Log in

Relevant bibliographies by topics / Hele-Shaw flow; Stokes flow

Contents

Journal articles
Dissertations / Theses
Books
Book chapters
Conference papers
Reports

Academic literature on the topic 'Hele-Shaw flow; Stokes flow'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the lists of relevant articles, books, theses, conference reports, and other scholarly sources on the topic 'Hele-Shaw flow; Stokes flow.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Journal articles on the topic "Hele-Shaw flow; Stokes flow"

1

RICHARDSON, S. "Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region." European Journal of Applied Mathematics 11, no. 3 (June 2000): 249–69. http://dx.doi.org/10.1017/s0956792500004149.

Full text

Abstract:

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

APA, Harvard, Vancouver, ISO, and other styles

2

Heussler, F. H. C., R. M. Oliveira, M. O. John, and E. Meiburg. "Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements." Journal of Fluid Mechanics 752 (July 2, 2014): 157–83. http://dx.doi.org/10.1017/jfm.2014.327.

Full text

Abstract:

AbstractGravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier–Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts, on the other hand, can move more slowly than the Poiseuille flow, resulting in more complex streamline patterns and the formation of a spike at the tip of the front, in line with earlier findings. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations of viscously and gravitationally unstable vertical displacements show a strong vorticity quadrupole along the length of the finger, similar to recent observations for neutrally buoyant flows. This quadrupole results in an inner splitting instability of vertically propagating fingers. Even though the quadrupole’s strength increases for larger destabilizing density differences, the inner splitting is delayed due to the presence of a secondary, outer quadrupole which counteracts the inner one. For large unstable density differences, the formation of a secondary, downward-propagating front is observed, which is also characterized by inner and outer vorticity quadrupoles. This front develops an anchor-like shape as a result of the flow induced by these quadrupoles. Increased spanwise wavelengths of the initial perturbation are seen to result in the formation of the well-known tip-splitting instability. For suitable initial conditions, the inner and tip-splitting instabilities can be seen to develop side by side, affecting different regions of the flow field.

APA, Harvard, Vancouver, ISO, and other styles

3

Pozrikidis, C. "The motion of particles in the Hele-Shaw cell." Journal of Fluid Mechanics 261 (February 25, 1994): 199–222. http://dx.doi.org/10.1017/s0022112094000315.

Full text

Abstract:

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

APA, Harvard, Vancouver, ISO, and other styles

4

RICHARDSON, S. "Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension." European Journal of Applied Mathematics 8, no. 4 (August 1997): 311–29. http://dx.doi.org/10.1017/s0956792597003057.

Full text

Abstract:

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are such flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are N such disks in a general position, the evolution of the fluid region is described by a conformal map involving 2N−1 time-dependent parameters whose variation is governed by N invariants and N−1 first order differential equations. When N=2, or when the problem enjoys some special features of symmetry, the moving boundary of the fluid domain during the motion can be determined by solving purely algebraic equations, the solution of a single differential equation being needed only to link a particular boundary shape to a particular time. The analysis is aided by exploiting a connection with Hele-Shaw free boundary flows when the zero-surface-tension model is employed. If the initial configuration for the Stokes flow problem can be produced by injection (or suction) at N points into an initially empty Hele-Shaw cell, as can the N-disk configuration referred to above, then so can all later configurations; the points where the fluid must be injected move, but the amount to be injected at each of the N points remains invariant. The efficacy of our solution procedure is illustrated by a number of examples, and we exploit the method to show that the free boundary in such a Stokes flow driven by surface tension alone may pass through a cusped state.

APA, Harvard, Vancouver, ISO, and other styles

5

Oliveira, Rafael M., and Eckart Meiburg. "Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations." Journal of Fluid Mechanics 687 (October 12, 2011): 431–60. http://dx.doi.org/10.1017/jfm.2011.367.

Full text

Abstract:

AbstractThree-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Péclet numbers.

APA, Harvard, Vancouver, ISO, and other styles

6

Lee, S. Y., R. Teodorescu, and P. Wiegmann. "Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent." Physica D: Nonlinear Phenomena 240, no. 13 (June 2011): 1080–91. http://dx.doi.org/10.1016/j.physd.2010.09.017.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

LIN, Y. L. "Large-time rescaling behaviours of Stokes and Hele-Shaw flows driven by injection." European Journal of Applied Mathematics 22, no. 1 (December 2, 2010): 7–19. http://dx.doi.org/10.1017/s0956792510000264.

Full text

Abstract:

In this paper, we give a precise description of the rescaling behaviours of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behaviour is given in terms of many conserved quantities as well.

APA, Harvard, Vancouver, ISO, and other styles

8

Demidov, A. S., J. P. Lohéac, and V. Runge. "Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours." Russian Journal of Mathematical Physics 23, no. 1 (January 2016): 35–55. http://dx.doi.org/10.1134/s1061920816010039.

