Abstract: In this study, out-of-plane free vibrations of a circular
rods is investigated theoretically. The governing equations for
naturally twisted and curved spatial rods are obtained using
Timoshenko beam theory and rewritten for circular rods. Effects of
the axial and shear deformations are considered in the formulations.
Ordinary differential equations in scalar form are solved analytically
by using transfer matrix method. The circular rods of the mass matrix
are obtained by using straight rod of consistent mass matrix. Free
vibrations frequencies obtained by solving eigenvalue problem. A
computer program coded in MATHEMATICA language is prepared.
Circular beams are analyzed through various examples for free
vibrations analysis. Results are compared with ANSYS results based
on finite element method and available in the literature.
Abstract: In this paper, the problem of steady laminar boundary
layer flow and heat transfer over a permeable exponentially
stretching/shrinking sheet with generalized slip velocity is
considered. The similarity transformations are used to transform the
governing nonlinear partial differential equations to a system of
nonlinear ordinary differential equations. The transformed equations
are then solved numerically using the bvp4c function in MATLAB.
Dual solutions are found for a certain range of the suction and
stretching/shrinking parameters. The effects of the suction parameter,
stretching/shrinking parameter, velocity slip parameter, critical shear
rate and Prandtl number on the skin friction and heat transfer
coefficients as well as the velocity and temperature profiles are
presented and discussed.
Abstract: Starting from nonlocal continuum mechanics, a
thermodynamically new nonlocal model of Euler-Bernoulli
nanobeams is provided. The nonlocal variational formulation is
consistently provided and the governing differential equation for
transverse displacement is presented. Higher-order boundary
conditions are then consistently derived. An example is contributed in
order to show the effectiveness of the proposed model.
Abstract: An analysis is carried out to investigate the effect of
magnetic field and heat source on the steady boundary layer flow and
heat transfer of a Casson nanofluid over a vertical cylinder stretching
exponentially along its radial direction. Using a similarity
transformation, the governing mathematical equations, with the
boundary conditions are reduced to a system of coupled, non –linear
ordinary differential equations. The resulting system is solved
numerically by the fourth order Runge – Kutta scheme with shooting
technique. The influence of various physical parameters such as
Reynolds number, Prandtl number, magnetic field, Brownian motion
parameter, thermophoresis parameter, Lewis number and the natural
convection parameter are presented graphically and discussed for non
– dimensional velocity, temperature and nanoparticle volume
fraction. Numerical data for the skin – friction coefficient, local
Nusselt number and the local Sherwood number have been tabulated
for various parametric conditions. It is found that the local Nusselt
number is a decreasing function of Brownian motion parameter Nb
and the thermophoresis parameter Nt.
Abstract: We have developed a new computer program in
Fortran 90, in order to obtain numerical solutions of a system
of Relativistic Magnetohydrodynamics partial differential equations
with predetermined gravitation (GRMHD), capable of simulating
the formation of relativistic jets from the accretion disk of matter
up to his ejection. Initially we carried out a study on numerical
methods of unidimensional Finite Volume, namely Lax-Friedrichs,
Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods
dependent on Riemann problems, applied to equations Euler in
order to verify their main features and make comparisons among
those methods. It was then implemented the method of Finite
Volume Centered of Nessyahu-Tadmor, a numerical schemes that
has a formulation free and without dimensional separation of
Riemann problem solvers, even in two or more spatial dimensions,
at this point, already applied in equations GRMHD. Finally, the
Nessyahu-Tadmor method was possible to obtain stable numerical
solutions - without spurious oscillations or excessive dissipation -
from the magnetized accretion disk process in rotation with respect
to a central black hole (BH) Schwarzschild and immersed in a
magnetosphere, for the ejection of matter in the form of jet over a
distance of fourteen times the radius of the BH, a record in terms
of astrophysical simulation of this kind. Also in our simulations,
we managed to get substructures jets. A great advantage obtained
was that, with the our code, we got simulate GRMHD equations in
a simple personal computer.
Abstract: Laplace transformations have wide applications in
engineering and sciences. All previous studies of modified Laplace
transformations depend on differential equation with initial
conditions. The purpose of our paper is to solve the linear differential
equations (not initial value problem) and then find the general
solution (not particular) via the Laplace transformations without
needed any initial condition. The study involves both types of
differential equations, ordinary and partial.
