Why the Boundary of a Round Drop Becomes a Curve of Order Four

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In cases in which the fluid was pure water, the top surface of the polished aluminium substrate was pretreated with a hydrophilic coating Lotus Leaf HydroPhil-S to improve the wettability and ensure precise and uniform pinning along the contact line. Several tests were performed to ensure that the solid particles from the coating remained on the substrate after the drop evaporation. The evaporation rate was also measured for drops placed on coated and uncoated surfaces.

The comparison, showing no difference, is presented in Fig. Finally, we also used the infrared camera to examine the thermal field in drops with and without the pedestal and hydrophilic coating. Once again, no difference was observed in either case. Note that the schematics are not to scale and the simulations are in 3D. Evaporation kinetics. We consider only the gas phase to investigate the evaporation kinetics of non-heated sessile drops of pure fluids. Under these conditions and for liquids with relatively low volatility as those considered in this work, it can be shown that the evaporation timescale is significantly larger than the timescale of species diffusion in the gas 1.

The process is, thus, quasi-stationary and can be accurately approximated by solving the problem of steady-state isothermal diffusion of vapour in the non-condensable gas air. A schematic of the model is presented in Fig. We thus solve. Here c is the vapour mass concentration in the air. This closes the boundary value problem. Since surface tension dominates gravity in these drops, the drop interface shape is the result of the surface energy minimization of the system. We thus use Surface Evolver 21 to calculate these.

As in the experiments, the contact line and the volume are prescribed, and therefore the resulting geometry is unique. A comparison between the experimental drop geometry and that calculated with Surface Evolver for the same contact area and volume is presented in Fig. It follows that the evaporation rate is defined as. Given the simplicity of the governing equation and the complexity of the 3D domain, open-source OpenFoam or commercial Comsol Multiphysics codes are the most efficient solvers for this problem.

We choose to use the latter, which allows finite element method FEM computations with unstructured tetrahedral meshing of the domain with finer resolution near the drop. The results of a mesh dependency test are presented in Table 1 and Fig. Finally, we investigate the effect of the domain size L.

The resulting evaporation rates decrease asymptotically to some value of. As L increases, the variation in tends to zero. Bottom view. According to Table 1 : a coarsest, b fine and c finest. Particle deposition. Above is a hydrodynamically passive gas region. Following the standard approach, the evaporation from the precursor film is taken to be negligible with respect to the macroscopic evaporation of the sessile drop.

The liquid is assumed to be sufficiently viscous that the hydrodynamic portion of the problem is governed by the Stokes equations subject to no-slip at the lower wall, and appropriate stress conditions at the interface,. Note that we have no disjoining pressure so that the drop will be perfectly wetting. This effectively describes a fully 3D problem in non-rectilinear domain including mixed boundary conditions. The flux J is then given by. Dropping the tilde decoration, this ultimately leads to the governing equations 26 ,. All that remains is to discern the form of the vapour mass concentration c whence J.

This can be simplified as follows. First, we note that this is posed in terms of the unscaled outer variable z. Second, rather than attempting to evaluate the problem as a Laplace problem subject to a mixed boundary condition, we follow the approach of Popov 11 and re-pose it as the solution to the Poisson problem. Therefore, in order to satisfy the first boundary condition we simply take. Finally, we note that to leading order in , so that J can be expressed as. We initialize our problem with the triangular profile shown in Fig.

We solve it be using an semi-implicit, operator-splitting strategy based on the method pL 1 of Witelski and Bowen The mesh and time step are varied to ensure convergence. We find that a typical flux profile is as given in Fig. In Fig. A combination of contact-line pinning and continuity drive capillary-induced flow towards the corners, convecting particles towards these apices. Upon complete evaporation, Fig. Parameter values are as in Fig. The parameter values are the same as those used to generate Fig. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Dynamics and universal scaling law in geometrically-controlled sessile drop evaporation. Cazabat, A. Evaporation of macroscopic sessile droplets. Soft Matter 6 , — Erbil, H. Evaporation of pure liquid sessile and spherical suspended drops: a review. Colloid Interface Sci.

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Transport and deposition patterns in drying sessile droplets. AIChE J. Deegan, R. Capillary flow as the cause of ring stains from dried liquid drops. Nature , — Contact line deposits in an evaporating drop. E 62 , — Hu, H. Analysis of the microfluid flow in an evaporating sessile droplet. Langmuir 21 , — Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet. Marangoni effect reverses coffee-ring depositions. B , — Picknett, R. Evaporation of sessile or pendant drops in still air. Evaporation of a sessile droplet on a substrate. Popov, Y.

Evaporative deposition patterns: spatial dimensions of the deposit. E 71 , 1—17 Sefiane, K. Self-excited hydrothermal waves in evaporating sessile drops. Sobac, B. Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate. Fluids 24 , Christy, J. Flow transition within an evaporating binary mixture sessile drop. Kim, H.

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  4. Why the Boundary of a Round Drop Becomes a Curve of Order Four!

Controlled uniform coating from the interplay of Marangoni flows and surface-adsorbed macromolecules. Tan, H. Evaporationtriggered microdroplet nucleation and the four life phases of an evaporating Ouzo drop. Natl Acad. USA , — Yoshino, M. Engineering surface and development of a new DNA micro array chip.

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Wear , — Calvert, P. Inkjet printing for materials and devices. Park, B. Direct writing of copper conductive patterns by ink-jet printing.

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Thin Solid Films , — Evaporation of sessile drops: a three-dimensional approach. Fluid Mech. The surface evolver. Characteristic angles in the wetting of an angular region: deposit growth. E 68 , Bennacer, R. Vortices, dissipation and flow transition in volatile binary drops.