Full text

APA, Harvard, Vancouver, ISO, and other styles

9

Llamoza, Johan, and Desiderio A. Vasquez. "Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities." Mathematical and Computational Applications 24, no. 2 (May 17, 2019): 51. http://dx.doi.org/10.3390/mca24020051.

Full text

Abstract:

Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stability analysis shows a transition to convection that depends on the density differences between reacted and unreacted fluids. A stabilizing density gradient can surpress the effects of diffusive instabilities. The two-dimensional numerical solutions of the nonlinear equations show an increase of speed due to convection. Brinkman’s equation lead to the same results as Darcy’s laws for narrow gap Hele–Shaw cells. For large gaps, modeling the hydrodynamics using Stokes’ flow lead to the same results.

APA, Harvard, Vancouver, ISO, and other styles

10

Zhu, Lailai, and François Gallaire. "A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field." Journal of Fluid Mechanics 798 (June 15, 2016): 955–69. http://dx.doi.org/10.1017/jfm.2016.357.

Full text

Abstract:

We adopt a boundary integral method to study the dynamics of a translating droplet confined in a Hele-Shaw cell in the Stokes regime. The droplet is driven by the motion of the ambient fluid with the same viscosity. We characterize the three-dimensional (3D) nature of the droplet interface and of the flow field. The interface develops an arc-shaped ridge near the rear-half rim with a protrusion in the rear and a laterally symmetric pair of higher peaks; this pair of protrusions has been identified by recent experiments (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501) and predicted asymptotically (Burgess & Foster, Phys. Fluids A, vol. 2 (7), 1990, pp. 1105–1117). The mean film thickness is well predicted by the extended Bretherton model (Klaseboer et al., Phys. Fluids, vol. 26 (3), 2014, 032107) with fitting parameters. The flow in the streamwise wall-normal middle plane is featured with recirculating zones, which are partitioned by stagnation points closely resembling those of a two-dimensional droplet in a channel. Recirculation is absent in the wall-parallel, unconfined planes, in sharp contrast to the interior flow inside a moving droplet in free space. The preferred orientation of the recirculation results from the anisotropic confinement of the Hele-Shaw cell. On these planes, we identify a dipolar disturbance flow field induced by the travelling droplet and its $1/r^{2}$ spatial decay is confirmed numerically. We pinpoint counter-rotating streamwise vortex structures near the lateral interface of the droplet, further highlighting the complex 3D flow pattern.

APA, Harvard, Vancouver, ISO, and other styles

More sources

Dissertations / Theses on the topic "Hele-Shaw flow; Stokes flow"

1

Cummings, Linda Jane. "Free boundary models in viscous flow." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.339364.

Full text

APA, Harvard, Vancouver, ISO, and other styles

2

Brown, Sarah Marie. "A Numerical Scheme for Mullins-Sekerka Flow in Three Space Dimensions." BYU ScholarsArchive, 2004. https://scholarsarchive.byu.edu/etd/136.

Full text

Abstract:

The Mullins-Sekerka problem, also called two-sided Hele-Shaw flow, arises in modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while smaller particles eventually dissolve. Single particles become spherical. This process is described by evolving harmonic functions within the two phases with the moving interface driven by the jump in the normal derivatives of the harmonic functions at the interface. The harmonic functions are continuous across the interface, taking on values equal to the mean curvature of the interface. This dissertation reformulates the three-dimensional problem as one on the two-dimensional interface by using boundary integrals. A semi-implicit scheme to solve the free boundary problem numerically is implemented. Numerical analysis tasks include discretizing surfaces, overcoming node bunching, and dealing with topology change in a toroidal particle. A particle (node)-cluster technique is developed with the aim of alleviating excessive run time caused by filling the dense matrix used in solving a system of linear equations.

APA, Harvard, Vancouver, ISO, and other styles

3

Azzopardi, Charles-Louis. "Dispositifs microfluidiques pour l’injection de fluides à travers un réseau de gouttes : application biocapteur." Thesis, Bourgogne Franche-Comté, 2018. http://www.theses.fr/2018UBFCD020/document.

