Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
John A. Smythe University of Washington Concept is to model the collapse and spread of a water droplet on a surface. This general case will then be modified to apply to liquid characteristics that are consistent with the materials of interest in this work. It follows then that mass tracking and solvent evaporation must be included. The final step is to apply the model to a non-planar surface. The first surface will be the case of a small depression just beyond the initial edge of the droplet. Assumption is that the beginning radius is larger than the balance between the hydrostatic force and the Laplace pressure. g 1 1 Consolidation gives: g 1 The diameter (or capillary length) is approximately 2.7 mm for water in air. This just provides a lower bound for the droplet size. The second concept is the surface energy for boundary 4. This is assumed to be sufficiently high that the droplet (water for the first case) will wet the surface at least to some degree. The initial intent is to examine conditions where the droplet relaxes but does not grow in r such that it will need to interact with boundary 6. Mass is also conserved and will need to be tracked as the interface moves. The contact edge is expected to “freeze” in a state known as an advancing front. 5 3 Subdomain 2 (air) 6 Droplet subdomain 3 z 4 r 1 Subdomain 1 (substrate) 7 2 Initial condition Note: boundary 4 can also be the edge of the domain space. Substrate is only needed if a heating or cooling effect is considered. The initial cases will just have a boundary at the line labeled 4. Gen Exam Preparation 1 of 13 John A. Smythe University of Washington 5 3 Subdomain 2 (air) z Collapsed droplet 6 4 r Subdomain 1 (substrate) 1 7 2 After small amount of time (< 1 Sec) The collapse event (dz and dr) is expected to be driven by body force of gravity and the advancing contact front. Velocity in r (ux) is non zero for cases of r not equal zero. Velocity in z (uz) is non zero for all cases of r until equilibrium conditions are met. The interfacial free energies for the three phases (Young circa 1830) as associated with the equilibrium. It is important to note that this relation does not take in to account volume. This may not be a significant limitation because the edge of the droplet in contact with solid surface should dominate. The capillary length is only a specific case of a spherical droplet in free space. lv cos sv sl cos γlv sv sl lv θ γsl γsv Where γlv, γsv and γsl refer to the interfacial energies of the liquid-vapor, solid-vapor and solidliquid interfaces respectively. Gen Exam Preparation 2 of 13 John A. Smythe University of Washington The fluid flow for water and air are both to be governed by incompressible Navier-Stokes. Though air is compressible, the pressure in this case is constant. u u u u u T p Fst g t u 0 Level-set is intended to be used to define the liquid-gas interface as it moves towards equilibrium. The governing equations (vector short form) for two phase flow subdomain settings: u u u I u u T F g n t u 0 The surface tension force acting on the interface between liquid and gas is σκδn. Curvature κ depends on second derivatives of Φ , n is the unit normal to the interface and δ is a Dirac delta function concentrated at the interface.. 1 u t The variables available for Sources/Sinks are the surface tension coefficient σ (N/m), gravity gz (m/s2), volume forces Fr (N/m3) and Fz (N/m3). The level set has variables γ (gamma) that is used as the reinitialization parameter (m/s) and ε (epsilon) that is the parameter that controls the interface thickness (m). Simplified test for model structure development is using the following geometry showing the droplet after initial wetting of the surface. Gen Exam Preparation 3 of 13 John A. Smythe University of Washington 4 3 Subdomain 2 (air) 6 7 1 z Droplet 2 r 5 Subdomain 1 (droplet) Geom and sub domains with Boundary numbers Boundary settings: Governing equation for type Interior boundary and condition Initial fluid interface (boundary 7): pI u u n 0 T Governing equation at top (4) and right (6) boundary (no slip) is u=0. Boundary 2 and 5 governing equation (Wall, Wetted wall) n u 0, t pI u u T n 0 Ffr / u, n n interface cos() In this case, β is the slip length (m) and θ is the contact angle (rad). Noting that 360 degrees (a circle) is 2π radians. The terms are defined in the coefficients tab in the boundary settings dialog when type is Wall and condition is Wetted wall. Gen Exam Preparation 4 of 13 John A. Smythe University of Washington Wall θ θ = contact angle Radians Degrees π/6 30 π/3 60 2π/3 120 5π/6 150 π 180 Boundary 1 and 3 are type Symmetry boundary and condition Axial symmetry. Governing equation is: r=0 Boundary 4 and 6 are type Wall and condition No slip. Governing equation is: u=0 Boundary 7 is the initial fluid interface. Governing equation (vector short form) is: pI u u n 0 T Gen Exam Preparation 5 of 13 John A. Smythe University of Washington The geometry for mod2_2-16 has an initial elliptical shape with intercepts at z = 2.5e-3 m and r = 4e-3 m. Reference points for tracking variables are also shown. 6 3 2 1 5 4 Geom 1 with sub domains and reference points. mod2_2-16 Non-conservative level set, tolerance 0.01, epsilon ~7e-4. Gen Exam Preparation 6 of 13 John A. Smythe University of Washington mod2_2-16 ST 0.02 Non-conservative level set, tolerance 0.01, epsilon 7.295e-4. Vf of fluid 2 is phi. ST 0.02 mod2_2-16 Non-conservative level set, tolerance 0.01, epsilon 7.295e-4. Gen Exam Preparation 7 of 13 John A. Smythe University of Washington 2 4 mod2_2-16 The geometry for sh2-17 has an initial spherical shape with intercepts at z = 3e-3 m and r = 3e3 m. Reference points for tracking variables are also shown. 3 6 2 1 Gen Exam Preparation 5 4 8 of 13 John A. Smythe University of Washington sh2-17 Non--conservative level set, tolerance 0.01, epsilon 7.295e-4. sh2-17 Non--conservative level set, tolerance 0.01, epsilon 7.295e-4. Gen Exam Preparation 9 of 13 John A. Smythe University of Washington sh2-17 Density tracking shows accumulation of mass at point 5 which is the bottom right corner. The wetted wall conditions are permitting accumulation as the droplet is oscillating. sh2-17 Non--conservative level set, tolerance 0.01, epsilon 7.295e-4. Gen Exam Preparation 10 of 13 John A. Smythe University of Washington sh2-17 Non--conservative level set, tolerance 0.01, epsilon 7.295e-4. The next objective was to understand the mass loss to point 5 and to reduce the change in phi. The tolerance was reduced from 0.01 to 0.005, epsilon was reduced to 3e-4 and a conservative level set was used. It shows an initial shift in z and then remains somewhat stable. Gen Exam Preparation 11 of 13 John A. Smythe University of Washington The results still show the apparent pinning at the initial r intercept of the initial fluid interface. Gen Exam Preparation 12 of 13 John A. Smythe University of Washington Appendix A Initial setup in COMSOL Chemical Engineering Module Momentum Transport Multiphase Flow Level Set Two-Phase Flow, Laminar Transient analysis (or Transient initialization) Dependent variables: u, v, p, phi Use of Get Initial Value Solve > Get Initial Value Then use Plot Parameters, select Min/Max tab and then enter the Expression to use. In this case, it was 5*epsilon/gamma. After initial solution is obtained, the analysis mode needs to be set back to transient initialization and time field needs to be updated in Solver Parameters>Time stepping area and Times edit field. Initialization is done using just a few time steps such as 0:1e-4:3e-3. This is sufficient to define the initial level set. The solution is then stored and used to run a time of interest. A relatively large mesh size is run initially to confirm expected behavior in a short time. Gen Exam Preparation 13 of 13