Modified Bessel Equation Research Articles

Entropy generation effects with thermal radiation on MHD Casson fluid flow in porous micro-annulus

The study of entropy generation analysis on natural convection non-Newtonian (Casson) fluid flow between two concentric cylinders/tubes is evaluated in this article. The influence of the external magnetic field normal to the flow direction with thermal radiation is analyzed on the porous surface of cylinders under the condition of Casson velocity slip and thermal jumps. The non-dimensional momentum equation of Casson fluid (CF) is solved analytically by using a modified Bessel equation containing rational number order of first and second kinds, and the energy equation is solved by using the Cauchy-Euler equation. This analysis aims to find out the loss of energy of CF flow within two cylinders and present the combinations of some parameters through which it can be minimized. Volume flow rate, entropy heat generation and irreversibility parameter are evaluated through different parameters such as the Casson parameter (CP), magnetic field parameter, Reynolds number, Brinkman number and Grashof number. Velocity is diminished for the increment of CP [Formula: see text] Volume flow rate descends with the enhancement of M and [Formula: see text] Entropy generation increased with elevation of [Formula: see text] and [Formula: see text] but reverse effects are found through higher values of [Formula: see text] Moreover, [Formula: see text] is reduced due to magnetic field and group parameter. At the end, a comparison of the velocity profile for [Formula: see text] with Abbas et al. (2019) and the volume flow rate for M with Jha et al. (2015) is established.

Generalized Solutions of Ordinary Differential Equations Related to the Chebyshev Polynomial of the Second Kind

In this work, we employed the Laplace transform of right-sided distributions in conjunction with the power series method to obtain distributional solutions to the modified Bessel equation and its related equation, whose coefficients contain the parameters ν and γ. We demonstrated that the solutions can be expressed as finite linear combinations of the Dirac delta function and its derivatives, with the specific form depending on the values of ν and γ.

Effect of Treatment Parameter on Oscillatory Flow of Blood Through an Atherosclerotic Artery with Heat Transfer

This research work has been carried out to investigate the influence of treatment parameter on flow of blood in a stenosed artery in the presence of magnetic field with heat transfer. The momentum equation governing the flow field has been solved by scaling it to dimensionless structure with the aid of some dimensionless parameters. The equations have been analytically solved using modified Bessel equation and by the method of undetermined coefficients in order to obtain the temperature profile and velocity profile of the blood flow. The characteristics of the flow have been derived for a certain set of values RT ; Da; \theta; Gr; Re; Pr; \omega; \delta involved in the model analysis and are presented graphically with the help of software Mathematica. Moreover the velocity of the blood is adopting a wavy pattern as the values of the parameters vary. The study can be useful in providing a perception of the treatment caused by the superfluous consumption of fatty foods hence decreasing the risk of cancer, hypertension and many heart related diseases.

ХАРАКТЕРИСТИКИ КОЛЬЦЕВОГО РЕБРА ПРЯМОУГОЛЬНОГО ПРОФИЛЯ С ЭНЕРГОВЫДЕЛЕНИЕМ

The heat conduction equation for an annular fin with an arbitrary profile in the presence of energy release in the fin is obtained in the article. The resulting equation differs from the approximate equation given in the literature by the presence of energy release and a more accurate determination of the length of the arc element. As boundary conditions, the temperature of the base of the fin is set, and at the end of the fin, heat exchange occurs according to the Newton - Richmann law with the environment. The equation for the fin of a rectangular profile is an inhomogeneous modified Bessel equation. Its solution contains the Bessel functions of the imaginary argument of the first and second kind of zero order. The efficiency of the fin and the heat flow through the base of the fin are determined. The energy release in the fin increases its efficiency compared to the efficiency of the fin in the absence of energy release, and also reduces the heat flow. The restriction by the values of energy release in the fin is found as condition for the applicability of the finning. The fin efficiency must be less than one. If the efficiency exceeds one, the fin plays the opposite role: the flow is directed in the reverse side. In the article, an expression is obtained for the surface build-up coefficient kh. When calculating the heating (cooling) of a body with a finned surface, the heat transfer coefficient should be increased by kh times.

The heat equation for nanoconstrictions in 2D materials with Joule self-heating

We consider the heat equation for monolayer two-dimensional materials in the presence of heat flow into a substrate and Joule heating due to electrical current. We compare devices including a nanowire of constant width and a bow tie (or wedge) constriction of varying width, and we derive approximate one-dimensional heat equations for them; a bow tie constriction is described by the modified Bessel equation of zero order. We compare steady state analytic solutions of the approximate equations with numerical results obtained by a finite element method solution of the two-dimensional equation. Using these solutions, we describe the role of thermal conductivity, thermal boundary resistance with the substrate and device geometry. The temperature in a device at fixed potential difference will remain finite as the width shrinks, but will diverge for fixed current, logarithmically with width for the bow tie as compared to an inverse square dependence in a nanowire.