Full text

Abstract:

La microfluidique, domaine de recherche qui a émergé il y a juste 20 ans, a permis de réduire les dimensions des dispositifs d’analyse biologique ouvrant la porte au concept de « laboratoire-sur-puce » (lab-on-chip). Les succès de cette approche sont déjà nombreux, depuis l’analyse du génome en passant par la réduction du coût des analyses médicales. L’utilisation de gouttes comme enceinte réactionnelle au sein de ces dispositifs est une évolution récente qui permet de réduire encore le volume des échantillons biologiques et d’augmenter la vitesse d’analyse en parallélisant les mesures.Notre équipe développe des capteurs acoustiques dédiés à la détection d’analytes biologiques en milieu liquide. Ce type de capteur a pour principal défaut de ne permettre qu’une mesure contraignant au remplacement de l’interface de biodétection pour une réutilisation éventuelle du capteur. Dès lors, ils utilisent majoritairement une chambre de détection tout ou partie jetable, même si quelques travaux de recherche ont pu montrer la régénération d’un capteur par traitement chimique.Nous proposons ici de s’affranchir des étapes lourdes de remplacement ou de traitement de l’interface de détection qui conduisent entre chaque mesure au démontage du dispositif de détection. Nous employons dans ce cas les gouttes non plus comme enceinte réactionnelle mais comme interface de détection mobile. Elles ont ainsi le potentiel d’être générées et fonctionnalisées directement dans le dispositif pour détecter un analyte spécifique et peuvent être simplement évacuées afin de régénérer l’interface pour effectuer une mesure différente.Les travaux présentés dans cette thèse visent plus particulièrement la capture sur gouttes fonctionnalisées dans ce type de capteur innovant. Ils exposent le développement, incluant la fabrication et la caractérisation, de ces dispositifs microfluidiques ainsi que le montage d’un banc de test expérimental dédié. Ce sujet est suivi de deux projets ancillaires de développement de dispositifs microfluidiques liés aux capteurs acoustiques et à l’utilisation de gouttes. Le premier vise à homogénéiser les vitesses d’écoulements dans une chambre réactionnelle. Le second exploite les propriétés de génération de gouttes pour réaliser un condensateur à capacité variable
Since two decades the research on microfluidics systems allowed creating devices for biological detection with regular improvement in compactness, functionality integration and quantity of biological sample, leading to the concept of lab-on-chip. This approach has resulted in dramatic changes in the biomedical field, for example, opening the possibility to perform genomic analysis or improving the medical analysis cost. Using droplet as reaction chamber is a recent evolution that leads to a decrease in biological sample volume and an increase in analysis speed by multiplexing.Our team develops acoustical sensors dedicated to detect biomarker of interest in liquids. The principal weakness of theses sensors lies in their need for replacement of the biodetection interface for performing a new measurement. Accordingly, they use a detection chamber partially or totally disposable. However, few research works showed reusability of sensor by regenerating the bioreceptor layer on the detection interface by chemical treatment.We are proposing to avoid the replacement or the chemical treatment of the detection interface that requires dismounting the device between measurements. We are using here droplets, not as reaction chambers but as movable detection interface. They can be generated and configured directly inside the device to detect a specific biomarker. Then, droplets can be easily evacuated and replaced through the device, which allows to chain measurement of various configurations without dismounting it.The research work conducted in this thesis focuses on the fluidic aspects of this innovative sensor. They show development, including realization and characterization, of theses microfluidic devices and its dedicated characterization setup. This project is followed by two ancillary works about development of microfluidic devices for acoustical sensors and droplets systems. The first one is aiming at the homogenization of the flow velocity inside a reaction chamber. The second one is exploiting property of droplet generation for the realization of a variable capacitance capacitor

APA, Harvard, Vancouver, ISO, and other styles

4

Maes, RENAUD POL. "Etude expérimentale de la digitation visqueuse de fluides miscibles en cellule de Hele-Shaw." Doctoral thesis, Universite Libre de Bruxelles, 2010. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/216585.

Full text

Abstract:

La digitation visqueuse est une instabilité hydrodynamique apparaissant lorsque, dans un milieu poreux, un fluide moins visqueux déplace un fluide plus visqueux. L'objectif de notre thèse est l'étude expérimentale des propriétés des motifs de digitation lorsque l'échantillon de fluide visqueux est de taille finie et lorsqu'une réaction chimique modifie la viscosité dans un milieu poreux modèle, en l'occurrence une cellule de Hele-Shaw. En particulier, notre étude a permis de quantifier la contribution de dispersion et de la digitation visqueuse, l'étalement dans l'espace d'échantillons de taille finie en fonction des paramètres expérimentaux (contraste de viscosité, vitesse de déplacement et taille de l'échantillon). Pour les fluides réactifs, nous analysons la digitation induite par une réaction A + B C dont le produit C est plus visqueux que les réactifs A et B, ceux-ci ayant la même viscosité. Nous mettons en évidence l'effet des concentrations en réactifs, du choix du fluide vecteur et du débit d'injection sur le motif de digitation.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

APA, Harvard, Vancouver, ISO, and other styles

5

ANJOS, Pedro Henrique Amorim. "Efeitos de molhamento nas instabilidades hidrodinâmicas e em forças adesivas em fluidos confinados." Universidade Federal de Pernambuco, 2015. https://repositorio.ufpe.br/handle/123456789/18074.

