The boundary element method is used to simulate the flow of an emulsion drop through a converging channel. The drop flow is governed by the Stokes equations since the Reynolds number is low. Integral equations are derived relating the velocity and stress fields on the drop surface and channel boundaries. The equations account for the viscosity ratio, capillary number, and drop shape. Nodes and quadratic elements are used to discretize the boundaries and solve the integral equations numerically. The results show the effect of parameters on the flow rate-pressure relation and drop shape dynamics.
The document discusses the plastic analysis of slabs using the kinematic method. It begins by introducing slabs and their structural modeling. It then covers yield lines and the assumption that they can be approximated as bisectors. Several examples are presented of determining the plastic moment capacity of different types of slabs, including squared, hexagonal, and circular slabs with various support conditions, through satisfying the principle of virtual work by equating external and internal work. The method is demonstrated on more complex slab layouts as well.
This document discusses a study investigating how shape factors in the volumetric balance model for surface irrigation vary over time, rather than remaining constant as traditionally assumed. The study combines the volumetric balance model with the kinematic wave model to derive equations showing how the surface and subsurface shape factors change in relation to time and other variables. The results show the subsurface shape factor varies more significantly over time compared to the surface shape factor. The new time-dependent approach is found to provide good agreement with field data on water advancement curves, demonstrating the impermanence of assuming constant shape factors.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
This document proposes a method for continuous time series alignment in human action recognition. It defines continuous versions of time series, warping paths, and the dynamic time warping (DTW) distance. The method finds the optimal continuous warping path by approximating solutions to a cost minimization problem. An experiment applies the continuous DTW to classify human activities from accelerometer data, achieving classification accuracy close to the discrete DTW method. The continuous approach solves issues with resampling data and has potential for improved approximations and optimization methods.
The document discusses the plastic analysis of slabs using the kinematic method. It begins by defining a slab as a structural element contained in a plane where the third dimension is much smaller than the other two. It then covers yield lines which divide slabs into rigid regions, assumptions of the yield line method, and provides examples of calculating the plastic moment capacity of slabs by equating external and internal work.
1) The document discusses the plastic analysis of slabs using the kinematic method. It describes analyzing different slab geometries including square, hexagonal, and circular slabs with various support conditions.
2) The kinematic method involves assuming a collapse mechanism defined by yield lines and calculating the external and internal work done. Equating the two gives the plastic moment capacity.
3) Examples are presented of analyzing different slab configurations and determining the correct collapse pattern by evaluating which yields the highest plastic moment capacity.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
This document discusses methods for progressive collapse analysis of structures. It covers elastic and plastic analysis, the kinematic method, and the incremental method. The kinematic method involves guessing a collapse pattern and calculating the load required to cause it, while the incremental method analyzes the structure incrementally as plastic hinges form sequentially. An example problem demonstrates the incremental method by calculating the loads at which successive plastic hinges form in a beam until full collapse.
The document discusses the plastic analysis of slabs using the kinematic method. It begins by introducing slabs and their structural modeling. It then covers yield lines and the assumption that they can be approximated as bisectors. Several examples are presented of determining the plastic moment capacity of different types of slabs, including squared, hexagonal, and circular slabs with various support conditions, through satisfying the principle of virtual work by equating external and internal work. The method is demonstrated on more complex slab layouts as well.
This document discusses a study investigating how shape factors in the volumetric balance model for surface irrigation vary over time, rather than remaining constant as traditionally assumed. The study combines the volumetric balance model with the kinematic wave model to derive equations showing how the surface and subsurface shape factors change in relation to time and other variables. The results show the subsurface shape factor varies more significantly over time compared to the surface shape factor. The new time-dependent approach is found to provide good agreement with field data on water advancement curves, demonstrating the impermanence of assuming constant shape factors.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
This document proposes a method for continuous time series alignment in human action recognition. It defines continuous versions of time series, warping paths, and the dynamic time warping (DTW) distance. The method finds the optimal continuous warping path by approximating solutions to a cost minimization problem. An experiment applies the continuous DTW to classify human activities from accelerometer data, achieving classification accuracy close to the discrete DTW method. The continuous approach solves issues with resampling data and has potential for improved approximations and optimization methods.
The document discusses the plastic analysis of slabs using the kinematic method. It begins by defining a slab as a structural element contained in a plane where the third dimension is much smaller than the other two. It then covers yield lines which divide slabs into rigid regions, assumptions of the yield line method, and provides examples of calculating the plastic moment capacity of slabs by equating external and internal work.
1) The document discusses the plastic analysis of slabs using the kinematic method. It describes analyzing different slab geometries including square, hexagonal, and circular slabs with various support conditions.
2) The kinematic method involves assuming a collapse mechanism defined by yield lines and calculating the external and internal work done. Equating the two gives the plastic moment capacity.
3) Examples are presented of analyzing different slab configurations and determining the correct collapse pattern by evaluating which yields the highest plastic moment capacity.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
This document discusses methods for progressive collapse analysis of structures. It covers elastic and plastic analysis, the kinematic method, and the incremental method. The kinematic method involves guessing a collapse pattern and calculating the load required to cause it, while the incremental method analyzes the structure incrementally as plastic hinges form sequentially. An example problem demonstrates the incremental method by calculating the loads at which successive plastic hinges form in a beam until full collapse.
The document discusses progressive collapse analysis of structures through elastic and plastic analysis methods. It describes the kinematic method and incremental method of plastic analysis to determine the formation of plastic hinges in frames as the load increases. The incremental method involves performing elastic analysis with an additional hinge at each load level until global collapse occurs. An example problem demonstrates applying the incremental method to calculate the load levels that result in formation of successive plastic hinges in a beam.
This document discusses plastic analysis of slabs. It presents six potential yield line patterns that could develop in slabs, with the cantilever pattern being most likely. It states that the minimum required plastic moment capacity (Mp) to build the slab is equal to twice the distributed load (q). An example is given where a slab with Mp of 100 kNm can support a maximum distributed load of 50 kN/m2 after accounting for the slab's own weight.
The document discusses plastic analysis of slabs for a structural analysis test. It examines failure patterns of cantilever and span slabs, including rectangular and triangular regions. It also addresses how adding a courtyard or applying a linear load along the perimeter would affect the failure patterns.
