Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material

We consider a heat conduction model arising in transient heat transfer through longitudinal fins of a heterogeneous (functionally graded) material. In this case, the thermal conductivity depends on the spatial variable. The heat transfer coefficient depends on the temperature and is given by the power law function. The resulting nonlinear partial differential equation is analyzed using both classical and nonclassical symmetry techniques. Both the transient state and the steady state result in a number of exotic symmetries being admitted by the governing equation. Furthermore, nonclassical symmetries are also admitted. Both classical and nonclassical symmetry analysis results in some useful reductions and some remarkable exact solutions are constructed.     

Steady one-dimensional heat flow in a longitudinal triangular and parabolic fin

We consider a heat transfer problem of a longitudinal fin with triangular and parabolic profiles. Both thermal conductivity and heat transfer coefficient are assumed to be temperature-dependent, and given by power laws. We construct exact solution when the problem is linearizable. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The obtained exact solutions satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as thermo-geometric fin parameter and the fin efficiency are analyzed.     

Steady heat transfer through a radial fin with rectangular and hyperbolic profiles

We construct some exact solutions for thermal diffusion in a fin with a rectangular profile and another with a hyperbolic profile. Both the thermal conductivity and the heat transfer coefficient are assumed to be temperature dependent. Moreover, the thermal conductivity and the heat transfer terms are given by the same power law in one case and distinct power laws in the other. A point transformation is introduced to linearize the problem when the power laws are equal. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The exact solutions obtained satisfy the realistic boundary conditions. The effects of applicable physical parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.     

Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls

A comprehensive numerical investigation on the natural convection in a rectangular enclosure is presented. The flow is induced due to the constant partial heating at lower half of the left vertical wall and partial cooling at upper half of the right vertical wall along with rest walls are adiabatic. In this investigation the Special attention is given to understand the effect of aspect ratio and heat source intensity i.e. Rayleigh number, Ra, on the fluid flow configuration as well as on the local and average heat transfer rates. The range of Rayleigh (Ra) and aspect ratio (A) is taken [10³, 10⁶] and [0.5, 4] respectively. The results are presented in terms of stream function (ψ), temperature (θ) and heat transfer rates (local Nusselt numbers Nu_L, and average Nusselt numbers Nu). The numerical experiments show that increasing of Ra implies the enhancement of thermal buoyancy force, which in turn increases the thermal convection in the cavity. As a result, the local as well as average heat transfer rate is expected to increase. The local transfer rate (Nu_L) is increases in the small region near the left vertical wall of the left wall of the cavity and after that it is decreases in the middle portion of heated region. And, it start to increase near to the middle point of left wall. It is also observed that the local heat transfer is increases as increases the aspect ratio. The average heat transfer rate (Nu) is increases as the aspect ratio A increases from 0.5 to 1 and beyond that it is decreases smoothly. It is also found that the heat transfer rate attains its maximum value at aspect ratio one.     

Analytic solutions for a rotating radial fin of rectangular and various convex parabolic profiles

The homotopy analysis method (HAM) is used to develop an analytical solution for the thermal performance of a radial fin of rectangular and various convex parabolic profiles mounted on a rotating shaft and losing heat by convection to its surroundings. The convection heat transfer coefficient is assumed to be a function of both the radial coordinate and the angular speed of the shaft. Results are presented for the temperature distribution, heat transfer rate, and the fin efficiency illustrating the effect of thickness profile, the ratio of outer to inner radius, and the angular speed of the shaft. Comparison of HAM results with the direct numerical solutions shows that the analytic results produced by HAM are highly accurate over a wide range of parameters that are likely to be encountered in practice.     

Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient

A homotopy analysis method (HAM) is used to develop analytical solution for the thermal performance of a straight fin of trapezoidal profile when both the thermal conductivity and the heat transfer coefficient are temperature dependent. Results are presented for the temperature distribution, heat transfer rate, and fin efficiency for a range of values of parameters appearing in the mathematical model. Since the HAM algorithm contains a parameter that controls the convergence and accuracy of the solution, its results can be verified internally by calculating the residual error. The HAM results were also found to be accurate to at least three places of decimal compared with the direct numerical solution of the mathematical model generated using a fourth–fifth-order Runge–Kutta–Fehlberg method. The HAM solution appears in terms of algebraic expressions which are not only easy to compute but also give highly accurate results covering a wide range of values of the parameters rather than the small values dictated by the perturbation solution.     

Numerical investigation of a multiserver retrial model

We consider a queueing model in which customers arrive in a Poisson stream to be served by one ofc servers. Each arriving customer enters a pool of active customers and starts generating requests for service at exponentially distributed time intervals at rate until he finds a free server and begins service. An analytical solution of this model is difficult and does not lend itself to numerical implementation. In this paper, we make a simplifying approximation, based on understanding of the physical behavior of the system, which yields an infinitesimal generator with a modified matrix-geometric equilibrium probability vector. That vector can be very efficiently computed even for high congestion levels. Illustrative numerical examples demonstrate the effectiveness of the approximation as well as the effect of the retrial rate on the system behavior for various levels of congestion. This study shows how numerical results for analytically intractable systems can be obtained by combining intuition with efficient algorithmic methods.This author's research was supported in part by Grants Nos. ECS-88-03061 from the National Science Foundation and AFOSR-88-0076 from the Air Force Office of Scientific Research.     

Numerical solution of the nonlinear problem of motion of a long rectangular plate in a variable magnetic field

The coupled problem of motion of a long rectangular plate in a variable magnetic field is analyzed in the framework of the geometrically nonlinear theory of thin shells. The proposed numerical procedure is applied to estimate the effect of external electromagnetic and mechanical fields on the stress- strain state of the plate.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 37–42, 1990.     

Numerical optimization methods for packing equal orthogonally oriented ellipses in a rectangular domain

Linear models are constructed for the numerical solution of the problem of packing the maximum possible number of equal ellipses of given size in a rectangular domain R. It is shown that the l _p metric can be used to determine the conditions under which ellipses with mutually orthogonal major axes (orthogonally oriented ellipses) do not intersect. In R a grid is constructed whose nodes generate a finite set T of points. It is assumed that the centers of the ellipses can be placed only at some points of T. The cases are considered when the major axes of all the ellipses are parallel to the x or y axis or the major axes of some of the ellipses are parallel to the x axis and the others, to the y axis. The problems of packing equal ellipses with centers in T are reduced to integer linear programming problems. A heuristic algorithm based on the linear models is proposed for solving the ellipse packing problems. Numerical results are presented that demonstrate the effectiveness of this approach.     

Numerical investigation of the Liebau phenomenon

The Liebau phenomenon is the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to system configuration. This paper is concerned with two different physical configurations and respective models. Comparison and derivation among the models is discussed. Accurate numerical schemes which solve these models are presented. By means of numerical simulations it is investigated under which conditions valveless pumping takes place.     

Numerical investigation of the Liebau phenomenon

The Liebau phenomenon is the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to system configuration. This paper is concerned with two different physical configurations and respective models. Comparison and derivation among the models is discussed. Accurate numerical schemes which solve these models are presented. By means of numerical simulations it is investigated under which conditions valveless pumping takes place.     

Finite-element investigation of the stress-strain state of an inhomogeneous rectangular plate

With the use of the finite-element method, the generalized plane stressed state of a rectangle of isotropic functionally gradient materials under the action of normal load is investigated. A finite-element model is constructed by the Bubnov–Galerkin method. The domain of the body is split into rectangular gradient elements that take into account dependences of Young’s modulus and Poisson’s ratios on coordinates. Numerical calculations are performed for the case where Young’s modulus is a polynomial function. The influence of the material gradientness and the sizes of the rectangle on its stress-strain state is analyzed.     