Full text

Abstract:

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-12-12T14:44:32Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação - Pedro Henrique Amorim Anjos.pdf: 9970150 bytes, checksum: 29bca81f576fbdecd909c0ab24d68ce5 (MD5)
Made available in DSpace on 2016-12-12T14:44:32Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação - Pedro Henrique Amorim Anjos.pdf: 9970150 bytes, checksum: 29bca81f576fbdecd909c0ab24d68ce5 (MD5) Previous issue date: 2015-02-05
CNPQ
Consideramos o movimento da interface entre dois fluidos viscosos, imiscíveis e incompressíveis confinados em uma célula de Hele-Shaw radial. Quando o fluido de baixa viscosidade é injetado no centro da célula e desloca radialmente o fluido de alta viscosidade, a instabilidade de Saffman-Taylor entra em ação levando ao crescimento de estruturas em formato de dedos. Dependendo da natureza dos fluidos envolvidos, eles podem molhar as paredes da célula de Hele-Shaw, deixando para trás um filme molhante de espessura finita. Diante disso, investigamos a influência da camada de fluido molhante, deixada pelo fluido deslocado, na dinâmica linear e fracamente não-linear do sistema. Mais especificamente, examinamos como a instabilidade da interface e os mecanismos de formação de padrões (bifurcação e competição de dedos) são afetados pela presença da camada de filme molhante no limite de baixo número de capilaridade. Estudamos também uma variação da instabilidade de Saffman-Taylor usual induzida pela injeção: o problema do levantamento em célula de Hele-Shaw. Nossos resultados analíticos indicam que o molhamento tem um impacto significativo nos padrões não-lineares resultantes. Ele é responsável por uma atenuação na variação do comprimento dos dedos, enquanto que induz o aparecimento de estruturas que apresentam dedos pequenos e largos, provenientes do fluido não molhante, alternados por dedos pequenos e finos pertencentes ao fluido molhante. Durante o processo de levantamento, medidas adquiridas pelo teste de adesão quantificam a força adesiva de um fluido viscoso confinado entre placas paralelas. Experimentos e intensas simulações numéricas indicam que a formação de dedos afeta a força de levantamento, promovendo uma diminuição na magnitude da mesma. Concluímos este trabalho propondo um modelo analítico que forneça a força de adesão considerando não só o efeito de dedos da interface, mas também a ação do molhamento e de estresses viscosos normais.
We consider the interfacial motion between two immiscible, incompressible, viscous fluids in the confined geometry of a radial Hele-Shaw cell. When the less viscous fluid is injected at the center and drives radially the more viscous fluid, the Saffman-Taylor instability takes place leading to the growth of fingerlike shapes. Depending on the nature of the fluids involved, they can wet the walls of the Hele-Shaw cell plates, leaving behind a film of finite thickness. In this framework, we investigate the influence of a thin wetting film trailing behind the displaced fluid on the linear and weakly nonlinear dynamics of the system. More specifically, we examine how the interface instability and the pattern formation mechanisms of finger tip-splitting and finger competition are affected by the presence of such a film in the low capillary number limit. We also examined a variant of the usual injection-driven Saffman-Taylor instability: the lifting Hele-Shaw cell problem. Our analytical results indicate that wettability has a significant impact on the resulting nonlinear patterns. It restrains finger length variability while inducing the development of structures presenting short, blunt penetrating fingers of the nonwetting fluid, alternated by short, sharp fingers of the wetting fluid. During the lifting process, probe-tack measurements evaluate the adhesion strength of viscous fluids confined between parallel plates. Existing meticulous experiments and intensive numerical simulations indicate that fingering formation affects the lifting force, making it to decrease in intensity. We conclude this work by proposing an analytical model that computes the lifting adhesion force by taking into account not only the effect of interfacial fingering, but also the action of wetting, and viscous normal stresses.

APA, Harvard, Vancouver, ISO, and other styles

6

Kreyenberg, Philipp Johannes [Verfasser], and Kurt [Akademischer Betreuer] Roth. "Flow Field Estimation of Active Solute Transport – Information Transfer from Synthetic Data to Hele-Shaw Cell Experiments Using Convolutional Neural Networks / Philipp Johannes Kreyenberg ; Betreuer: Kurt Roth." Heidelberg : Universitätsbibliothek Heidelberg, 2019. http://d-nb.info/1200021592/34.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

Beauvier, Edouard. "Propagation d'un front de réaction-diffusion dans un écoulement cellulaire multi-échelle." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4728/document.