Abstract: Taro Scarab beetles (Papuana uninodis, Coleoptera:
Scarabaeidae) inflict severe damage on important root crops and
plants such as Taro or Cocoyam, yam, sweet potatoes, oil palm and
coffee tea plants across Africa and Asia resulting in economic
hardship and starvation in some nations. Scoliid wasps and
Metarhizium anisopliae fungus - bio-control agents; are shown to be
able to control the population of Scarab beetle adults and larvae using
a newly created simulation model based on non-linear ordinary
differential equations that track the populations of the beetle life
cycle stages: egg, larva, pupa, adult and the population of the scoliid
parasitoid wasps, which attack beetle larvae. In spite of the challenge
driven by the longevity of the scarab beetles, the combined effect of
the larval wasps and the fungal bio-control agent is able to control
and drive down the population of both the adult and the beetle eggs
below the environmental carrying capacity within an interval of 120
days, offering the long term prospect of a stable and eco-friendly
environment; where the population of scarab beetles is: regulated by
parasitoid wasps and beneficial soil saprophytes.
Abstract: This study proposes the transformation of nonlinear
Magnetic Levitation System into linear one, via state and feedback
transformations using explicit algorithm. This algorithm allows
computing explicitly the linearizing state coordinates and feedback
for any nonlinear control system, which is feedback linearizable,
without solving the Partial Differential Equations. The algorithm is
performed using a maximum of N-1 steps where N being the
dimension of the system.
Abstract: A theoretical study has been presented to describe the boundary layer flow and heat transfer on an exponentially shrinking sheet with a variable wall temperature and suction, in the presence of magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically using the shooting method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented through graphs and tables for several sets of values of the parameters. The effects of the governing parameters on the flow and heat transfer characteristics are thoroughly examined.
Abstract: The mixed convection stagnation point flow toward a vertical plate is investigated. The external flow impinges normal to the heated plate and the surface temperature is assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the mixed convection parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.
Abstract: In this paper, we apply the Exp-function method to
Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara
equation is the combination of the Rosenau and standard Kawahara
equations and Rosenau-KdV equation is the combination of the
Rosenau and standard KdV equations. These equations are nonlinear
partial differential equations (NPDE) which play an important role
in mathematical physics. Exp-function method is easy, succinct and
powerful to implement to nonlinear partial differential equations
arising in mathematical physics. We mainly try to present an
application of Exp-function method and offer solutions for common
errors wich occur during some of the recent works.
Abstract: The coefficient diagram method is primarily an algebraic control design method whose objective is to easily obtain
a good controller with minimum user effort. As a matter of fact, if a
system model, in the form of linear differential equations, is known,
the user only need to define a time-constant and the controller order.
The later can be established regarding the expected disturbance type
via a lookup table first published by Koksal and Hamamci in 2004.
However an inaccuracy in this table was detected and pointed-out in
the present work. Moreover the above mentioned table was expanded
in order to enclose any k order type disturbance.
Abstract: In this paper, an analytical study is made for the dynamic behavior of human brain tissue under transient loading. In this analytical model the Mooney-Rivlin constitutive law is coupled with visco-elastic constitutive equations to take into account both the nonlinear and time-dependent mechanical behavior of brain tissue. Five ordinary differential equations representing the relationships of five main parameters (radial stress, circumferential stress, radial strain, circumferential strain, and particle velocity) are obtained by using the characteristic method to transform five partial differential equations (two continuity equations, one motion equation, and two constitutive equations). Analytical expressions of the attenuation properties for spherical wave in brain tissue are analytically derived. Numerical results are obtained based on the five ordinary differential equations. The mechanical responses (particle velocity and stress) of brain are compared at different radii including 5, 6, 10, 15 and 25 mm under four different input conditions. The results illustrate that loading curves types of the particle velocity significantly influences the stress in brain tissue. The understanding of the influence by the input loading cures can be used to reduce the potentially injury to brain under head impact by designing protective structures to control the loading curves types.
