Inverse Problems with Unknown Boundary Conditions and Final Overdetermination for Time Fractional Diffusion-Wave Equations in Cylindrical Domains

Inverse problems to reconstruct a solution of a time fractional diffusion-wave equation in a cylindrical domain are studied. The equation is complemented by initial and final conditions and partly given boundary conditions. Two cases are considered: (1) full boundary data on a lateral hypersurface of the cylinder are given, but the boundary data on bases of the cylinder are specified in a neighborhood of a final time; (2) boundary data on the whole boundary of the cylinder are specified in a neighborhood of the final time, but the cylinder is either a cube or a circular cylinder. Uniqueness of solutions of the inverse problems is proved. Uniqueness for similar problems in an interval and a disk is established, too.

A MIXED PROBLEM FOR A DEGENERATE MULTIDIMENSIONAL ELLIPTIC EQUATION

This article shows the unique solvability and obtains an explicit form of the classical solution of the mixed prob-lem in a cylindrical domain for a model degenerate multidimensional elliptic equation. The correctness of boundary value problems in the plane for elliptic equations by the method of the theory of ana-lytic functions of a complex variable has been well studied. The first boundary value problem or the Dirichlet problem for multidimensional elliptic equations with degeneration on the boundary has been sufficiently analyzed. However, as we know, the mixed problem for the indicated equations has been studied very little.

Finite Element Study of Bio-Convective Stefan Blowing Ag-MgO/Water Hybrid Nanofluid Induced by Stretching Cylinder Utilizing Non-Fourier and Non-Fick’s Laws

In the present framework, an analysis on nanofluid magneto-transport phenomena over an extending cylinder influenced by gyrotactic behavior of algal suspension, is made using the Cattaneo–Christov heat flux (non-Fourier) and mass flux (non-Fick’s) concept in modified Buongiorno’s model. Two dimensional incompressible MHD hybrid nanofluid which comprises chemically reactive hybrid nanomaterials (Ag-MgO NPs) and Stefan blowing effect along with multiple slips is considered. The experimental correlations with their dependency on initial nanoparticle volume fraction are used for viscosity and thermal conductivity of nanofluids. Similarity transformation is used to convert the governing PDE’s into non-linear ODE’s along with boundary conditions, which are solved using the Galerkin Finite Element Method (GFEM). The mesh independent test with different boundary layer thickness (ξ∞) has been conducted by taking both linear and quadratic shape functions to achieve a optimal desired value. The results are calculated for a realistic range of physical parameters. The validation of FEM results shows an excellent correlation with MATLAB bvp5c subroutine. The warmth exhibitions are assessed through modified version of Buongiorno’s model which effectively reflects the significant highlights of Stefan blowing, slip, curvature, free stream, thermophoresis, Brownian motion and bio-convection parameters. The present study in cylindrical domain is relevant to novel microbial fuel cell technologies utilizing hybrid nanoparticles and concept of Stefan blowing with bioconvection phenomena.

On the Choice of Interface Parameters in Robin–Robin Loosely Coupled Schemes for Fluid–Structure Interaction

We consider two loosely coupled schemes for the solution of the fluid–structure interaction problem in the presence of large added mass effect. In particular, we introduce the Robin–Robin and Robin–Neumann explicit schemes where suitable interface conditions of Robin type are used. For the estimate of interface Robin parameters which guarantee stability of the numerical solution, we propose a new strategy based on the optimization of the reduction factor of the corresponding strongly coupled (implicit) scheme, by means of the optimized Schwarz method. To check the suitability of our proposals, we show numerical results both in an ideal cylindrical domain and in a real human carotid. Our results showed the effectiveness of our proposal for the calibration of interface parameters, which leads to stable results and shows how the explicit solution tends to the implicit one for decreasing values of the time discretization parameter.