Full text

Abstract:

La propagation d'un front de réaction-diffusion est étudiée expérimentalement dans un écoulement cellulaire multiéchelle. Le front est produit par réaction autocatalytique en solution. L'écoulement est réalisé en géométrie de Hele-Shaw par électroconvection, son caractère multiéchelle étant réalisé par l'action combinée de deux nappes d'aimants d'échelles différentes. La géométrie du front et sa vitesse moyenne de propagation sont déterminées pour une large gamme d'intensité des vortex de chaque échelle. Elles sont confortées par une simulation numérique de l'avancée du domaine brulé dans le domaine frais. L'effet de la nature multiéchelle de l'écoulement sur la vitesse moyenne du front est compris par une méthode de renormalisation dont la validation est fournie par l'obtention d'un courbe maitresse pour l'ensemble des données
The propagation of a reaction-diffusion front is experimentally studied in a multi-scale cellular flow. The front is produced by an autocatalytic chemical reaction in an aqueous solution. The flow is generated by electroconvection and its multi-scale nature is induced by overlaying magnets of different scales. This enables an independent tune of the flow intensity at each scale. The geometry and the mean velocity of the front have been determined over a large range of scale intensities. These features are confirmed by a numerical simulation based on a burnt and fresh domain dynamics, the burnt domain expanding across the fresh one. The effect of the multi-scale nature of the flow on the mean front velocity is recovered by a renormalisation method validated by a collapse of the data onto a single curve

APA, Harvard, Vancouver, ISO, and other styles

8

Zhang, Fengshou. "Pattern formation in fluid injection into dense granular media." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/43716.

Full text

Abstract:

Integrated theoretical and experimental analysis is carried out in this work to investigate the fundamental failure mechanisms and flow patterns involved in the process of fluid injection into dense granular media. The experimental work is conducted with aqueous glycerin solutions, utilizing a novel setup based on a Hele-Shaw cell filled with dense dry sand. The two dimensional nature of the setup allows direct visualization and imaging analysis of the real-time fluid and grain kinematics. The experimental results reveal that the fluid flow patterns show a transition from simple radial flow to a ramified morphology while the granular media behaviors change from that of rigid porous media to localized failure that lead to development of fluid channels. Based on the failure/flow patterns, four distinct failure/flow regimes can be identified, namely, (i) a simple radial flow regime, (ii) an infiltration-dominated regime, (iii) a grain displacement-dominated regime, and (iv) a viscous fingering-dominated regime. These distinct failure/flow regimes emerge as a result of competition among various energy dissipation mechanisms, namely, viscous dissipation through infiltration, dissipation due to grain displacements, and viscous dissipation through flow in thin channels and can be classified based on the characteristic times associated with fluid injection, hydromechanical coupling and viscoelastoplasticity. The injection process is also analyzed numerically using the discrete element method (DEM) coupled with two fluid flow scheme, a fixed coarse grid scheme based on computational fluid dynamics (CFD) and a pore network modeling scheme. The numerical results from the two complementary methods reproduce phenomena consistent with the experimental observations and justify the concept of associating the displacement regimes with the partition among energy dissipation mechanisms. The research in this work, though fundamental in nature, will have direct impacts on many engineering problems in civil, environmental and petroleum engineering such as ground improvement, environmental remediation and reservoir stimulation.

APA, Harvard, Vancouver, ISO, and other styles

9

Mostefai, Mohamed Sadek. "Déduction rigoureuse de l'équation de Reynolds à partir d'un système modélisant l'écoulement à faible épaisseur d'un fluide micropolaire, et étude de deux problèmes à frontière libre : Hele-Shaw généralisé et Stephan à deux phases pour un fluide non newtonien." Saint-Etienne, 1997. http://www.theses.fr/1997STET4019.

Full text

Abstract:

Dans le chapitre 1, on considère le modèle micropolaire de Navier-Stokes avec conditions de bords de type Dirichlet non homogènes en dimension deux. On donnera un résultat d'existence d'une solution faible en utilisant le théorème du point fixe de Leray-Schauder, puis on prouvera l'unicité de la solution faible du problème sous certaines hypothèses. On établiera une justification mathématique de l’équation de Reynolds généralisé à partir de ce modèle là. On étudiera ensuite la forme de l'équation de Reynolds suivant le choix de la viscosité et des données initiales. Dans le chapitre 2, nous considérons le modèle de Hele-Shaw généralisé dans une cellule laminaire, qui consiste à injecter du fluide, avec un débit non constant w 0, à travers un trou de frontière 1, situé sur l'une des deux surfaces ; et à tenir compte que l'une des surfaces a une géométrie quelconque et animée d'un mouvement relatif vertical. En introduisant un changement de variable de type Baiocchi, le problème initial se ramène à l'étude d'une inéquation variationnelle avec terme de Volterra. L'existence d'une solution pour cette dernière est donnée par le théorème du point fixe de Banach. Des résultats de régularité en espace pour la solution seront prouvés en introduisant un problème pénalisé et en utilisant la méthode de Rothe (semi-discrétisation en temps), puis on montrera que la dérivée par rapport à t de la solution de l'inéquation variationnelle est dans l#(0, t, h#2()), ce dernier résultat nous permet de revenir au problème initial. Dans le chapitre 3, on considère un problème de Stefan à deux phases avec convection. Le problème est gouverné par un système couple non linéaire, comprenant la loi de Darcy pour un fluide non newtonien et l'équation d'équilibre d'énergie avec second membre dans l#1. Pour prouver l'existence de solutions du problème faible on introduira une famille de solutions approchées (#, p#), > 0, définies sur le domaine entier , en insérant une fonction de pénalité convenable dans l'équation de pression. On considère ensuite séparement les problèmes en # et p#, respectivement, et en utilisant le principe de point fixe de Schauder, on montre l'existence de couples solutions (#, p#) du problème approché, pour tout > 0. En faisant tendre vers zéro, on montre que les solutions du problème approché convergent vers une limite (, p) qui est une solution faible du problème variationnel. On montre aussi que la fonction est continue d'où le domaine où > 0 est un ensemble ouvert, et l'interface des deux phases est définie a posteriori comme l'ensemble de niveau = 0. On établira, enfin, quelques relations entre les solutions faibles et classiques, dans le cas d’une courbe assez régulière