Abstract: In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential
(VID) equation is considered. The method is developed by means
of the Legendre wavelet approximation and collocation method. The
properties of Legendre wavelet together with Gaussian integration
method are utilized to reduce the problem to the solution of nonlinear
programming one. Some numerical examples are given to confirm the
accuracy and ease of implementation of the method.
Abstract: In this paper, Semi-orthogonal B-spline scaling
functions and wavelets and their dual functions are presented
to approximate the solutions of integro-differential equations.The
B-spline scaling functions and wavelets, their properties and the
operational matrices of derivative for this function are presented to
reduce the solution of integro-differential equations to the solution of
algebraic equations. Here we compute B-spline scaling functions of
degree 4 and their dual, then we will show that by using them we have
better approximation results for the solution of integro-differential
equations in comparison with less degrees of scaling functions
Abstract: In this paper, we have investigated the free convection MHD flow due to heat and mass transfer through porous medium bounded by an infinite vertical non-conducting porous plate with time dependent suction under the influence of uniform transverse magnetic field of strength H0. When Temperature (T) and Concentration (C) at the plate is oscillatory with time about a constant non-zero mean. The velocity distribution, the temperature distribution, co-efficient of skin friction and role of heat transfer is investigated. Here the partial differential equations are involved. Exact solution is not possible so approximate solution is obtained and various graphs are plotted.
Abstract: A theoretical approach to consider formation of chemical compound layer at the interface between initial substances A and B due to the interfacial interaction and diffusion is developed. It is considered situation when speed of interfacial interaction is large enough and diffusion of A-atoms through AB-layer is much more then diffusion of B-atoms. Atoms from A-layer diffuse toward B-atoms and form AB-atoms on the surface of B-layer. B-atoms are assumed to be immobile. The growth kinetics of the AB-layer is described by two differential equations with non-linear coupling, producing a good fit to the experimental data. It is shown that growth of the thickness of the AB-layer determines by dependence of chemical reaction rate on reactants concentration. In special case the thickness of the AB-layer can grow linearly or parabolically depending on that which of processes (interaction or the diffusion) controls the growth. The thickness of AB-layer as function of time is obtained. The moment of time (transition point) at which the linear growth are changed by parabolic is found.
Abstract: The problem of magnetohydrodynamics boundary layer flow and heat transfer on a permeable stretching surface in a second grade nanofluid under the effect of heat generation and partial slip is studied theoretically. The Brownian motion and thermophoresis effects are also considered. The boundary layer equations governed by the PDE’s are transformed into a set of ODE’s with the help of local similarity transformations. The differential equations are solved by variational finite element method. The effects of different controlling parameters on the flow field and heat transfer characteristics are examined. The numerical results for the dimensionless velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically. The comparison confirmed excellent agreement. The present study is of great interest in coating and suspensions, cooling of metallic plate, oils and grease, paper production, coal water or coal-oil slurries, heat exchangers technology, materials processing exploiting.
Abstract: The present analysis considers the steady stagnation point flow and heat transfer towards a permeable shrinking sheet in an upper-convected Maxwell (UCM) electrically conducting fluid, with a constant magnetic field applied in the transverse direction to flow and a local heat generation within the boundary layer, with a heat generation rate proportional to (T-T)p Using a similarity transformation, the governing system of partial differential equations is first transformed into a system of ordinary differential equations, which is then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results are obtained for the flow and thermal fields for various values of the stretching/shrinking parameter λ, the magnetic parameter M, the elastic parameter K, the Prandtl number Pr, the suction parameter s, the heat generation parameter Q, and the exponent p. The results indicate the existence of dual solutions for the shrinking sheet up to a critical value λc whose value depends on the value of M, K, and s. In the presence of internal heat absorption (Q
Abstract: In this paper, the steady laminar three-dimensional boundary layer flow and heat transfer of a copper (Cu)-water nanofluid in the vicinity of a permeable shrinking flat surface in an otherwise quiescent fluid is studied. The nanofluid mathematical model in which the effect of the nanoparticle volume fraction is taken into account is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c from Matlab. Dual solutions (upper and lower branch solutions) are found for the similarity boundary layer equations for a certain range of the suction parameter. A stability analysis has been performed to show which branch solutions are stable and physically realizable. The numerical results for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles are obtained, presented and discussed in detail for a range of various governing parameters.