flow inlets was observed with the initial conditions corresponding to the three flow regimes. Similar study was carried out for a five-inlet case and the dynamic behavior of individual reservoirs was observed. The change of flow regimes in the system over time with reservoir draining was evident and the different time-scales involved were identified.   

Streamline Patterns and Eddies in Slipping Stokes Flow

Streamline topologies are analyzed in the vicinity of boundaries in the limit of Stokes flow with Navier slip boundary conditions for some simple flows involving two- and three-dimensional configurations. It is found that the streamline pattern transformations, and consequently the flow fields are sensitive to the non-dimensional slip parameter $\lambda$. For two-dimensional flows, the separated/attached eddies - that are known to exist in the no-slip case at the contour - get destroyed or pushed away from the boundary as the slip is varied. Analysis of flow generated by a point force (stokeslet) inside a spherical container reveals that when the stokeslet is positioned at the center of the container, the eddy pattern - that is noted in the no-slip case - undergoes a series of transformations due to slip variations and eventually disappears. Furthermore, the parameter $\lambda$ dictates the locations of the stagnation point and the point of zero vorticiy in the flow domain. Our analytical solution indicates that the {\it co-existence of a stagnation point ($r_{stag}$) and a point of zero vorticity ($r_{\Omega=0}$) in the flow region is necessary for the occurrence of closed eddies}. The results may be of some interest in small scale hydrodynamics in which Stokes flow occurs.   

Analysis of Slip Boundary Condition in Single and Multi-Phase Flows

Over the past two decades several studies have been conducted to understand the molecular mechanism of slip in fluids at the boundary and to better understand the contact point singularity in two phase flow. Although for single phase flows, researchers have looked into the effects of unsteady flow in gases; in liquids, most of the study has been limited to steady flows. In this paper we use molecular dynamic simulations to study slip in an unsteady flow. An unsteady slip model is established, the non-dimensionalizing of which leads to a universal curve for boundary slip. The universal curve gives the slip length for a given shear rate and gradient of shear rate, for steady and unsteady flow. We also identify a non-dimensional number which is defined as the ratio of phase speed to local speed of sound that explains the mechanism responsible for the transition of slip boundary condition from finite to perfect slip. The slip boundary condition is further studied for steady and unsteady multi-phase flows. Emphasis is placed on observing the slip at the wall at the fluid-fluid interface. We establish a universal curve for slip boundary condition for multi-phase flow, for steady and unsteady flows.   

Effective slip identities for viscous flow over arbitrary patterned surfaces

For a variety of applications, most recently microfluidics, the ability to control fluid motions using surface texturing has been an area of ongoing interest. In this talk, we will develop several identities relating to the construction of effective slip boundary conditions for patterned surfaces. The effective slip measures the apparent slip of a fluid layer flowing over a patterned surface when viewing the flow far from the surface. In specific, shear flows of tall fluid layers over periodic surfaces (surfaces perturbed from a planar no-slip boundary by height and/or hydrophobicity fluctuations) are governed by an effective slip matrix that relates the vector of far-field shear stress (applied to the top of the fluid layer) to the effective slip velocity vector that emerges from the flow. Of particular note, we will demonstrate several general rules that describe the effective slip matrix: (1) that the effective slip matrix is always symmetric, (2) that the effective slip over any hydrophobically striped surface implies a family of related results for slip over other striped surfaces, and (3) that when height or hydrophobicity fluctuations are small, the slip matrix can be approximated directly using a simple formula derived from the surface pattern.