This document contains information about a KVPY exam held on November 1st, 2015 for class 11 students. It provides 15 multiple choice questions in Mathematics and 1 multiple choice question in Physics related to calculating the mass of an object given measurements of applied force and acceleration with uncertainties. The questions cover topics like polynomials, factorials, geometry, and trigonometry. The document also lists the contact information for FIITJEE, the organization that administered the exam.
1. The document contains 8 questions from 2 parts (Part A and Part B) on the topic of fluid mechanics for a 4th semester engineering examination.
2. Question 1a defines key fluid mechanics terms like specific weight, dynamic viscosity, kinematic viscosity, surface tension, and capillarity. Question 1b calculates the pressure difference between two points in a horizontal pipe using differential manometer readings.
3. Question 2a states and proves Pascal's law. Question 2b derives a relationship between the length and diameter of a wooden cylinder that must float in water based on its specific gravity.
This document covers topics in structural analysis including:
1. Elastic analysis of grids using the compatibility method and slope deflection method.
2. Plastic analysis of beams, frames, and slabs.
3. Introduction to second order analysis of structures.
The document provides examples of applying the slope deflection method to analyze grids, including determining degrees of freedom, points of zero moment, drawing shear and moment diagrams, and checking joint equilibrium. It also discusses differences between analyzing determinate and indeterminate grids.
The document contains 18 multi-part questions from an exam for the Brazilian Naval Academy in 2015. The questions cover topics such as calculus, geometry, trigonometry, complex numbers, and systems of equations.
This document contains a summary of the GATE 2014 exam for Civil Engineering. It includes:
1) An analysis of the exam showing the percentage of questions from different topics. The highest percentage of questions came from geotechnical engineering (12%) and fluid mechanics (12%).
2) A breakdown of the questions in Set 1 of the exam paper showing the topics covered, number of questions, and total marks for each subject area.
3) The document provides the question paper and answer keys for reference.
The document discusses the plastic analysis of slabs using the yield line method. It introduces slabs and their structural modeling, defines yield lines and their assumptions as being the bisectors of slab elements. It then presents the yield line patterns for cantilever and simply supported slabs under plastic loading conditions and calculates their plastic moment capacities to determine the controlling failure mechanism. Finally, it assigns further practice determining yield line patterns and plastic moments.
The document discusses two methods for progressive collapse analysis: the kinematic method and incremental method. The incremental method involves performing elastic analysis with an increasing number of hinges inserted until global collapse occurs. This leads directly to the failure pattern with minimal calculations. The kinematic method instead assumes the failure mechanism and plastic hinge locations and solves for the critical load. The document provides an example of using the incremental method to analyze a frame, determining that two plastic hinges will form at supports D and E, and calculating the corresponding displacements and load at collapse.
The document discusses the plastic analysis of slabs. It introduces the concept of yield lines, which are lines of plastic hinging that divide the slab into rigid regions. It states that yield lines are assumed to follow the bisectors of angles between supports. The document also describes the virtual work principle of external work equaling internal work that is used to analyze slab collapse mechanisms defined by yield lines. Finally, it provides some examples of how different slabs may break along yield lines.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
B spline collocation solution for an equation arising in instability phenomenoneSAT Journals
This document presents a numerical solution to an equation describing instability phenomenon in double phase flow through a homogeneous porous medium. The governing nonlinear partial differential equation is derived based on Darcy's law and equations of mass conservation for the two immiscible fluids. The equation is then nondimensionalized and solved using the collocation method with cubic B-spline finite elements. The numerical solution provides the saturation distribution of the injected fluid as a function of both time and distance through the porous medium.
This document summarizes a study analyzing aerodynamic wing sections at ultra-low Reynolds numbers below 10,000. Computational fluid dynamics (CFD) techniques are used to model steady and unsteady flows. For steady flows, different wing section geometries are analyzed by varying parameters like thickness, camber, and leading edge shape. For unsteady flows, wing motions like heaving, pitching, flapping, and hovering are modeled. Strouhal numbers, reduced frequencies, and Reynolds numbers are determined to characterize the unsteady aerodynamics. The finite element method is used to solve the Navier-Stokes equations with a fractional step method for the incompressible, laminar flows.
In this paper, linear graphical method, moment method and inverse function method are first applied in the laboratory test of one dimensional sand column device, determining the longitudinal dispersion coefficient. The longitudinal dispersions for five groups of sand taken from 20cm below the ground surface in the Oil Refinery of China Petroleum Ningxia Filial are obtained. On this basis, the problems within the calculation process when the three kinds of methods are applied into actual data were discussed. It can be readily concluded that the three values of dispersion coefficients are approximate, and the errors caused by the subjective factors of artificial mapping and numerical reading were avoided. The inverse function method is recommended to apply for the high accuracy, sample calculation process, less known conditions and better linearity.
This document summarizes a numerical simulation study of flow past a circular cylinder in a channel at varying ratios of tunnel height to cylinder diameter (H/D). Two computational fluid dynamics codes, 3D PURLES and OpenFOAM, were used to simulate the flow at a Reynolds number of 40. The simulations showed a decrease in wake length and a shift of flow separation downstream at smaller H/D ratios. Grid resolution and H/D ratios from 2-30 were investigated. The results from both codes were consistent and confirmed the effects of tunnel walls in changing flow characteristics around the cylinder.
1) Computational techniques are essential for accurately simulating high-speed coating flows that conventional asymptotic models cannot capture. Accessing smaller spatio-temporal scales through computation and experiment is needed to identify the true physics.
2) Gas dynamics, particularly the mean free path of gas molecules, play a key role in phenomena like air entrainment in coating flows. At reduced pressures, the longer mean free path can delay air entrainment.
3) There is still debate around wetting because different models can describe experiments reasonably well using different parameter values. Fully resolving interfaces and accessing microscales may be needed to definitively identify the governing physics.
This document summarizes a study that used sigmoidal parameterization and Metropolis-Hasting (MH) inversion to estimate seismic velocity models from traveltime data. The key points are:
1) Sigmoidal functions were used to parameterize discontinuous velocity fields, allowing for sharp variations while maintaining continuity.
2) Ray tracing and the MH algorithm were used to invert traveltime data and estimate model parameters.
3) Tests on synthetic models showed the MH method produced higher resolution velocity models that better fit the observed traveltime data, compared to other global optimization methods like very fast simulated annealing.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
This document summarizes a study that estimates artery wall stiffness using a data assimilation method applied to a 3D fluid-structure interaction (FSI) model. The study presents:
1) An FSI model coupling fluid (Navier-Stokes) and solid (elastodynamic) equations on a moving domain, with coupling conditions at the interface.
