Motion of a Viscous Drop With a Moving Contact Line

1. Motion of a Viscous Drop With a Moving Contact Line A contact line is defined at the intersection of a solid surface with the interface between two immiscible fluids. When one fluid displaces another immiscible fluid along a solid surface, the process is called dynamic wetting and a "moving" contact line (one whose position relative to the solid changes in time) often appears. The physics of dynamic wetting controls such natural and industrial processes as spraying of paints and insecticides, dishwashing, film formation and rupture in the eye and in the alveoli, application of coatings, printing, drying and imbibition of fibrous materials, oil recovery from porous rocks, and microfluidics. The spreading of a liquid over a smooth surface is a complicated free boundary problem characterized by the presence of a moving contact line. The motion of the liquid during the spreading process is controlled by the fluid dynamics occurring in the immediate neighborhood of the contact line. This fundamental spreading process is of vital importance and is an area of active research. A contact line is defined at the intersection of a solid surface with the interface between two immiscible fluids. When one fluid displaces another immiscible fluid along a solid surface, the process is called dynamic wetting and a "moving" contact line (one whose position relative to the solid changes in time) often appears. The physics of dynamic wetting controls such natural and industrial processes as spraying of paints and insecticides, dishwashing, film formation and rupture in the eye and in the alveoli, application of coatings, printing, drying and imbibition of fibrous materials, oil recovery from porous rocks, and microfluidics. The spreading of a liquid over a smooth surface is a complicated free boundary problem characterized by the presence of a moving contact line. The motion of the liquid during the spreading process is controlled by the fluid dynamics occurring in the immediate neighborhood of the contact line. This fundamental spreading process is of vital importance and is an area of active research.

2. 2-D viscous drop resting on bottom of a paralleled channel. Two viscous immiscible incompressible fluids in horizontal parallel-walled channel. The two fluids are separated by an interface with surface tension ? and contact angle ? with the wall. (Note there are two contact angles both measured from within the drop) The contact angle can be constant or allowed to depend on the slip velocity. The parallel-walled channel has height d and a periodic solution of period L is sought The origin is set at the bottom left corner, x and y directions shown Navier slip condition employed along the bottom wall to eliminate the stress singularity at the contact line. Uc- slip velocity of the contact point. 2-D viscous drop resting on bottom of a paralleled channel. Two viscous immiscible incompressible fluids in horizontal parallel-walled channel. The two fluids are separated by an interface with surface tension ? and contact angle ? with the wall. (Note there are two contact angles both measured from within the drop) The contact angle can be constant or allowed to depend on the slip velocity. The parallel-walled channel has height d and a periodic solution of period L is sought The origin is set at the bottom left corner, x and y directions shown Navier slip condition employed along the bottom wall to eliminate the stress singularity at the contact line. Uc- slip velocity of the contact point.

3. Purpose of This Study To develop a front tracking numerical method to solve the 2-D N-S equations in a two-phase region. To illustrate the effects of certain parameters on the dynamics of drop motion. To present a convergent numerical method to solve for the motion of an interface with a contact line.

4. Difficulties Faced Boundary condition on the wall; No slip condition introduces a non-integrable stress singularity at the contact line. The actual contact angle is not easily measured. Can relax this unphysical problem by introducing slip along the solid wall. Different mechanisms have been proposed for removing singularity near the contact line (a) Action of long-range Van-Der-Waals forces or shear thinning rheologies. (b)positive slippage. There area number of models of slip, here they chose the Navier slip condition (has been used successfully for many other problems). The method could be modified easily if another condition is to be used. Experimentally-measured contact line is the �apparent contact angle�, as opposed to the actual contact angle, which is difficult to measure. Studying the dependence of the contact angle gives dependence of spreading rate on various physical parameters. This is an area of active research. Can relax this unphysical problem by introducing slip along the solid wall. Different mechanisms have been proposed for removing singularity near the contact line (a) Action of long-range Van-Der-Waals forces or shear thinning rheologies. (b)positive slippage. There area number of models of slip, here they chose the Navier slip condition (has been used successfully for many other problems). The method could be modified easily if another condition is to be used. Experimentally-measured contact line is the �apparent contact angle�, as opposed to the actual contact angle, which is difficult to measure. Studying the dependence of the contact angle gives dependence of spreading rate on various physical parameters. This is an area of active research.