Multicriteria optimization of a rectangular composite plate subjected to longitudinal thermal stresses and buckling in shear loading

The multicriteria optimization of the structure and geometry of a laminated anisotropic composite plate subjected to thermal and shear loading is considered. From the known properties of the monolayer and given values of variable structural parameters, the thermoelastic properties of the layered composite are determined. The optimization criteria — the critical shear load and the longitudinal thermal stresses — depend on two variable design parameters of composite properties and temperature. In the space of optimization criteria, the domain of allowable solutions and the Pareto-optimal subregion are found. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 85–92, January–February, 2007.     

Numerical implementation of the spline-collocation method for bending of rectangular plates

Different approaches are considered to the application of the spline-collocation method to bending of rectangular plates. The effect of the choice of collocation nodes on the order of accuracy of the method is investigated. Numerical results obtained with different arrangements of collocation points are reported.Institute of Mechanics, Ukrainian Academy of Sciences. Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 93–100, 1989.     

Numerical modelling of 3D sloshing experiments in rectangular tanks

This work encompasses numerical and experimental studies of three-dimensional (3D) sloshing problems. The two-fluid viscous flow, which is solved within a stabilized finite element context, involves liquid and gaseous phases. The free surface is captured with a level set (LS) method, including the bounded renormalization with continuous penalization technique, to avoid the well-known spreading of the marker function. Specifically, this technique is improved with a volume-preserving algorithm for long-term analyses. To verify the numerical model, the responses of free-sloshing cases are compared with analytical solutions and other results computed using a Lagrangian technique. These simulations assess the influence of considering two-dimensional (2D) and 3D analyses, as well as the effects of depth and viscosity. This work presents data obtained from a forced sloshing experiment that is specifically devoted to 3D free surface behaviour. Free surface evolution measurements are used to validate the numerical method. Moreover, the effect of the initial conditions used to promote 3D behaviour in the modelling is evaluated.     

Numerical investigation of a liquid displacing a gas in thin porous layers

This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which occurs e.g. during the filling process of a lithium-ion battery with an electrolyte. The investigation is based upon the local volume-averaged Navier-Stokes equations, using a Volume-of-Fluid method to treat the interface. For the flow the wall effect and capillary forces have to be considered. Capillary rise experiments are used to determine the permeability. Since the layers are thin and the characteristic size of the particles is comparatively large, friction with the electrode is taken into account with respect to the mobility of the contact line. The implemented models are validated against analytical results, showing a good agreement. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical investigation of electro-osmotic flows and chaotic stirring in rectangular cavities

Two-dimensional, time-independent and time-dependent electro-osmotic flows driven by a uniform electric field in a closed rectangular cavity with uniform and non-uniform zeta potential distributions along the cavity’s walls are investigated theoretically. First, we derive an expression for the one-dimensional velocity and pressure profiles for a flow in a slender cavity with uniform (albeit possibly different) zeta potentials at its top and bottom walls. Subsequently, using the method of superposition, we compute the flow in a finite length cavity whose upper and lower walls are subjected to non-uniform zeta potentials. Although the solutions are in the form of infinite series, with appropriate modifications, the series converge rapidly, allowing one to compute the flow fields accurately while maintaining only a few terms in the series. Finally, we demonstrate that by time-wise periodic modulation of the zeta potential, one can induce chaotic advection in the cavity. Such chaotic flows can be used to stir and mix fluids. Since devices operating on this principle do not require any moving parts, they may be particularly suitable for microfluidic devices.     

The single-mode approximation in the problem of the propagation of a plane longitudinal wave in an elastic medium with a periodic system of rectangular defects

The approach suggested in [1, 2] is applied to the problem of the propagation of a plan longitudinal wave in an elastic medium containing a periodic system of rectangular defects. Explicit analytical representations for the scattering coefficients as well as a refined low-frequency solution are derived using a uniform approximation of the single-mode type. A comparison of the results with solutions obtained by other methods is given.