2) A reduced order unscented Kalman filter data assimilation procedure to estimate uncertain parameters like Young's modulus from displacement measurements.
3) Experimental validation using a silicone tube where estimated stiffness matched independent tests, and application to a clinical case of aorta stiffness estimation.
The document discusses progressive collapse analysis of structures through elastic and plastic analysis methods. It describes the kinematic method and incremental method of plastic analysis to determine the formation of plastic hinges in frames as the load increases. The incremental method involves performing elastic analysis with an additional hinge at each load level until global collapse occurs. An example problem demonstrates applying the incremental method to calculate the load levels that result in formation of successive plastic hinges in a beam.
This document discusses plastic analysis of slabs. It presents six potential yield line patterns that could develop in slabs, with the cantilever pattern being most likely. It states that the minimum required plastic moment capacity (Mp) to build the slab is equal to twice the distributed load (q). An example is given where a slab with Mp of 100 kNm can support a maximum distributed load of 50 kN/m2 after accounting for the slab's own weight.
The document discusses plastic analysis of slabs for a structural analysis test. It examines failure patterns of cantilever and span slabs, including rectangular and triangular regions. It also addresses how adding a courtyard or applying a linear load along the perimeter would affect the failure patterns.
This document contains information about a KVPY exam held on November 1st, 2015 for class 11 students. It provides 15 multiple choice questions in Mathematics and 1 multiple choice question in Physics related to calculating the mass of an object given measurements of applied force and acceleration with uncertainties. The questions cover topics like polynomials, factorials, geometry, and trigonometry. The document also lists the contact information for FIITJEE, the organization that administered the exam.
1. The document contains 8 questions from 2 parts (Part A and Part B) on the topic of fluid mechanics for a 4th semester engineering examination.
2. Question 1a defines key fluid mechanics terms like specific weight, dynamic viscosity, kinematic viscosity, surface tension, and capillarity. Question 1b calculates the pressure difference between two points in a horizontal pipe using differential manometer readings.
3. Question 2a states and proves Pascal's law. Question 2b derives a relationship between the length and diameter of a wooden cylinder that must float in water based on its specific gravity.
This document covers topics in structural analysis including:
1. Elastic analysis of grids using the compatibility method and slope deflection method.
2. Plastic analysis of beams, frames, and slabs.
3. Introduction to second order analysis of structures.
The document provides examples of applying the slope deflection method to analyze grids, including determining degrees of freedom, points of zero moment, drawing shear and moment diagrams, and checking joint equilibrium. It also discusses differences between analyzing determinate and indeterminate grids.
The document contains 18 multi-part questions from an exam for the Brazilian Naval Academy in 2015. The questions cover topics such as calculus, geometry, trigonometry, complex numbers, and systems of equations.
This document contains a summary of the GATE 2014 exam for Civil Engineering. It includes:
1) An analysis of the exam showing the percentage of questions from different topics. The highest percentage of questions came from geotechnical engineering (12%) and fluid mechanics (12%).
2) A breakdown of the questions in Set 1 of the exam paper showing the topics covered, number of questions, and total marks for each subject area.
3) The document provides the question paper and answer keys for reference.
The document discusses the plastic analysis of slabs using the yield line method. It introduces slabs and their structural modeling, defines yield lines and their assumptions as being the bisectors of slab elements. It then presents the yield line patterns for cantilever and simply supported slabs under plastic loading conditions and calculates their plastic moment capacities to determine the controlling failure mechanism. Finally, it assigns further practice determining yield line patterns and plastic moments.
The document discusses two methods for progressive collapse analysis: the kinematic method and incremental method. The incremental method involves performing elastic analysis with an increasing number of hinges inserted until global collapse occurs. This leads directly to the failure pattern with minimal calculations. The kinematic method instead assumes the failure mechanism and plastic hinge locations and solves for the critical load. The document provides an example of using the incremental method to analyze a frame, determining that two plastic hinges will form at supports D and E, and calculating the corresponding displacements and load at collapse.
The document discusses the plastic analysis of slabs. It introduces the concept of yield lines, which are lines of plastic hinging that divide the slab into rigid regions. It states that yield lines are assumed to follow the bisectors of angles between supports. The document also describes the virtual work principle of external work equaling internal work that is used to analyze slab collapse mechanisms defined by yield lines. Finally, it provides some examples of how different slabs may break along yield lines.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
This document summarizes a numerical simulation of flow between two rotating coaxial frustum cones. The simulation found that the fluid does not flow directly out of the outlet, but flows upward to a certain height, generating a vortex area with large velocity and pressure. This reflux area moves upward as the Reynolds number increases. For small frustum inclinations, the flow becomes unstable with very large velocity and pressure, even at small rotation rates. The study compares the flow properties to Taylor-Couette flow between rotating cylinders.
Implementation of finite volume method in creeping flow around a circular cyl...eSAT Journals
Abstract
A Finite Volume Method have been performed in simulation of creeping flow around a circular cylinder contained between plates. By adopting the SIMPLE algorithm the governing equations are solved together with Papanastasious regularization. Apparent viscosity is calculated on each iteration by using forward difference operation. Yield surfaces are studied over the range of Oldroyd number 10≤Od≤〖10〗^4. The model results are found to be in good agreement with obtained results of the other method.
Keywords: Creeping flow, Finite Volume Method, Yielded and unyielded zone, Bingham number.
B spline collocation solution for an equation arising in instability phenomenoneSAT Journals
This document presents a numerical solution to an equation describing instability phenomenon in double phase flow through a homogeneous porous medium. The governing nonlinear partial differential equation is derived based on Darcy's law and equations of mass conservation for the two immiscible fluids. The equation is then nondimensionalized and solved using the collocation method with cubic B-spline finite elements. The numerical solution provides the saturation distribution of the injected fluid as a function of both time and distance through the porous medium.
This document summarizes a study analyzing aerodynamic wing sections at ultra-low Reynolds numbers below 10,000. Computational fluid dynamics (CFD) techniques are used to model steady and unsteady flows. For steady flows, different wing section geometries are analyzed by varying parameters like thickness, camber, and leading edge shape. For unsteady flows, wing motions like heaving, pitching, flapping, and hovering are modeled. Strouhal numbers, reduced frequencies, and Reynolds numbers are determined to characterize the unsteady aerodynamics. The finite element method is used to solve the Navier-Stokes equations with a fractional step method for the incompressible, laminar flows.
In this paper, linear graphical method, moment method and inverse function method are first applied in the laboratory test of one dimensional sand column device, determining the longitudinal dispersion coefficient. The longitudinal dispersions for five groups of sand taken from 20cm below the ground surface in the Oil Refinery of China Petroleum Ningxia Filial are obtained. On this basis, the problems within the calculation process when the three kinds of methods are applied into actual data were discussed. It can be readily concluded that the three values of dispersion coefficients are approximate, and the errors caused by the subjective factors of artificial mapping and numerical reading were avoided. The inverse function method is recommended to apply for the high accuracy, sample calculation process, less known conditions and better linearity.
This document summarizes a numerical simulation study of flow past a circular cylinder in a channel at varying ratios of tunnel height to cylinder diameter (H/D). Two computational fluid dynamics codes, 3D PURLES and OpenFOAM, were used to simulate the flow at a Reynolds number of 40. The simulations showed a decrease in wake length and a shift of flow separation downstream at smaller H/D ratios. Grid resolution and H/D ratios from 2-30 were investigated. The results from both codes were consistent and confirmed the effects of tunnel walls in changing flow characteristics around the cylinder.
1) Computational techniques are essential for accurately simulating high-speed coating flows that conventional asymptotic models cannot capture. Accessing smaller spatio-temporal scales through computation and experiment is needed to identify the true physics.
2) Gas dynamics, particularly the mean free path of gas molecules, play a key role in phenomena like air entrainment in coating flows. At reduced pressures, the longer mean free path can delay air entrainment.
3) There is still debate around wetting because different models can describe experiments reasonably well using different parameter values. Fully resolving interfaces and accessing microscales may be needed to definitively identify the governing physics.
This document summarizes a study that used sigmoidal parameterization and Metropolis-Hasting (MH) inversion to estimate seismic velocity models from traveltime data. The key points are:
1) Sigmoidal functions were used to parameterize discontinuous velocity fields, allowing for sharp variations while maintaining continuity.
2) Ray tracing and the MH algorithm were used to invert traveltime data and estimate model parameters.
3) Tests on synthetic models showed the MH method produced higher resolution velocity models that better fit the observed traveltime data, compared to other global optimization methods like very fast simulated annealing.
INRIA-USFD-KCL- Identification of artery wall stiffness - 2014Cristina Staicu
This document summarizes a study that estimates artery wall stiffness using a data assimilation method applied to a 3D fluid-structure interaction (FSI) model. The study presents:
1) An FSI model coupling fluid (Navier-Stokes) and solid (elastodynamic) equations on a moving domain, with coupling conditions at the interface.
2) A reduced order unscented Kalman filter data assimilation procedure to estimate uncertain parameters like Young's modulus from displacement measurements.
3) Experimental validation using a silicone tube where estimated stiffness matched independent tests, and application to a clinical case of aorta stiffness estimation.
Within the framework of the theory of plane steady filtration of an incompressible fluid according to Darcy’s law, two limiting schemes modeling the filtration flows under the Joukowski tongue through a soil massive spread over an impermeable foundation or strongly permeable confined water bearing horizon are considered.
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
1) The document presents a solution to the Burger's equation, which describes the longitudinal dispersion phenomenon that occurs when miscible fluids flow through porous media.
2) The solution is obtained by applying the Sumudu transform to reduce the Burger's equation to a heat equation, which is then solved using the homotopy perturbation method.
3) The final solution obtained for concentration C is expressed as a function of distance x and time t, representing the concentration profile for longitudinal dispersion at any point in the porous media and at any time.
Talk presented on GAMM 2019 Conference in Vienna, Austria.
Parallel algorithm for uncertainty quantification in the density driven subsurface flow. Estimate risks of subsurface flow pollution.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNS...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
Determination of the Probability Size Distribution
of Solid Particles in a Technical Water by Tomantschger Kurt W*, Petrović Dragan V and Radojević Rade L in Evolutions in Mechanical Engineering
1) The document compares different simplified models for simulating rivulet flow down a slowly varying substrate to a direct numerical simulation using computational fluid dynamics (CFD).
2) Three simplified models are considered: constant contact angle and varying width, constant width and varying contact angle, and varying both contact angle and width.
3) The CFD simulation uses the lubrication approximation to directly solve the Navier-Stokes equations, while the simplified models make additional assumptions to obtain analytical solutions.
4) Comparison of the results from the different methods show that the model allowing both contact angle and width to vary provides the closest agreement to the CFD simulation.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
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5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...
Cunha CILAMCE 2016
1. FLOW OF AN EMULSION DROP THROUGH A CONVERGING
CHANNEL USING THE BOUNDARY ELEMENT METHOD
Lucas Hildebrand Pires da Cunha
´Eder Lima de Albuquerque
lucashilde@gmail.com
eder@unb.br
Department of Mechanical Engineering, Universidade de Bras´ılia
Campus Universit´ario Darcy Ribeiro, 70910-900, DF, Bras´ılia, Brazil
Ivan Rosa de Siqueira
ivan@lmmp.mec.puc-rio.br
Laboratory of Microhydrodynamics and Flow in Porous Media - LMMP, Department of Me-
chanical Engineering, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro
R. Marquˆes de S˜ao Vincente - 225 G´avea, 22451-900, RJ, Rio de Janeiro, Brazil
Abstract. The Boundary Element Method is employed to simulate the flow of a planar emulsion
drop through a converging channel. The physical flow parameters allow to describe it using
Stokes equations. A verification on the method accuracy and convergence is done related to
the mesh and time increment refinement. The pump pressure answer, during the drop flow to
keep the flow rate, is studied in relation to the capillary number, the viscosity ratio and the drop
initial diameter.
Keywords: Emulsion, Planar drop, Porous media, Boundary Element Method.
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
2. Flow of an emulsion drop through a converging channel using the Boundary Element Method
1 INTRODUCTION
The use of oil-water emulsions as a mobility control agent in enhanced oil recovery pro-
cesses in order to achieve a more efficient sweep of the petroleum reservoir is a widespread
theme in the current literature (Alvarado & Manrique, 2010). The macroscopic behavior of
these methods is intrinsically related to the pore-scale flow. Since the drop diameter has the
same order of magnitude of the pore throats dimensions, the emulsion cannot be treated as a
singe phase non-Newtonian fluid. Otherwise, this complex liquid must be modeled as a biphasic
mixture of two immiscible viscous Newtonian fluids. Several experimental results and theoret-
ical analysis have shown that for the emulsion flow, the pressure drop at a fixed flow rate is
raised by two distinct mechanisms: a viscous effect related to the presence of the high viscosity
oil phase and a capillary effect related to the deformation of the droplet interface as it flows
through the porous throat (Roca & Carvalho, 2013). Within this context, the present work aims
to perform a numerical investigation of the flow of an emulsion drop though a porous media
using the Boundary Element Method (BEM).
The model considers the flow of a single planar oil droplet through a converging chan-
nel representing the porous throat, and can provide important qualitative results in this type of
study. The flow is free from inertial effects and thereby it is governed by the Stokes’ incom-
pressible equations. In this sense, a boundary integral formulation based on line integrals over
the drop surface and the channel boundaries relating both velocity and stress fields can be ob-
tained. Several numerical aspects of the method are discussed in details, including the use of
quadratic continuous boundary elements, the interpolation of the unknown fields using second-
order shape function and, the discretization and numerical solution of the integral equations.
The results show the effects of physical and geometric parameters, such as the fluids viscosity
ratio, capillary number and droplet initial diameter on the flow rate-pressure drop relation and
on the droplet shape in the flow.
2 MATHEMATICAL MODELING
A scheme of the problem is represented in Fig. 1. Both phases are composed by incom-
pressible Newtonian liquids with the same density ρ. The drop is initially spherical with diam-
eter a and viscosity λµ, and it is immersed in other fluid with the viscosity equal to µ. Here, H
and h are the height of the entrance and exit (throat) of the channel, and P0 and PL are the inlet
and outlet pressure, respectively. Each fluid occupies a region Ω limited by a contour Γ, and the
subscripts i and o are used to correspond to the inner and outer fluids, respectively.
Due the physical parameters associated to the inner and outer flows, the inertial forces are
neglectable comparing to viscous ones (Re 1).
2.1 Stokes flow integral representation
According to Kim & Karrila (1991), the flow of a Newtonian incompressible fluid free
from inertial effects is governed by the Stokes’ equations shown in Eq. (1) and Eq. (2), where
σ is the stress tensor field and u is the velocity field, respectively.
· u = 0. (1)
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
3. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
Ω (t)
Ω (t)
Γ(t)
Γ
o
i
i
o
ρ, λμ
a h
H
ρ, μ
u(x)
P0
PL x
x
2
1
Figure 1: Scheme of the planar drop dispersed in other immiscible fluid flowing through a converging
channel. The drop is composed by the i fluid while the continuous one is compose by the o fluid. Each fluid
occupies a region Ω limited by a contour Γ.
· σ = 0. (2)
The stress tensor, σ, is given by,
σ(x) = −p(x)I + 2µD, (3)
where I is the unit tensor, p(x) is the mechanical pressure and D = 1
2
( u + uT
).
The Stokes flow integral representation in a domain Ω limited by a contour Γ can be ob-
tained utilizing the fundamental solution of Stokes flow created by a force point in an infinite
fluid domain, presented by Ladyzheskaya (1969), and the Lorentz Reciprocity Theorem (Kim
& Karrila, 1991; Pozrikids, 1992). After all, the velocity on a point x0 can be obtained using,
x0 ∈ Ω, u(x0)
x0 ∈ Γ, c(x0)u(x0)
x0 ∈ Ω, 0
=
1
4πµ Γ
J (x − x0) · σ(x) · ˆn dΓ(x)
−
1
4π Γ
u(x) · K(x − x0) · ˆn dΓ(x), (4)
where c(x0) is a constant resultant from the Dirac Delta function integration and depend on the
contour geometry, ˆn is the contour normal direction and, J and K are both Green functions,
given by
J (x − x0) = I log
1
|x − x0|
+
(x − x0)(x − x0)
|x − x0|2
, (5)
and
K(x − x0) = −4
(x − x0)(x − x0)(x − x0)
|x − x0|4
. (6)
2.2 Flow integral representation on the interface
The relation between the inner and outer flows are obtained using the boundary conditions
on the interface. Due the continuity in velocity, the two flows should have the same velocity
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
4. Flow of an emulsion drop through a converging channel using the Boundary Element Method
over the shared points, such that
ui(x) = uo(x) , x ∈ Γi(t). (7)
There is no continuity in the stress tensor on the interface, but it is possible to calculate
the stress jump over the normal direction, ∆f(x, t) = [σo(x, t) − σi(x, t)] · ˆn(x, t), using the
Young-Laplace equation,
∆f(x, t) = σκ(x, t)ˆn(x, t) , x ∈ Γi(t), (8)
where, σ is the interfacial tension between the fluids and κ is the surface curvature.
Using the velocity and tension relations over the fluids interface, it is possible to calculate
the velocity over Γi and Γo by the dimensionless1
Eqs. (9) and (10), respectively.
x0 ∈ Γi(t), c(x0, t)(λ + 1)ui(x0, t) =
1
4π Γo
J (x − x0) · to(x, t) dΓ
−
1
4π Γo
uo(x, t) · K(x − x0) · ˆn(x, t) dΓ
+
1
4π
Ca−1
Γi(t)
J (x − x0) · κ(x, t)ˆn(x, t) dΓ
−
1 − λ
4π
a
H Γi(t)
ui(x, t) · K(x − x0) · ˆn(x, t) dΓ.
(9)
x0 ∈ Γo, c(x0, t)uo(x0, t) =
1
4π Γo
J (x − x0) · to(x, t) dΓ
−
1
4π Γo
uo(x, t) · K(x − x0) · ˆn(x, t) dΓ
+
1
4π
Ca−1
Γi(t)
J (x − x0) · κ(x, t)ˆn(x, t) dΓ
−
1 − λ
4π
a
H Γi(t)
ui(x, t) · K(x − x0) · ˆn(x, t) dΓ.
(10)
The capillary number, Ca, present in Eqs. (9) and (10) is one of the most important param-
eters in emulsion rheology studies. It represents the relation between viscous forces and surface
forces associated to interfacial tension, and it is given by Ca = µU/σ. Still in these equations,
the normal vector over the channel contour points to the outside, while over the interface it
points to the inside.
3 NUMERICAL METHODOLOGY
The solution of the problem is given by a boundary integral over the contour of the channel
and over the drop interface. Therefore, we employed the BEM to numerically solve the problem.
1
The dimensionless variables are defined as: x0 = x0/H, x = x /H, ui = ui/U, uo = uo/U, to =
Hto/µU, κ = aκ , J = J , K = HK , dΓ = dΓ /H over Γo and dΓ = dΓ /a over Γi(t), where denotes the
dimensional variable.
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
5. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
The contours were divided in No + Ni elements, being No for the outer contour (Γo) and Ni for
the inner one (Γi). It was chosen the continuous quadratic element. In this case, every element is
composed by three nodes, sharing the two nodes on the extremes with the neighboring elements.
The use of shape functions allows to interpolate any vector variable by its values on the nodes,
x =
x1
x2
=
N1 0 N2 0 N3 0
0 N1 0 N2 0 N3
x
(1)
1
x
(1)
2
x
(2)
1
x
(2)
2
x
(3)
1
x
(3)
2
= N(ξ) · x(n), (11)
where x
(n)
d is the variable value on the node n and direction d, ξ is the local spatial coordinate
of the element which goes form −1 to 1, and Ns is the value of the shape functions s for ξ,
N1 =
ξ
2
(ξ − 1), (12)
N2 = (1 − ξ)(1 + ξ) (13)
and
N3 =
ξ
2
(ξ + 1). (14)
The discretization of the contours in elements and use of quadratic shape functions allow
to rewrite Eqs. (9) and (10), respectively, as
xm ∈ Γi(t), cm
(λ + 1)um
i (t) =
1
4π
No
n=1 ∆Γn
o
J mn
· N(ξ) dΓ · to(n)(t)
−
1
4π
No
n=1 ∆Γn
o
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ · uo(n)(t)
+
1
4π
Ca−1
No+Ni
n=No+1 ∆Γn
i
J mn
· κn
(ξ, t)ˆnn
(ξ, t) dΓ
−
1 − λ
4π
a
H
No+Ni
n=No+1 ∆Γn
i
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ · ui(n)(t),
(15)
and
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
6. Flow of an emulsion drop through a converging channel using the Boundary Element Method
xm ∈ Γo, cm
um
o (t) =
1
4π
No
n=1 ∆Γn
o
J mn
· N(ξ) dΓ · to(n)(t)
−
1
4π
No
n=1 ∆Γn
o
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ · uo(n)(t)
+
1
4π
Ca−1
No+Ni
n=No+1 ∆Γn
i
J mn
· κn
(ξ, t)ˆnn
(ξ, t) dΓ
−
1 − λ
4π
a
H
No+Ni
n=No+1 ∆Γn
i
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ · ui(n)(t).
(16)
The subscripts m and n determine that the functions are evaluated on the points xm and
xn, respectively. Khayat et al. (1997) showed that it is possible to compact these two equations
in a linear system like H ·U = G·T +B, where H and G are matrices containing the system
coefficients (purely geometrical), U and T are vectors containing the nodes values of velocity
and tension, respectively, and B is an independent vector associated to the stress jump on the
interface. The linear system can be expressed as
No+Ni
n=1
Hmn(t) · u(n)(t) =
No+Ni
n=1
Gmn · t(n)(t) +
No+Ni
n=1
Bmn(t). (17)
Finally, the coefficients H, G and B of the linear system are given by:
Hmn(t) =
cm
δmn +
1
4π ∆Γn
o
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ, n ∈ [1, No];
cm
(λ + 1)δmn +
1 − λ
4π
a
H ∆Γn
i
Kmn
· ˆnn
(ξ, t) · N(ξ) dΓ,
n ∈ [No + 1, No + Ni];
(18)
Gmn =
1
4π ∆Γn
o
J mn
· N(ξ) dΓ, n ∈ [1, No];
0, n ∈ [No + 1, Ni + No];
(19)
and
Bmn(t) =
0, n ∈ [1, No];
1
4π
Ca−1
∆Γn
i
J mn
· κn
(ξ, t)ˆnn
(ξ, t) dΓ,
n ∈ [No + 1, Ni + No].
(20)
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
7. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
To each node should be applied one boundary condition for each direction, velocity or
tension. On the channel entrance section is applied the parabolic profile of velocity for a pre-
determined mean velocity; On the walls of the channel, there is a no-slip condition, so u(x, t) =
0; For the channel exit section is defined a pressure equal to zero, so t(x, t) = 0; Over the fluids
interface it is used the Young-Laplace equation, which defines the vector B.
Solving the system, it will be obtained the tension and velocity over all nodes. The drop
surface is evolved using an Euler first-order equation at each node on the interface, so ∆x0 =
u(x0, t) ∆t, for a pre-determined time step ∆t.
4 RESULTS
This section brings all results obtained in this work, including discussions about it. All
results were obtained using a MATLAB R
code, considering the same channel geometry, pre-
sented in Fig. 2. The channel dimensions are function of the entrance height, H, and it is
composed by eight straight segments (1 to 8). The initial drop center is always at the same
point, xc = (0.4H, 0.5H). The drop surface, initially round, is defined by two arcs segments (9
and 10), and the channel convergence ratio is 2:1.
1
2
3
4
5
6
7
8 910
(0, 0)
(0, H)
(0.8H, 0)
(0.8H, H)
(1.2H, 0.25H)
(1.2H, 0.75H)
(2.5H, 0.25H)
(2.5H, 0.75H)(0.4H, 0.5H + R)
(0.4H, 0.5H − R)
Figure 2: Channel geometry, showing the 10 segments defining the channel and drop contours.
4.1 Mesh convergence
The contour discretization was done using the parameter Ne. Following Fig. 2, the seg-
ments 2, 4 and 6 are divided in Ne elements, the segments 1, 7 and 8 are divided in 2Ne
elements, and the segments 3, 5, 9 and 10 are divided in 3Ne elements. The method accuracy,
in relation to Ne, was studied for: the absolute error for the outlet flow rate, |Qout(t)−Qin|/Qin,
being Qin the flow rate imposed at the channel entrance; the drop area error, |A(t) − A0|/A0,
being A0 the initial drop area; and the convergence for the pump pressure curve to keep the flow
rate constant. This study used the parameters: Ca = 0.25, λ = 10, a = 0.55 and ∆t = 0.01.