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Sternling, C. Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AlChE J. Espin, L. Droplet spreading and absorption on rough, permeable substrates. Wray, A. Langmuir 30 , — Jackson, J. Classical Electrodynamics Wiley Witelski, T. Adi schemes for higher-order nonlinear diffusion equations. Download references. Lee, V. Solotych and A. Scammell at the University of Maryland in the experimental part of this study.

The project was conceived by P. The simulations were performed by P. The findings were discussed by all the authors. The paper was written by P. Correspondence to P. AVI kb. This work is licensed under a Creative Commons Attribution 4. Reprints and Permissions. Journal of Colloid and Interface Science International Journal of Heat and Mass Transfer Physica D: Nonlinear Phenomena Experimental Thermal and Fluid Science By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Advanced search. Skip to main content. Subjects Applied mathematics Fluid dynamics. It is important to note here that in absence of stabilized production trends the technique cannot be expected to give reliable results. The technique is not necessarily grounded in fundamental theory but is based on empirical observation of production decline. There are theoretical equivalent to these decline processes. It can be demonstrated that under conditions such as constant well back pressure, equation of fluid flow through porous media under boundary dominated flow are equivalent to exponential decline.

However for our purpose it is the empirical nature of this term which has a greater significance since it allows the technique to be applied to multiple fluid streams even ratios!

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This book concerns the problem of evolution of a round oil spot surrounded by water when oil is extracted from a well inside the spot. It turns out. ykoketomel.ml: Why the Boundary of a Round Drop Becomes a Curve of Order Four (University Lecture Series) (): A. N. Varchenko, P. I. Etingof: .

The basic assumption in this procedure is that whatever causes controlled the trend of a curve in the past will continue to govern its trend in the future in a uniform manner. His work was further extended by other researchers to include special cases.

Following section gives a historical perspective of work done on the subject;. The major application of DCA in the industry today is still based on equations and curves described by Arps. Arps applied the equation of Hyperbola to define three general equations to model production declines. Arps did not provide physical reasons for the three types of decline. Clearly all wells do not exhibit exponential behavior during depletion. In many cases a more gradual hyperbolic decline is observed where rate time performance is better than estimated from exponential solutions implying that hyperbolic decline results from natural and artificial driving energies that slow down pressure depletion.

Hyperbolic decline is observed when reservoir drive mechanism is solution gas cap drive, gas cap expansion or water drive. It is also possible where natural drive is supplemented by injection of water gas. The type of decline and its characteristic shape is a major feature of DCA.

We shall be talking more about this as we go further. The various types of declines experienced by a well are documented in the Fig 1 and Fig 2. Time showing various types of declines on Cartesian plot.

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Time showing various types of declines on Semilog plot. Note change in shapes of curves. Pending permission approval. Observe the change in Shapes of curve from Cartesian to logarithmic; this is very helpful in identification of type of decline. Reservoir types with exponential declines [1] :. The decline rate is not a constant but changes with time, since the data plots as a curve on semi-log paper. The hyperbolic exponent b is the rate of change of the decline rate with respect to time. It is evident from Eq 1 that a large value of b close to 1 has a dominant effect on shape of the curve q vs.

This causes and maintains the shape of the curve during this time to be essentially flat. For a given set of values for q and b the short term shape for the curve is not largely effected by the value of b but the long terms shape is. This implies that in short term all decline curves; exponential, hyperbolic and harmonic give similar results. However due the very same reasons make it extremely difficult to determine the value of b. The problem is aggravated if the data is noisy which is often the case making it possible to fit a wide range of b values to the same dataset. However since the value of b has large impact in the late time, it will lead to different estimates of EUR.

Reliability in estimation of b increases with maturity of production data. The value of b captures a large number of physical events and processes. A large body of publications are dedicated to this topic. Based on what has been covered so far, the engineer performing a DCA analysis needs to be aware of the following:. However one more factor, also extremely important at this stage is to determine type of decline. Since the signature of shape may not be apparent on a log q vs.

As shown in Figures 8 to 11 Shapes of curves for the same data plotted in different ways helps determine the type of declines. A major use of decline curve analysis is made in estimation of reserves. Even for the assets where history matched simulation models are available, a cross check with DCA is normally made to give increased confidence in numbers. The fact that DCA does not have a theoretical basis is an asset here since financial institutions are more acceptable to DCA estimates than other more technical methodologies.

A major difference when applying DCA for estimation of reserves arises understandably due the very nature of definitions of reserves and financial implications associated with the process. The ultimate recovery numbers become more important than the profiles. Application of constraints in the production system, operating costs, capital costs and well behavior itself all need to be put into right perspective to come up with reliable estimations. An in-depth description of application of DCA to reserves estimation is outside the purview of this guideline, however some typical situation and their treatment are discussed in section II of this chapter.

While everything else remains same, estimation of reserves does come up with several typical situations to which there are no ready answers. Some of these situations are listed out below for reference. The solutions to these problems could vary from engineer to engineer or organization to organization. Some of the best practices have however been compiled and can be found in production forecasting principles and definition. Decline analysis and forecasts generated based on such analysis whether production profiles or reserves should be fundamentally grounded in good understanding of the factors that control this behavior.

Specifically always arbitrarily using an exponential decline approach for water drive, solution gas drive and gravity drainage systems is neither technically not empirically justified. However whenever such phenomenon is observed, usually non reservoir factors are at play. In order to estimate future waterflood performance we need to examine what is controlling the oil decline rate. After substantial water breakthrough, the rate is usually controlled by.

RF vs. As a special case, Roland Horne..

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Reservoirs producing with high watercut, high GOR need to be analyzed using ratio plots in conjunction with conventional plots to ensure there is no overestimation of volumes based on rate plots only. Log WOR vs. Np, Log GOR vs. Np, Watercut vs.