Analytical and numerical studies for Seiches in a closed basin with bottom friction

A standing wave oscillation in a closed basin, known as a seiche, could cause destruction when its period matches the period of another wave generated by external forces such as wind, quakes, or abrupt changes in atmospheric pressure. It is due to the resonance phenomena that allow waves to have higher amplitude and greater energy, resulting in damages around the area. One condition that might restrict the resonance from occurring is when the bottom friction is present. Therefore, a modified mathematical model based on the shallow water equations will be used in this paper to investigate resonance phenomena in closed basins and to analyze the effects of bottom friction on the phenomena. The study will be conducted for several closed basin shapes. The model will be solved analytically and numerically in order to determine the natural resonant period of the basin, which is the period that can generate a resonance. The computational scheme proposed to solve the model is developed using the staggered grid finite volume method. The numerical scheme will be validated by comparing its results with the analytical solutions. As a result of the comparison, a rather excellent compatibility between the two results is achieved. Furthermore, the impacts that the friction coefficient has on the resonance phenomena are evaluated. It is observed that in the prevention of resonances, the bottom friction provides the best performance in the rectangular type while functioning the least efficient in the triangular basin. In addition, non-linearity effect as one of other factors that provide wave restriction is also considered and studied to compare its effect with the bottom friction effect on preventing resonance.     

Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles

We describe an analytic method for the computation of equilibrium shapes for two-dimensional vesicles characterized by a Helfrich elastic energy. We derive boundary value problems and solve them analytically in terms of elliptic functions and elliptic integrals. We derive solutions by prescribing length and area, or displacements and angle boundary conditions. The solutions are compared to solutions obtained by a boundary integral equation-based numerical scheme. Our method enables the identification of different configurations of deformable vesicles and accurate calculation of their shape, bending moments, tension, and the pressure jump across the vesicle membrane. Furthermore, we perform numerical experiments that indicate that all these configurations are stable minima.     

Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.     

Analytical and finite-element study of optimal strain distribution in various beam shapes for energy harvesting applications

Owing to the increasing demand for harvesting energy from environmental vibration for use in self-powered electronic applications, cantilever-based vibration energy harvesting has attracted considerable interest from various parties and has become one of the most common approaches to converting redundant mechanical energy into electrical energy. As the output voltage produced from a piezoelec-tric material depends largely on the geometric shape and the size of the beam, there is a need to model and compare the performance of cantilever beams of differing geometries. This paper presents the study of strain distribution in various shapes of cantilever beams, including a convex and concave edge profile elliptical beam that have not yet been discussed in any prior literature. Both analytical and finite-element models are derived and the resultant strain distributions in the beam are computed based on a MATLAB solver and ANSYS finite-element analysis tools. An optimum geome-try for a vibration-based energy harvesting system is verified. Finally, experimental results comparing the power density for triangular and rectangular piezoelectric beams are also pre-sented to validate the findings of the study, and the claim, as suggested in the literature, is verified.     

Green's function for thermopiezoelectric plates with holes of various shapes

Summary Thermoelectroelastic problems for holes of various shapes embedded in an infinite matrix are considered in this paper. Based on Lekhnitskii's formalism, the technique of conformal mapping and the exact electric boundary conditions on the hole boundary, the thermoelectroelastic Green's function has been obtained analytically in terms of a complex potential. As an application of the proposed function, the problem of an infinite plate containing a crack and a hole is analysed. A system of singular integral equations for the unknown temperature discontinuity and the discontinuity of elastic displacement and electric potential (EDEP) defined on crack faces is developed and solved numerically. Numerical results for stress and electric displacement (SED) intensity factors of the crack-hole system are presented to illustrate the application of the proposed formulation. Received 7 October 1998; accepted for publication 26 January 1999     

Existence theory for damped gravity waves in a closed rectangular basin

We study existence and uniqueness of solutions of the equations for the free surface motion of an incompressible, irrotational fluid in a rectangular basin subject to vertical oscillation. After adding artificial damping, which leaves the flow irrotational but correctly represents the physical rate of energy loss at high wave numbers, we prove global existence and uniqueness results in the appropriate Sobolev spaces, provided that the initial data and forcing amplitudes have sufficiently small norms. Convergence of spatially discretized (finite-dimensional) projections is also discussed.     