All results are presented in Figs. 3 to 5. The parameter x* is given by the coordinate x1 of the
extreme right node of the drop (xr
1) divided by the channel length, x*= xr
1/2.5H.
The plot in Fig. 3 shows that the absolute error associated to the outlet flow rate decreases
considerably with the mesh refinement, and it stays more stable over x*. For Ne = 3, this
error is around 7.0%, decreasing to 1.0% for Ne = 9. Analyzing Fig. 4, the error associated
to the area is basically invariant to mesh refinement. The drop area error is associated to the
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
8. Flow of an emulsion drop through a converging channel using the Boundary Element Method
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1
Flowrateabsoluteerror(%)
x*
Ne = 3
Ne = 5
Ne = 7
Ne = 9
Figure 3: Absolute error obtained for the outlet flow rate in function of Ne.
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8 1
Areaabsolteerror(%)
x*
Ne = 3
Ne = 5
Ne = 7
Ne = 9
Figure 4: Absolute error obtained for the drop area in function of Ne.
instantaneous shape, which depends essentially to the time step, justifying this fact. But, all
errors are low and hit the maximum value around 0.4% at the relative position of x*= 0.7 (after
the drop enters the throat). Finally, the pressure curves in Fig. 5 show that the pressure depends
on the mesh discretization, and they collapse for Ne bigger than 7.
After all these analysis, and considering the balance between accuracy and computational
time, Ne = 7 was chosen for the other simulations in this work.
4.2 Method stability related to the time step
The time step, ∆t, study was done using the following parameters: Ca = 0.25, λ = 10,
a = 0.55 and Ne = 7. The results are shown in Figs. 6 to 8.
Figure 6 shows that the flow rate error is unresponsive to the temporal refinement employed
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
9. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
140
150
160
170
180
190
0.2 0.4 0.6 0.8 1
ΔP
x*
Ne = 3
Ne = 5
Ne = 7
Ne = 9
Figure 5: Pump pressure curve obtained in function of Ne.
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.2 0.4 0.6 0.8 1
Flowrateabsoluteerror(%)
x*
Δt = 0.1
Δt = 0.05
Δt = 0.01
Δt = 0.005
Figure 6: Absolute error obtained for the outlet flow rate in function of ∆t.
to the Euler method. In fact this error only depends on the mesh discretization. Anyway, the
numeric error remained low and between 1.4% and 1.6%. The curves tend to collapse when
the ∆t is reduced. In contrast, the area error is affected a lot by the time step refinement, as it
is shown in Fig. 7. For ∆t = 0.1 the maximum error is around 4.75%, reducing to 0.25% for
∆t = 0.005. Finally, the pressure curve tends to collapse when ∆t is decreased. In fact, the
curves for ∆t = 0.01 and ∆t = 0.005 are considerably close, as shown in Fig. 8. Due these
analysis, it was chosen ∆t = 0.01 for the next simulations.
4.3 Capillary number effect
Capillary number is one the main parameters in the study of mechanics and rheology of
emulsions, being defined as the viscous and inertial forces ratio, originated by the interfacial
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
10. Flow of an emulsion drop through a converging channel using the Boundary Element Method
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.2 0.4 0.6 0.8 1
Areaabsoluteerror(%)
x*
Δt = 0.1
Δt = 0.05
Δt = 0.01
Δt = 0.005
Figure 7: Absolute error obtained for the drop area in function of ∆t.
145
150
155
160
165
170
175
180
185
190
0.2 0.4 0.6 0.8 1
ΔP
x*
Δt = 0.1
Δt = 0.05
Δt = 0.01
Δt = 0.005
Figure 8: Pump pressure curve obtained in function of ∆t.
tension between two fluids. Here, it is defined as Ca = µU/σ. The interfacial tension acts to
maintain the drop with a round geometry. The channel geometry forces the drop to deform due
to the flow extensional character. In other hand, the capillary forces try to restore your spherical
original geometry. As a result, the flow increase the pump pressure, forcing the drop to enter the
convergence and keep a constant flow rate. This pump pressure increase can recover trapped oil
in neighboring ganglia. Figures 9 and 10 show moments of the drop through the channel for:
Ca = 1.0 and Ca = 0.0625; λ = 30 and a = 0.55.
Analyzing Figs. 9 and 10, smaller Ca values results in rounder drops, smoother shapes.
The Young-Laplace equation shows that bigger σ and bigger curvatures cause higher tensions,
pushing the drop surface to its curvature center, tending to keep the drop rounder.
Figure 11 brings moments of the drop for Ca = 0.0625, λ = 10 and a = 0.55, and
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
11. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
Figure 9: Moments of the drop flowing through the channel, for Ca = 1.
Figure 10: Moments of the drop flowing through the channel, for Ca = 0.0625.
Figure 11: Moment of the drop through the channel, for Ca = 0.0625, λ = 10 and a = 0.55.
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
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12. Flow of an emulsion drop through a converging channel using the Boundary Element Method
shows that the drop assumed an unexpected geometry, indicating a method instability in one or
more instants of the simulation. This instability can happen for three reasons (or a combination
of them): A high curvature somewhere on the drop surface created a high tension, resulting
in a fast evolution for which the time-step was not refined enough to capture the deformation
smoothly. Discontinuities on the surface, as it is shown in Fig. 12, results in wrong curvature
calculations. Curvatures centered outside the drop were not expected in the formulation, being
one of the instabilities causes. The first and the second mentioned causes can be corrected
refining even more the time-step and the mesh, respectively. The third cause requires a method
to recognize concave and convex surfaces. There are, in the literature, some techniques to
control the simulations avoiding instabilities. Yan et al. (2006) calculated the time-step in each
iteration using a maximum node displacement allowed. Another authors prefer to calculate
the time-step in function of the drop relaxation characteristic time (Oliveira, 2007). It was not
found in the literature a method to treat the discontinuities over the drop surface for continuous
quadratic elements.
1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Figure 12: Drop surface discontinuities after a high deformation, for Ca = 0.25 and λ = 10.
With the drop deformation, the nodes over the surface are no longer equally distributed,
they tend to be concentrated in the front and back part of the drop. This fact does not seem to be
a problem for the method stability and accuracy, because these two regions present the surface
higher curvatures, in other words, they need more careful treatment in the method. However,
Wrobel et al. (2009) showed a node relocation method, used in each iteration, to keep the nodes
equally spaced during the simulation.