APA, Harvard, Vancouver, ISO, and other styles

10

Jonsson, Karl. "Two Problems in non-linear PDE’s with Phase Transitions." Licentiate thesis, KTH, Matematik (Avd.), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-223562.

Full text

Abstract:

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem.

QC 20180222

APA, Harvard, Vancouver, ISO, and other styles

More sources

Books on the topic "Hele-Shaw flow; Stokes flow"

1

Glimm, James. The laminar to slug flow transition in Hele-Shaw flow. New York: Courant Institute of Mathematical Sciences, New York University, 1989.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

2

Tanveer, Saleh. Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell. Part II: Stability. Hampton, Va: ICASE, 1989.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

3

Baker, Gregory. A well-posed numerical method to track isolated conformal map singularities in Hele-Shaw flow. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1995.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

4

Baker, Gregory. A well-posed numerical method to track isolated conformal map singularities in Hele-Shaw flow. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1995.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

5

Tanveer, Saleh. Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell. Part I: Velocity selection. Hampton, Va: ICASE, 1989.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

6

Interfacial wave theory of pattern formation: Selection of dendritic growth and viscous fingering in Hele-Shaw flow. Berlin: Springer, 1998.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

7

Institute for Computer Applications in Science and Engineering., ed. Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1989.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

8

Interfacial Wave Theory of Pattern Formation: Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow. Springer, 2011.

Find full text

APA, Harvard, Vancouver, ISO, and other styles

Book chapters on the topic "Hele-Shaw flow; Stokes flow"

1

Maher, J. V. "Fingering Patterns in Hele-Shaw Flow." In Springer Proceedings in Physics, 252–57. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/978-3-642-93301-1_32.

Full text

APA, Harvard, Vancouver, ISO, and other styles

2

Shelley, Michael J., Raymond E. Goldstein, and Adriana I. Pesci. "Topological Transitions in Hele-Shaw Flow." In Singularities in Fluids, Plasmas and Optics, 167–88. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2022-7_13.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

Bernal, Francisco, and Manuel Kindelan. "Meshless Simulation of Hele-Shaw Flow." In Progress in Industrial Mathematics at ECMI 2006, 677–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-71992-2_113.

Full text

APA, Harvard, Vancouver, ISO, and other styles

4

Schwartz, Leonard W. "Recent Developments in Hele-Shaw Flow Modeling." In Numerical Simulation in Oil Recovery, 249–61. New York, NY: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4684-6352-1_16.

Full text

APA, Harvard, Vancouver, ISO, and other styles

5

Meyer, Gunter H. "Front Tracking for the Unstable Hele-Shaw and Muskat Problems." In Flow in Porous Media, 129–37. Basel: Birkhäuser Basel, 1993. http://dx.doi.org/10.1007/978-3-0348-8564-5_12.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Chen, Ching-Yao, Shu-Wei Wang, and Yu-Chia Liu. "Fingering Instabilities in a Miscible Rotating Hele-Shaw Flow." In Computational Fluid Dynamics 2002, 473–78. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-642-59334-5_70.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

Sharma, Mayank, Shikhar Jaiswal, Sumit Singh Rajput, Shreya Tripathi, Subiksha C, and G. Vinayagamurthy. "Flow Visualization of Two-Dimensional Bodies Using Hele-Shaw Apparatus." In Lecture Notes in Mechanical Engineering, 841–52. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0698-4_92.

Full text

APA, Harvard, Vancouver, ISO, and other styles

8

Parlange, J. Yves, Tammo S. Steenhuis, Ling Li, D. A. Barry, and Frank Stagnitti. "Column flow in stratified soils and fingers in Hele-Shaw cells: A review." In Environmental Mechanics: Water, Mass and Energy Transfer in the Biosphere, 79–85. Washington, D. C.: American Geophysical Union, 2002. http://dx.doi.org/10.1029/129gm08.