5. Formulation of the Problem . Front tracking features high resolution of physical quantities at the material interface, thus giving a more accurate solution to the physical problem. It eliminates mass diffusion across the interface, reduces mesh orientation effects, and reduces diffusion of vorticity from the interface (where it is deposited by a shock wave in the RM instability) into the interior. . Front tracking features high resolution of physical quantities at the material interface, thus giving a more accurate solution to the physical problem. It eliminates mass diffusion across the interface, reduces mesh orientation effects, and reduces diffusion of vorticity from the interface (where it is deposited by a shock wave in the RM instability) into the interior.

6. Formulation of the Problem Navier Stokes and continuity equations govern the motion of the two fluids. u, p ? and ? are the velocity, pressure, dynamic viscosity and density. The position vector x has two scalar components x and y, and the velocity vector u has components u and v. No slip on the top wall, but we cannot use that on bottom wall or we get non-integrable stress singularity. is the slip coefficient, assumed to be a small positive constant. Set to be 0.02 in this study�large enough to allow selection of a mesh size that can manage computations in a reasonable time. Navier Stokes and continuity equations govern the motion of the two fluids. u, p ? and ? are the velocity, pressure, dynamic viscosity and density. The position vector x has two scalar components x and y, and the velocity vector u has components u and v. No slip on the top wall, but we cannot use that on bottom wall or we get non-integrable stress singularity. is the slip coefficient, assumed to be a small positive constant. Set to be 0.02 in this study�large enough to allow selection of a mesh size that can manage computations in a reasonable time.

7. Interfacial Conditions on y=h(x,t) Here n is the unit normal from fluid 1 to fluid 2, t is the unit tangent vector on the interface, T is the stress tensor, ? is the surface tension between two fluids, and ? is twice the mean curvature of the interface. The square brackets denote a jump in the quantity at the interface. These conditions represent The jump in the normal stress. The continuity of tangential stress. 3 & 4. The continuity of velocity.Here n is the unit normal from fluid 1 to fluid 2, t is the unit tangent vector on the interface, T is the stress tensor, ? is the surface tension between two fluids, and ? is twice the mean curvature of the interface. The square brackets denote a jump in the quantity at the interface. These conditions represent The jump in the normal stress. The continuity of tangential stress. 3 & 4. The continuity of velocity.

8. Condition at Contact Line The first relation has been used by many authors, and gives results which agree with experimental data. The second one has the advantage of being simpler, and we have one less parameter to specify. Both give similar predictions, and either one can be easily implemented in the code used.The first relation has been used by many authors, and gives results which agree with experimental data. The second one has the advantage of being simpler, and we have one less parameter to specify. Both give similar predictions, and either one can be easily implemented in the code used.

9. Front Tracking Method

10. Front Tracking Method 2 grids- one grid for N-S equations, the second for description of the interface. A smearing of the interface is allowed; To account properly for surface tension. The interface is tracked as part of the solution; Allows explicit enforcement of contact angle conditions. Front-tracking has been used for a number of problems in multiphase flow. Smearing is used in Phase Field method, but here, the model equations are the same everywhere, so the smoothing is only a numerical technique. Front-tracking has been used for a number of problems in multiphase flow. Smearing is used in Phase Field method, but here, the model equations are the same everywhere, so the smoothing is only a numerical technique.

11. Reformulation of the Problem

12. Effect of the interface now occurs as delta function on RHS of the N-S equation. The integral over S is over the whole fluid-fluid interface S. The aim is to write a smooth version of the delta function term (Peskin 1977), and compute the unknown velocities u and pressure p from the N-S equation on a regular computation grid. Effect of the interface now occurs as delta function on RHS of the N-S equation. The integral over S is over the whole fluid-fluid interface S. The aim is to write a smooth version of the delta function term (Peskin 1977), and compute the unknown velocities u and pressure p from the N-S equation on a regular computation grid.

13. Algorithm

14. First grid: MAC (marker and cell) grid for velocity and density. Second grid: separate grid aligned with the interface; Treated as a set of arcs (elements) formed by connecting neighborhood points. Grid points on the interface move with the local flow field and maintain the desired contact angle at each step. In all results, the grid size is denoted by h and is the same in x and y directions. In all results, the grid size is denoted by h and is the same in x and y directions.