Analytical Solution of a Non-isothermal Flow in Cylindrical Geometry

This study proposes analytical solution to the problem of transport in a Newtonian fluid within a cylindrical domain. The flow is assumed to be dominated along the channel axis, and is taken to be axi-symmetric. No-slip boundary condition is considered for velocity while the temperature and concentration have Dirichlet boundary values. The resulting problem is transformed into a set of non-trivial variable coefficient differential equations in a cylindrical geometry. By adopting the series solution method of Frobenius, the closed-form analytical solutions are derived for the flow variables. We conduct an analysis of the derived model, and showed that, indeed, the flow variables are axi-symmetric. We also state and prove another theorem to show that the derived concentration model is positivity preserving – meaning that it yields positive concentration - provided the boundary value is non-negative. Finally, we present graphical results for the flow variables and discuss the effect of the relevant flow parameters. The results showed that (i) an increase in the cooling parameter, reduces the fluid velocity, (ii) the temperature decreases as the cooling parameter increases and (iii) an increase in the injection parameter, leads to increase in the concentration.

Entropy Analysis on a Three-Dimensional Wavy Flow of Eyring–Powell Nanofluid: A Comparative Study

The thermal management of a system needs an accurate and efficient measurement of exergy. For optimal performance, entropy should be minimized. This study explores the enhancement of the thermal exchange and entropy in the stream of Eyring–Powell fluid comprising nanoparticles saturating the vertical oriented dual cylindrical domain with uniform thermal conductivity and viscous dissipation effects. A symmetrical sine wave over the walls is used to induce the flow. The mathematical treatment for the conservation laws are described by a set of PDEs, which are, later on, converted to ordinary differential equations by homotopy deformations and then evaluated on the Mathematica software tool. The expression of the pressure rise term has been handled numerically by using numerical integration by Mathematica through the algorithm of the Newton–Cotes formula. The impact of the various factors on velocity, heat, entropy profile, and the Bejan number are elaborated pictorially and tabularly. The entropy generation is enhanced with the variation of viscous dissipation but reduced in the case of the concentration parameter, but viscous dissipation reveals opposite findings for the Newtonian fluid. From the abovementioned detailed discussion, it can be concluded that Eyring–Powell shows the difference in behavior in the entropy generation and in the presence of nanoparticles due to the significant dissipation effects, and also, it travels faster than the viscous fluid. A comparison between the Eyring-Powell and Newtonian fluid are also made for each pertinent parameter through special cases. This study may be applicable for cancer therapy in biomedicine by nanofluid characteristics in various drugs considered as a non-Newtonian fluid.

LINEAR CONJUGATION PROBLEM WITH MULTIPOINT NONLOCAL CONDITION FOR A PARABOLIC-HYPERBOLIC EQUATION IN CYLINDRICAL DOMAIN

The linear conjugation problem with multipoint nonlocal condition in the time variable for a mixed parabolic-hyperbolic equation of the second order in a cylindrical domain, which is Cartesian product of the time segment and the spatial multidimensional torus, is investigated. The conditions of the existence and uniqueness of а solution to the problem in the scale of Sobolev spaces are obtained. It has been proved that these conditions fulfill for almost all (with respect to the Lebesgue measure) values of the left node of the multipoint condition.

Radiofrequency ablation combined with conductive fluid-based dopants (saline normal and colloidal gold): computer modeling and ex vivo experiments

Background The volume of the coagulation zones created during radiofrequency ablation (RFA) is limited by the appearance of roll-off. Doping the tissue with conductive fluids, e.g., gold nanoparticles (AuNPs) could enlarge these zones by delaying roll-off. Our goal was to characterize the electrical conductivity of a substrate doped with AuNPs in a computer modeling study and ex vivo experiments to investigate their effect on coagulation zone volumes. Methods The electrical conductivity of substrates doped with normal saline or AuNPs was assessed experimentally on agar phantoms. The computer models, built and solved on COMSOL Multiphysics, consisted of a cylindrical domain mimicking liver tissue and a spherical domain mimicking a doped zone with 2, 3 and 4 cm diameters. Ex vivo experiments were conducted on bovine liver fragments under three different conditions: non-doped tissue (ND Group), 2 mL of 0.9% NaCl (NaCl Group), and 2 mL of AuNPs 0.1 wt% (AuNPs Group). Results The theoretical analysis showed that adding normal saline or colloidal gold in concentrations lower than 10% only modifies the electrical conductivity of the doped substrate with practically no change in the thermal characteristics. The computer results showed a relationship between doped zone size and electrode length regarding the created coagulation zone. There was good agreement between the ex vivo and computational results in terms of transverse diameter of the coagulation zone. Conclusions Both the computer and ex vivo experiments showed that doping with AuNPs can enlarge the coagulation zone, especially the transverse diameter and hence enhance sphericity.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .