Mathematical Analysis of Continuum Mechanics and Industrial Applications

2017 ◽  
Keyword(s):  
Continuum Mechanics ◽  
Mathematical Analysis ◽  
Industrial Applications

scholarly journals Diffusion–reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study

2020 ◽  
Vol 66 (5) ◽  
pp. 1131-1152 ◽  
Author(s):  
Alex Viguerie ◽  
Alessandro Veneziani ◽  
Guillermo Lorenzo ◽  
Davide Baroli ◽  
Nicole Aretz-Nellesen ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Continuum Mechanics ◽  
Mathematical Analysis ◽  
Numerical Study ◽  
Interdisciplinary Collaboration ◽  
Compartmental Models ◽  
Diffusion Reaction ◽  
Pde Model ◽  
Fundamental Equations ◽  
Detailed Derivation

Abstract The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models, in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.

scholarly journals GALERKIN LEAST-SQUARES APPROXIMATIONS FOR FLOWS OF CASSON FLUIDS THROUGH AN EXPANSION

2006 ◽  
Vol 5 (2) ◽  
pp. 82
Author(s):  
F. S. F. Zinani ◽  
S. Frey
Keyword(s):  
Least Squares ◽  
Continuum Mechanics ◽  
Complex Geometry ◽  
Shear Thinning ◽  
High Viscosity ◽  
Industrial Applications ◽  
Material Behavior ◽  
Sudden Expansion ◽  
Shear Thinning Fluids ◽  
Viscosity Function

Among non-Newtonian fluid models, purely viscous constitutive equations play an important role in industrial applications regardless their lack of accuracy in non-viscometric flows. In this work we are concerned with the flow of viscoplastic shear-thinning fluids in complex geometry. Viscoplastic fluids are those that behave as extremely high viscosity materials when submitted to low stresses and that flow when submitted to stresses higher than a yield stress value. Usually, they also present shearthinning behavior. Fluids such as molten chocolate, xanthan gum solutions, blood, wastewater sludges, muds, and polymer solutions present viscoplastic shear-thinning features. In order to approximate numerically viscoplastic shear-thinning flows we first describe a mechanical model based on continuum mechanics conservation laws of mass and momentum. The description of material behavior is such as to respect certain principles of objectivity and generality in continuum mechanics. The Generalized Newtonian Liquid constitutive equation with Casson viscosity function is able to predict viscoplasticity and shear-thinning. The numerical approximation of the equations is performed by a finite element method. To prevent the model from pathologies known for the classic Galerkin method, we employ a stabilized method based on a Galerkin least-squares (GLS) scheme, which is designed to circumvent Babuška-Brezzi condition and deal with the asymmetry of the advective operator. We present approximations for the flow through a planar 4:1 sudden expansion. We investigate the influence of Reynolds and Casson numbers on the flow dynamics.

Density waves in disk galaxies

1967 ◽  
Vol 31 ◽  
pp. 313-317 ◽  
Author(s):  
C. C. Lin ◽  
F. H. Shu
Keyword(s):  
Mathematical Analysis ◽  
Spiral Structure ◽  
Stellar System ◽  
Collective Modes ◽  
Disk Galaxies ◽  
Spiral Pattern ◽  
Density Waves ◽  
The Galaxy ◽  
Detailed Mathematical Analysis

Density waves in the nature of those proposed by B. Lindblad are described by detailed mathematical analysis of collective modes in a disk-like stellar system. The treatment is centered around a hypothesis of quasi-stationary spiral structure. We examine (a) the mechanism for the maintenance of this spiral pattern, and (b) its consequences on the observable features of the galaxy.

The Application of Ultra-Thin Sectioning Techniques to Non-Biological Samples

1968 ◽  
Vol 26 ◽  
pp. 332-333
Author(s):  
C. F. Oster
Keyword(s):  
Biological Sciences ◽  
Epoxy Resin ◽  
Cross Sections ◽  
Biological Samples ◽  
Industrial Applications ◽  
Photographic Film ◽  
Silver Particles ◽  
Thin Sectioning ◽  
Embedding Medium

Although ultra-thin sectioning techniques are widely used in the biological sciences, their applications are somewhat less popular but very useful in industrial applications. This presentation will review several specific applications where ultra-thin sectioning techniques have proven invaluable.The preparation of samples for sectioning usually involves embedding in an epoxy resin. Araldite 6005 Resin and Hardener are mixed so that the hardness of the embedding medium matches that of the sample to reduce any distortion of the sample during the sectioning process. No dehydration series are needed to prepare our usual samples for embedding, but some types require hardening and staining steps. The embedded samples are sectioned with either a prototype of a Porter-Blum Microtome or an LKB Ultrotome III. Both instruments are equipped with diamond knives.In the study of photographic film, the distribution of the developed silver particles through the layer is important to the image tone and/or scattering power. Also, the morphology of the developed silver is an important factor, and cross sections will show this structure.

The mensuration of interfaces

1992 ◽  
Vol 50 (1) ◽  
pp. 216-217
Author(s):  
W.M. Stobbs
Keyword(s):  
Electron Microscopy ◽  
50Th Anniversary ◽  
Numerical Data ◽  
Industrial Applications ◽  
Dynamical Theory ◽  
Large Strains ◽  
Reasonable Approach ◽  
Lattice Resolution ◽  
Analogue To Digital ◽  
The Way

I do not have access to the abstracts of the first meeting of EMSA but at this, the 50th Anniversary meeting of the Electron Microscopy Society of America, I have an excuse to consider the historical origins of the approaches we take to the use of electron microscopy for the characterisation of materials. I have myself been actively involved in the use of TEM for the characterisation of heterogeneities for little more than half of that period. My own view is that it was between the 3rd International Meeting at London, and the 1956 Stockholm meeting, the first of the European series , that the foundations of the approaches we now take to the characterisation of a material using the TEM were laid down. (This was 10 years before I took dynamical theory to be etched in stone.) It was at the 1956 meeting that Menter showed lattice resolution images of sodium faujasite and Hirsch, Home and Whelan showed images of dislocations in the XlVth session on “metallography and other industrial applications”. I have always incidentally been delighted by the way the latter authors misinterpreted astonishingly clear thickness fringes in a beaten (”) foil of Al as being contrast due to “large strains”, an error which they corrected with admirable rapidity as the theory developed. At the London meeting the research described covered a broad range of approaches, including many that are only now being rediscovered as worth further effort: however such is the power of “the image” to persuade that the above two papers set trends which influence, perhaps too strongly, the approaches we take now. Menter was clear that the way the planes in his image tended to be curved was associated with the imaging conditions rather than with lattice strains, and yet it now seems to be common practice to assume that the dots in an “atomic resolution image” can faithfully represent the variations in atomic spacing at a localised defect. Even when the more reasonable approach is taken of matching the image details with a computed simulation for an assumed model, the non-uniqueness of the interpreted fit seems to be rather rarely appreciated. Hirsch et al., on the other hand, made a point of using their images to get numerical data on characteristics of the specimen they examined, such as its dislocation density, which would not be expected to be influenced by uncertainties in the contrast. Nonetheless the trends were set with microscope manufacturers producing higher and higher resolution microscopes, while the blind faith of the users in the image produced as being a near directly interpretable representation of reality seems to have increased rather than been generally questioned. But if we want to test structural models we need numbers and it is the analogue to digital conversion of the information in the image which is required.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

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