Figure 13 shows the capillary number influence over the pump pressure2
curve. Smaller
capillary number represents higher pump pressure. It can be noticed that the greater pressure
increase happens between x*= 0.4 and x*= 0.6, when the drop is passing through the conver-
gence (higher deformations). After the drop enters in the channel, the pump pressure oscillates
around the value reached in the constriction region.
The relative point x*= 0.59 was adopted as reference to do a deeper analysis on capillary
number influence over the pressure behavior. For different viscosity ratio values, this relation is
shown in Fig. 14. This plot basically shows a linear relation between Ca−1
and ∆P/P∗
.
4.4 Viscosity ratio effects
The subject, now, will be the study of viscosity ratio, λ, effects over the drop flow through
the converging channel. Figure 15 shows the shape of 5 different drops at the same moment.
For all cases Ca = 0.25 and a = 0.55, but the viscosity ratios are different.
2
P∗
is the pressure difference obtained for the flow in the channel without the drop presence, where P∗
= 149.
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Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
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13. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
1
1.1
1.2
1.3
1.4
1.5
1.6
0.2 0.4 0.6 0.8 1
ΔP/P*
x*
Ca = 1
Ca = 0.5
Ca = 0.25
Ca = 0.125
Ca = 0.0625
Figure 13: Pump pressure curves obtained in function of Ca, for λ = 20.
y = 0.0051x + 1.3504
R² = 0.9444
y = 0.0075x + 1.2511
R² = 0.9993
y = 0.0132x + 1.1397
R² = 0.9964
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0 2 4 6 8 10 12 14 16
ΔP/P*(x*≈0.59)
1/Ca
λ=30
λ=20
λ=10
Figure 14: Pump pressure for x*= 0.59 in function of Ca−1
for different λ values.
From Fig. 15, higher viscosity ratios reduce the drop deformation. In fact, the higher is
the viscosity ratio, the higher is the droplet deformation resistance. Note that, while capillary
forces tend to make the drop round, the viscosity ratios act on your deformation resistance.
The Fig. 16 clearly shows the influence of viscosity ratio over ∆P/P∗
curves. Higher val-
ues for λ imply higher values for pump pressure. All curves present a similar shape. Figure 17
presents the obtained values to ∆P/P∗
for the relative drop position x*= 0.59 for different λ.
This plot shows that the relation between ∆P/P∗
and λ is basically linear and positive.
4.5 Initial drop diameter effects
Finally, the initial drop diameter, a, effects in the flow was studied. Figures 18 to 20
show the drop form for different sizes through the channel. For all cases, λ = 10 and Ca =
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
14. Flow of an emulsion drop through a converging channel using the Boundary Element Method
Figure 15: Drop shapes for the same instant. From the left to the right, λ = 10, 15, 20, 30 and 40, respectively.
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
0.2 0.4 0.6 0.8 1
ΔP/P*
x*
λ = 40
λ = 30
λ = 20
λ = 15
λ = 10
Figure 16: Pump pressure in function of λ, for Ca = 0.25.
y = 0.0082x + 1.1123
R² = 0.9974
1.15
1.2
1.25
1.3
1.35
1.4
1.45
10 15 20 25 30 35 40
ΔP/P*(x*≈0.59)
λ
Figure 17: Pump pressure for x*= 0.59 in function of λ, for Ca = 0.25.
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15. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
0.25. Bigger drops need to deform more to pass through the channel throat, confirming the
high extensional flow character close to the constriction. Consequently, the pump pressure is
increased, as it is shown in Fig. 21. Figure 22 brings the maximum pressure obtained in the
flow in function of the drop initial diameter.
Figure 18: Drop shapes through the channel, for a = 0.70.
Figure 19: Drop shapes through the channel, for a = 0.50.
Figure 20: Drop shapes through the channel, for a = 0.30.
Figure 21 confirms the increase for the pump pressure as bigger the drop initial diameter
is. This phenomenon is consequence of the higher drop strain rate when it is crossing the
constriction, increased with the drop diameter. In Fig. 20, the last moment exposed for this
drop size showed an instability, which did not happen to the others two cases. Smaller radii
are associated to intense curvatures, and how it was said before, curvature is the parameter
which creates more problems for the method. Finally, Fig. 22 shows a relation, approximately
quadratic, between the maximum ∆P/P∗
and a.
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
16. Flow of an emulsion drop through a converging channel using the Boundary Element Method
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0.2 0.4 0.6 0.8 1
ΔP/P*
x*
a=0.70
a=0.60
a=0.50
a=0.40
a=0.30
Figure 21: Pump pressure in function of the drop initial diameter a, for Ca = 0.25 and λ = 10.
y = 1.0835x2 - 0.4276x + 1.1681
R² = 0.9988
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0.3 0.4 0.5 0.6 0.7
ΔP/P*max
a
Figure 22: Maximum pump pressure in function of a, for Ca = 0.25 and λ = 10.
5 CONCLUSIONS
The Boundary Element Method was used to study the effect of physical and geometrical
parameters on the flow of an emulsion drop through a converging channel. First, the conver-
gence and accuracy of the method were studied in relation to the mesh and time step refinement.
The two parameters used for that error study were the numerical error for the outlet flow rate
and drop area, both considering the mass conservation and the fluids incompressibility. By these
results, the parameter for the number of elements chose was Ne = 7, which presents an error
around 1.5% for the outlet flow rate, and a time step ∆t = 0.01, presenting smaller errors than
0.5% for the drop area. After the determination of these two simulation parameters, the study of
the capillary number (Ca), viscosity ratio (λ) and drop initial diameter (a) effects were carried
out. In relation to the drop shapes, smaller Ca makes the drop rounder, while bigger λ makes
CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016
17. Cunha L.H.P., Albuquerque E.L., Siqueira I.R.
the drop more resistant to deformations. As bigger it is a, more deformation is suffered by the
drop to enter the constriction. The pump pressure increases for smaller Ca, bigger λ and bigger
a. The relation between Ca−1
and ∆P/P∗
is basically linear and positive. The same happens
between λ and ∆P/P∗
. While, between a and ∆P/P∗
is still positive, but quadratic.
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CILAMCE 2016
Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering
Suzana Moreira ´Avila (Editor), ABMEC, Bras´ılia, DF, Brazil, November 6-9, 2016