15. For each time step, all grid points (except the contact point) are moved by: CONTACT POINT DETAILS: How to march in time- This is done by fitting a quadratic polynomial through the two points next to the contact point to get the position of the contact point at time (n+1)dt. Extrapolating the curve at the contact point to satisfy the contact angle condition can generate numerical oscillations of the interface near the contact point as it travels�oscillations disappear with mesh refinement. CONTACT POINT DETAILS: How to march in time- This is done by fitting a quadratic polynomial through the two points next to the contact point to get the position of the contact point at time (n+1)dt. Extrapolating the curve at the contact point to satisfy the contact angle condition can generate numerical oscillations of the interface near the contact point as it travels�oscillations disappear with mesh refinement.

16. Projection Method for MAC Mesh

17. MAC mesh: The dark lines represent the fluid cell volumes we can divide the space into, and the dashed lines are the superimposed MAC mesh for the fluid-flow computation. The arrows show the location where the fluid velocities are determined, and the black circles show the nodes where the pressures are determined. (note: we do not require BC�s for Pressure) The earliest numerical method devised for time-dependent, free-surface, flow problems was the Marker-and-Cell (MAC) method (see Ref. 1965). This scheme is based on a fixed, Eulerian grid of control volumes. The location of fluid within the grid is determined by a set of marker particles that move with the fluid, but otherwise have no volume, mass or other properties. Grid cells containing markers are considered occupied by fluid, while those without markers are empty (or void). A free surface is defined to exist in any grid cell that contains particles and that also has at least one neighboring grid cell that is void. The location and orientation of the surface within the cell was not part of the original MAC method. Evolution of surfaces was computed by moving the markers with locally interpolated fluid velocities. Some special treatments were required to define the fluid properties in newly filled grid cells and to cancel values in cells that are emptied. The application of free-surface boundary conditions consisted of assigning the gas pressure to all surface cells. Also, velocity components were assigned to all locations on or immediately outside the surface in such a way as to approximate conditions of incompressibility and zero surface shear stress. The extraordinary success of the MAC method in solving a wide range of complicated free-surface flow problems is well documented in numerous publications. One reason for this success is that the markers do not track surfaces directly, but instead track fluid volumes. Surfaces are simply the boundaries of the volumes, and in this sense surfaces may appear, merge or disappear as volumes break apart or coalesce. A variety of improvements have contributed to an increase in the accuracy and applicability of the original MAC method. For example, applying gas pressures at interpolated surface locations within cells improves the accuracy in problems driven by hydrostatic forces, while the inclusion of surface tension forces extends the method to wider class of problems (see Refs. 1969,1975). In spite of its successes, the MAC method has been used primarily for two-dimensional simulations because it requires considerable memory and CPU time to accommodate the necessary number of marker particles. Typically, an average of about 16 markers in each grid cell is needed to insure an accurate tracking of surfaces undergoing large deformations. Another limitation of marker particles is that they don�t do a very good job of following flow processes in regions involving converging/diverging flows. Markers are usually interpreted as tracking the centroids of small fluid elements. However, when those fluid elements get pulled into long convoluted strands, the markers may no longer be good indicators of the fluid configuration. This can be seen, for example, at flow stagnation points where markers pile up in one direction, but are drawn apart in a perpendicular direction. If they are pulled apart enough (i.e., further than one grid cell width) unphysical voids may develop in the flow. The earliest numerical method devised for time-dependent, free-surface, flow problems was the Marker-and-Cell (MAC) method (see Ref. 1965). This scheme is based on a fixed, Eulerian grid of control volumes. The location of fluid within the grid is determined by a set of marker particles that move with the fluid, but otherwise have no volume, mass or other properties. Grid cells containing markers are considered occupied by fluid, while those without markers are empty (or void). A free surface is defined to exist in any grid cell that contains particles and that also has at least one neighboring grid cell that is void. The location and orientation of the surface within the cell was not part of the original MAC method. Evolution of surfaces was computed by moving the markers with locally interpolated fluid velocities. Some special treatments were required to define the fluid properties in newly filled grid cells and to cancel values in cells that are emptied. The application of free-surface boundary conditions consisted of assigning the gas pressure to all surface cells. Also, velocity components were assigned to all locations on or immediately outside the surface in such a way as to approximate conditions of incompressibility and zero surface shear stress. The extraordinary success of the MAC method in solving a wide range of complicated free-surface flow problems is well documented in numerous publications. One reason for this success is that the markers do not track surfaces directly, but instead track fluid volumes. Surfaces are simply the boundaries of the volumes, and in this sense surfaces may appear, merge or disappear as volumes break apart or coalesce. A variety of improvements have contributed to an increase in the accuracy and applicability of the original MAC method. For example, applying gas pressures at interpolated surface locations within cells improves the accuracy in problems driven by hydrostatic forces, while the inclusion of surface tension forces extends the method to wider class of problems (see Refs. 1969,1975). In spite of its successes, the MAC method has been used primarily for two-dimensional simulations because it requires considerable memory and CPU time to accommodate the necessary number of marker particles. Typically, an average of about 16 markers in each grid cell is needed to insure an accurate tracking of surfaces undergoing large deformations. Another limitation of marker particles is that they don�t do a very good job of following flow processes in regions involving converging/diverging flows. Markers are usually interpreted as tracking the centroids of small fluid elements. However, when those fluid elements get pulled into long convoluted strands, the markers may no longer be good indicators of the fluid configuration. This can be seen, for example, at flow stagnation points where markers pile up in one direction, but are drawn apart in a perpendicular direction. If they are pulled apart enough (i.e., further than one grid cell width) unphysical voids may develop in the flow.