Analytical and numerical solution along with PC spreadsheets modeling for a composite fin

Heat transfer through composite fins is investigated by both analytical and numerical methods. In this regard, governing differential equations of the two dimensional fin and one dimensional cladding are studied to examine the effect of Biot number and ratio of thermal conductivities of the fin material to the cladding, on the dimensionless temperature profiles. The results show that one dimensional analysis, traditionally used in fin analysis, is not applicable for composite fins, particularly when the conductivity ratio of the composite fin materials is low. In addition, the use of spreadsheet programs in solving the fin problem is investigated in somewhat more detail with regard to the solution as well as presentation of the graphical results.     

Analytical study of supersonic isentropic flow in a channel of constant cross section and an adjacent laval nozzle. Comparison with numerical and experimental results

An analytic study of plane supersonic flow based on the use of Prandtl-Meyer invariants is presented. Flow in both a channel of constant cross section and an adjacent Laval nozzle with a corner point is considered. The analytical results are compared with numerical and experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 158–169, July–August, 1994.     

Analytical and numerical study of buoyancy-driven convection in a vertical enclosure filled with nanofluids

This paper presents an analytical and numerical study of natural convection of nanofluids contained in a rectangular enclosure subject to uniform heat flux along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number Ra, the Prandtl number Pr, the aspect ratio of the cavity A and the solid volume fraction of nanoparticles, Φ. Three types of nanoparticles are taken into consideration: Cu, Al₂O₃ and TiO₂. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. In the first part of the analytical study, a scale analysis is made for the boundary layer regime situation. In the second part, an analytical solution based on the parallel flow approximation is reported for tall enclosures (A ≫ 1). In the boundary layer regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Solutions for the flow fields, temperature distributions and Nusselt numbers are obtained explicitly in terms of the governing parameters of the problem. A numerical study of the same phenomenon, obtained by solving the complete system of the governing equations, is also conducted. A good agreement is found between the analytical predictions and the numerical simulations.     

Coupled instability of compressed thin-walled rods with a closed transverse cross section

Dnepropetrovsk Civil Engineering Institute, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 29, No. 2, pp. 62–68, February, 1993.     

Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures

This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.     

Damage model with delay effect: Analytical and numerical studies of the evolution of the characteristic damage length

This paper studies the delayed damage model in a one-dimensional transient analysis. It is well-known that kind of model prevents the mesh dependency when it is used in a finite element code. The model problem concerns a clamped uniaxial damage elastic bar submitted to a step load at its extremity. In order to guarantee the correct behaviour of the model and to be able to choose the appropriate mesh size before performing a finite element calculation, an analytical evaluation of the size of the damaged zone called characteristic length is given and compared to the converged numerical results. Finally, a two-dimensional example is treated with a damage with or without the delay effect.     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.     

Analytical and numerical studies of free vibration frequencies of ferromagnetic plates with arbitrary electric conductance in a transverse magnetic field in the 3D setting

The problem on the vibrations of a magnetoelastic ferromagnetic plate was studied in [1–5] from the viewpoint of the averaging approach, i.e., on the basis of the classical Kirchhoff hypothesis. In [6–10], a new approach proposed for elastic plates in [11] was used to derive dispersion relations for magnetoelastic plates. In [10], the 3D approach was used to obtain the ferromagnetic plate vibration frequencies; in the case of a transverse magnetic field, the equations of the perturbed motion of the plates were written out with the initial stresses taken into account [5, 12] but without considering the initial strains. In [13, 14], the problems on the vibrations of conducting plates in a magnetic field were studied.In the present paper, we derive the dispersion equations, which are asymptotic equations for small magnetic fields and exact equations for the initial stresses and strains related by Hooke’s law. The corresponding numerical computations are also performed.