Variational model of scoliosis
Igor Popov, Nikita Lisitsa, Yuri Baloshin, Mikhail Dudin, and Stepan Bober
Available online 07 November 2017

Abstract
Scoliosis, being one of the most widespread spine diseases among children, has been studied extensively throughout the history of medicine, yet there is no clear understanding of its initiating factors and the mechanogenesis of the monomorphic three-dimensional deformation due to its polyetiological nature. We present a novel mathematical model of the process of emergence of three-dimensional deformation of human spine based on variational principles. Typical scoliosis geometry is assumed to be described as minimal curves of a particular energy functional, which are shown to closely resemble actual scoliosis. The numerical properties of the first stage of scoliosis are investigated, which is shown to have the highest influence on the development of the disease.

Mathematics Subject Classification
92C10; 74L15

Keywords
spine, model, variational method

DOI
https://doi.org/10.2298/TAM170818012P

An effect of a purely dissipative process of microstresses on plane strain gradient plasticity problems
Adebowale Borokinni, Odunayo Fadodun, and Adegbola Akinola
Available online 02 March 2018

Abstract
This article considers a plane strain gradient plasticity theory of the Gurtin--Anand model [M. Gurtin, L. Anand, A theory of strain gradient plasticity for isotropic, plastically irrotational materials Part I: Small deformations, J. Mech. Phys. Solids 53 (2005), 1624--1649] for an isotropic material undergoing small deformation in the absence of plastic spin. It is assumed that the system of microstresses is purely dissipative, so that the free energy reduces to a function of the elastic strain, while the microstresses are only related to the plastic strain rate and gradient of the plastic strain rate via the constitutive relations. The plane strain problem of the Gurtin-Anand model for a purely dissipative process gives rise to elastic incompressibility. A weak formulation of the flow rule is derived, making the plane strain problem suitable for finite element implementation.

Mathematics Subject Classification
74C10

Keywords
plane strain gradient, microstresses, flow rule, weak formulation

DOI
https://doi.org/10.2298/TAM171017002B

Asymptotic solution for the Darcy-Brinkman-Boussinesq flow in a pipe with helicoidal shape
Igor Pažanin
Available online 05 July 2018

Abstract
We study the fluid flow and heat transfer in a helical pipe filled with a sparsely packed porous medium. Motivated by the engineering applications, pipe's thickness and the distance between two coils of the helix have the same small order of magnitude, whereas the fluid inside the pipe is assumed to be cooled (or heated) by the exterior medium. After writing the dimensionless Darcy-Brinkman-Boussinesq system in curvilinear coordinates, we employ the multi-scale expansion technique to formally derive the effective model valid for small Brinkman-Darcy number. The obtained asymptotic solution is given in the explicit form which is important with regards to numerical simulations. Comparison with our previous results on the straight-pipe flow is also provided.

Mathematics Subject Classification
35B40; 35Q35; 76S05

Keywords
helical pipe, Darcy-Brinkman-Boussinesq system, Newton cooling condition, curvilinear coordinates, asymptotic approximation

DOI
https://doi.org/10.2298/TAM180424008P

On the influence of turbulent kinetic energy level on accuracy of 𝑘 − 𝜀 and LRR turbulence models
Djordje M. Novković, Jela M. Burazer, Aleksandar S. Ćoćić, and Milan R. Lečić
Available online 13 July 2018

Abstract
This paper presents research regarding the influence of turbulent kinetic energy (TKE) level on accuracy of Reynolds averaged Navier--Stokes (RANS) based turbulence models. A theoretical analysis of influence TKE level on accuracy of the RANS turbulence models has been performed according to the Boussinesq hypothesis definition. After that, this theoretical analysis has been investigated by comparison of numerically and experimentally obtained results on the test case of a steady-state incompressible swirl-free flow in a straight conical diffuser named Azad diffuser.~Numerical calculations have been performed using the OpenFOAM CFD software and first and second-order closure turbulence models. TKE level, velocity profiles and Reynolds stresses have been calculated downstream in four different cross sections of the diffuser. Certain conclusions about modeling turbulent flows by $k-\varepsilon$ and LRR turbulence models have been made by comparing the velocity profiles, TKE distribution and Reynolds stresses on the selected cross sections.

Mathematics Subject Classification
76F10

Keywords
CFD, diffuser, OpenFOAM, RANS, turbulent kinetic energy

DOI
https://doi.org/10.2298/TAM171201009N