Full text

APA, Harvard, Vancouver, ISO, and other styles

9

Boulais, Étienne, and Thomas Gervais. "Hele-Shaw Flow Theory in the Context of Open Microfluidics: From Dipoles to Quadrupoles." In Open-Space Microfluidics: Concepts, Implementations, Applications, 63–82. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA, 2018. http://dx.doi.org/10.1002/9783527696789.ch4.

Full text

APA, Harvard, Vancouver, ISO, and other styles

10

Fowler, Andrew. "The Scientific Legacy of George Gabriel Stokes." In George Gabriel Stokes, 197–216. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198822868.003.0011.

Full text

Abstract:

The scientific legacy of George Gabriel Stokes is considered. Certain aspects of Stokes’s research work are reviewed and related to more recent fields of research. These include the Navier–Stokes equations and other approaches to rational continuum mechanics, the issue of existence of solutions, the boundary no-slip condition; Stokes flow and the issue of pendulum drag; the Hele-Shaw cell, viscous fingering, wavelength selection in pattern formation; moving contact lines; the highest water wave, rogue waves, the NLS equation; Stokes lines, exponential asymptotics, dendrite growth, slow manifods, and diffraction.

APA, Harvard, Vancouver, ISO, and other styles

Conference papers on the topic "Hele-Shaw flow; Stokes flow"

1

Costalonga, Maxime, Hassan Peerhossaini, and Philippe Brunet. "Low Frequency Acoustic Streaming in a Hele-Shaw Cell." In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting collocated with the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/fedsm2014-21384.

Full text

Abstract:

When an acoustic wave propagates in a fluid, it can generate a second order flow which characteristic time is much longer than the period of the wave. Within a range of frequency between 1 and several hundred Hz, a relatively simple and versatile way to generate streaming flow is to put a vibrating object in the fluid. The flow develops vortices in the viscous boundary layer located in the vicinity of the source of vibrations, which in turns leads to an outer irrotational streaming denoted as Rayleigh streaming. Due to that the flow originates from non-linear time-irreversible terms of the Navier-Stokes equation, this phenomenon can be used to move fluids and even to generate efficient mixing at low Reynolds number, for instances in confined geometries. Here we report an experimental study of such streaming flow in a Hele-Shaw cell of 2 millimeters span using long exposure flow visualization and PIV measurements. Our study is especially focused on the effects of acoustic frequency and amplitude on flow dynamics. It is shown that some features of this flow can be predicted by simple scaling arguments, invoking a balance between viscous dissipation in the boundary layer and inertia term, and that acoustic streaming facilitates the generation of vortices.

APA, Harvard, Vancouver, ISO, and other styles

2

Gier, Jochen, Sabine Ardey, and Adam Heisler. "Analysis of Complex Three-Dimensional Flow in a Three-Stage LP Turbine by Means of Transitional Navier-Stokes Simulation." In ASME Turbo Expo 2000: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/2000-gt-0645.

Full text

Abstract:

The complex three-dimensional flow field in a highly loaded three-stage LPT is analysed on the basis of a steady three-dimensional flow simulation. The quality of the simulation concerning this configuration is demonstrated by means of a comparison with extensive experimental data gathered in a turbine test rig. For an accurate representation of the transitional character of the turbine flow a modified version of the Abu-Ghannam Shaw transition model is employed in the TRACE_S Navier-Stokes code in connection with a two-equation turbulence model. The flow field of this highly loaded turbine is characterised by complex secondary flow pattern as well as local separation and reattachment zones. The need and applicability of transition modelling is demonstrated by a comparison with a fully turbulent calculation and experimental flow visualisation. The basic flow structure is described in terms of several characteristic quantities and discussed in detail. For further analysis variations of the point of operation and the geometry also based on experiments are included in this investigation.

APA, Harvard, Vancouver, ISO, and other styles

3

Corvera, Eugenia, Hong Guo, and David Jasnow. "Hele-Shaw flow with anisotropic surface tension." In CAM-94 Physics meeting. AIP, 1995. http://dx.doi.org/10.1063/1.48765.

Full text

APA, Harvard, Vancouver, ISO, and other styles

4

Petkovic-Duran, K., and Y. Zhu. "Capillary flow in microfluidic Hele-Shaw cells." In Smart Nano-Micro Materials and Devices, edited by Saulius Juodkazis and Min Gu. SPIE, 2011. http://dx.doi.org/10.1117/12.903213.