18. MAC mesh used along with second-order centered-difference scheme in space, first-ordered forward Euler in time. Time stepping of momentum equations split into two pieces: Introduction of an intermediate variable u*. Determination of velocity at time (n+1) via u* and the pressure gradient.

19. Velocities satisfy the no-slip BC on top wall, and slip BC on bottom wall. Periodic BCs applied at the upstream and downstream boundaries. If the channel is horizontal, then the solutions are symmetric- allows computations on only half the domain. The major cost of algorithm comes from solving Poisson Equation for pressure. Solution of the linear system Ax=b needs to be solved, where the matrix A is the discretized operator on LHS. Matrix A is sparse and symmetric�a conjugate gradient method used to solve Poisson equation.Velocities satisfy the no-slip BC on top wall, and slip BC on bottom wall. Periodic BCs applied at the upstream and downstream boundaries. If the channel is horizontal, then the solutions are symmetric- allows computations on only half the domain. The major cost of algorithm comes from solving Poisson Equation for pressure. Solution of the linear system Ax=b needs to be solved, where the matrix A is the discretized operator on LHS. Matrix A is sparse and symmetric�a conjugate gradient method used to solve Poisson equation.

20. Solving the Poisson Equation The Poisson equation is solved by the conjugate gradient method�(using NETLIB, along with a code matrix multiplication algorithm and preconditioner in SLAP column format).

21. Mass Conservation & Convergence Test Numerical results for percentage mass loss of a drop (half ellipse with set dimensions) were compared different mesh sizes�found that mass of the drop was well conserved. With mesh-refinement, mass loss decreased and converged. Reason for introducing slip at bottom wall is to relax the non-integrable stress singularity at the contact line . The Navier slip condition can result in an integrable stress (finite energy dissipation), but it can still produce a singular stress, but with a bounded tangential stress. The results show that the introduction of the slip reduced the singular behavior in the neighborhood of the contact line�this means that the numerical results can be expected to predict accurately the dynamics of the drop-evolution problem.Reason for introducing slip at bottom wall is to relax the non-integrable stress singularity at the contact line . The Navier slip condition can result in an integrable stress (finite energy dissipation), but it can still produce a singular stress, but with a bounded tangential stress. The results show that the introduction of the slip reduced the singular behavior in the neighborhood of the contact line�this means that the numerical results can be expected to predict accurately the dynamics of the drop-evolution problem.

22. Results

23. Results A systematic investigation was done on the effects of the physical parameters on the motion of the drop. Results model accurately the behavior- see handout�

24. Conclusion

25. Results found to be consistent with experiment. Computations illustrate accurately the flow field around the contact line. Inertial effects significantly affect the motion of the contact line. The effects of density, viscosity, surface tension, gravity and contact angle on the dynamics of drop motion have been illustrated. I had trouble figuring out why gravity effects were important. The NASA Glenn Research Center is in the process of developing flight hardware to conduct a microgravity experiment to study the microscale phenomena in the vicinity of the moving contact line. In the absence of gravity, the region dominated by capillary force is enlarged, allowing detailed observations of flow and meniscus shape. I had trouble figuring out why gravity effects were important. The NASA Glenn Research Center is in the process of developing flight hardware to conduct a microgravity experiment to study the microscale phenomena in the vicinity of the moving contact line. In the absence of gravity, the region dominated by capillary force is enlarged, allowing detailed observations of flow and meniscus shape.