Full text

APA, Harvard, Vancouver, ISO, and other styles

5

Yamamoto, Takehiro. "Finger Growth in Surfactant Solution in Hele-Shaw Cells." In FLOW DYNAMICS: The Second International Conference on Flow Dynamics. AIP, 2006. http://dx.doi.org/10.1063/1.2204505.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Boronin, S., A. Osiptsov, and J. Desroches. "Flows of particle-laden Bingham fluids in a Hele-Shaw cell." In MULTIPHASE FLOW 2013. Southampton, UK: WIT Press, 2013. http://dx.doi.org/10.2495/mpf130121.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

Reinker, Felix, Marek Kapitz, and Stefan aus der Wiesche. "Boiling Under Hele-Shaw Flow Conditions: The Occurrence of Viscous Fingering." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-62886.

Full text

Abstract:

Boiling and bubble dynamics were experimentally investigated in a Hele-Shaw flow cell using pure water at atmospheric pressure as working fluid. The resulting vapor bubble shapes were recorded by means of a high-speed camera for several plate spacings and heating power levels. It was found that viscous fingering phenomena of vapor bubbles occurred only under very special boiling conditions and cell parameters. The evaporation front velocity was identified as a major parameter for the onset of viscous fingering. The observed basic viscous fingering dynamics was in reasonable agreement with theoretical analyses. In addition to that classical viscous large fingering, small-scale evaporation instability was observed leading to microscopic roughening of accelerating evaporation fronts. This instability might be explicitly related to evaporative heat and mass transfer effects across the fast-moving phase interface.

APA, Harvard, Vancouver, ISO, and other styles

8

Ngo, C. C., and C. G. Peinder. "Flow Simulation of Radiant Floor Heating System Using Hele-Shaw Analogy." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-38733.

Full text

Abstract:

Radiant floor heating systems are becoming increasingly popular in green building designs. Typically, solar or geothermal energy is employed as a source for such hydronic heating systems. Buried heating pipe system can be used for heating both residential and industrial spaces as well as defrosting snow on walkways, driveways and sport fields. Most of the heating pipes considered in such applications are buried in a porous medium (i.e., insulation layer or soils). Hele-Shaw cells with different pipe spacing were constructed to simulate different floor heating configurations. The objective of the present experimental study is to examine the flow field within porous medium using the Hele-Shaw analogy. The flow visualization experiment was set up to investigate how a change in pipe spacing and pipe temperature would affect the flow patterns from the heated pipes. Using time-elapsed photographs, one observes that the flow fields for different pipe spacings with different buoyancy strengths display distinct characteristics.

APA, Harvard, Vancouver, ISO, and other styles

9

Dénos, R., T. Arts, G. Paniagua, V. Michelassi, and F. Martelli. "Investigation of the Unsteady Rotor Aerodynamics in a Transonic Turbine Stage." In ASME Turbo Expo 2000: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/2000-gt-0435.

Full text

Abstract:

The paper focuses on the unsteady pressure field measured around the rotor mid-span profile of the VKI Brite transonic turbine stage. The understanding of the complex unsteady flow field is supported by a quasi-3D unsteady Navier-Stokes computation using a k-? turbulence model and a modified version of the Abu-Ghannam and Shaw correlation for the onset of transition. The agreement between computational and experimental results is satisfactory. They both reveal the dominance of the vane-shock in the interaction. For this reason, it is difficult to identify the influence of vane-wake ingestion in the rotor passage from the experimental data. However, the computations allow to draw some useful conclusions in this respect. The effect of the variation of the rotational speed, the stator-rotor spacing and the stator trailing edge coolant flow ejection is investigated and the unsteady blade force pattern is analyzed.

APA, Harvard, Vancouver, ISO, and other styles

10

Eulitz, Frank, and Karl Engel. "Numerical Investigation of Wake Interaction in a Low Pressure Turbine." In ASME 1998 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/98-gt-563.

Full text

Abstract:

A time-accurate Reynolds-averaged Navier-Stokes solver has been extended for a phenomenological study of wake/bladerow interaction in a low pressure turbine near midspan. To qualitatively account for unsteady laminar-turbulent boundary layer transition, a variant of the Abu-Ghanam Shaw transition correlation has been coupled with the Spalart-Allmaras one-equation turbulence model. The method is shown to be capable of capturing separated-flow and wake-induced transition, as well as becalming and relaminarization effects. The model turbine investigated consists of three stator and two rotor rows. Instantaneous Mach number and eddy-viscosity plots are presented to monitor the wake migration and interaction with downstream boundary layers. Especially on the suction sides, very large fluctuations of the skin friction coefficient are observed. Effects of the near and far wakes are identified.

APA, Harvard, Vancouver, ISO, and other styles

Reports on the topic "Hele-Shaw flow; Stokes flow"

1

Zeybek, M., and Y. C. Yortsos. Long waves in parallel flow in Hele-Shaw cells. Office of Scientific and Technical Information (OSTI), April 1991. http://dx.doi.org/10.2172/5920848.

Full text

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!