The Constructive Solution of the Fractal Composite Reservoir with Stress-Sensitivity Formation

The concentric two-zone composite reservoir model is a boundary value problems (BVPs) of modified Bessel equations. In this paper, we propose a constructive method to solve the BVPs for the system of modified Bessel equations with Robin mixed outer boundary condition and apply it to solve a two-zone fractal composite reservoir seepage model with stress-sensitivity formation. By using Pedrosa variable substitution, regular perturbation technique, Laplace transform, and Stehfest numerical inversion technique, the unified expression for the solutions of the reservoir model with three outer boundary (infinite, impermeable, and constant pressure) conditions is constructed. Type curves of bottom-hole pressure and pressure derivative are drawn, and sensitivity analysis of reservoir parameters are carried out. In comparison with the traditional approach, the solutions of this model are simple and regular, with continued fraction form, the constructive method is efficient and easy to operate. The application of this method avoids the complicated and trivial derivative operation and the use of Cramer’s rule to solve the system of linear equations. It can help to better understand the relationship between the solutions of the reservoir model and the inner and outer boundary conditions. The constructive method can be applied not only to solve the fractal composite reservoir model but also to solve more general reservoir model, BVPs of fluid diffusion, heat conduction, and so on.

Analytical solutions for sediment concentration in waves based on linear diffusivity

Two kinds of analytical solutions are derived through Laplace transform for the equation that governs wave-induced suspended sediment concentration with linear sediment diffusivity under two kinds of bottom boundary conditions, namely the reference concentration (Dirichlet) and pickup function (Numann), based on a variable transformation that is worked out to transform the governing equation into a modified Bessel equation. The ability of the two analytical solutions to describe the profiles of suspended sediment concentration is discussed by comparing with different experimental data. And it is demonstrated that the two analytical solutions can well describe the process of wave-induced suspended sediment concentration, including the amplitude and phase and vertical profile of sediment concentration. Furthermore, the solution with boundary condition of pickup function provides better results than that of reference concentration in terms of the phase-dependent variation of concentration.

The Design of a Frame-Like ZnO FBAR Sensor for Achieving Uniform Mass Sensitivity Distributions.

In this paper, an infinite circular ZnO thin film bulk acoustic resonator (FBAR) with a frame-like electrode operating at the thickness-extensional (TE) mode is studied. Two-dimensional scalar differential equations established for the problem in the Cartesian coordinate system are successfully solved by transforming them into normal Bessel equations and modified Bessel equations in the cylindrical coordinate system. Resonant frequencies and vibration distributions are obtained for this frame-like FBAR sensor. A nearly uniform mass sensitivity distribution in the active area is achieved by designing proper electrode size and mass ratio of the driving electrode to the ZnO film. Numerical results show that compared with the reported ring electrode FBAR sensor, the novel frame-like electrode FBAR can achieve a maximum optimization ratio (up to 97.90%) on the uniformity of the mass sensitivity distribution in the active area under the same structural parameters, which is also higher than the optimization ratio 77.63% obtained by the reported double-ring electrode design. Moreover, the mechanism to achieve a very uniform mass sensitivity distribution in the active area by the frame-like electrode is explained in detail according to dispersion curves. Namely, when the resonant frequency of the FBAR sensor is close to the cut-off frequency of the active region in the dispersion curve, the mass sensitivity distribution is nearly uniform. These conclusions provide a theoretical guidance for the design and optimization of ZnO FBAR mass sensors with high performance.

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software.

Unsteady flow of a Burgers’ fluid with Caputo fractional derivatives: A hybrid technique

Abstract The shear stress and velocity field related to the oscillatory motion of a fractional Burgers’ fluid model in an infinitely circular cylinder are examined using modified Bessel equation and Laplace transformation. The fluid is at rest initially and after t = 0 + , because of shear, it instantly starts to move along the axis of cylinder. The procured results are expressed in modified Bessel fields I 0 ( · ) and I 1 ( · ) and fulfill both initial and boundary conditions. Inverse Laplace transformation has been found numerically using Matlab software. In the end, numerical simulations have been performed to analyze the behavior of fractional parameter α , similarity parameter β , relaxation time λ 1 , retardation time λ 3 , radius of the circular cylinder R and material parameter λ 2 on our obtained solutions of velocity field and shear stress. The comparison between numerical and exact results are also presented in graphical and tabular form.

Semi Analytical Solutions for Fractional Oldroyd-B Fluid Through Rotating Annulus

In this paper, semi analytical solutions for velocity field and tangential stress correspond to fractional Oldroyd-B fluid, in an annulus, are acquired by Laplace transforms and modified Bessel equation. In the beginning, cylinders are stationary, motion is produced after t = 0 when both cylinders start rotating about their common axis. The governing equations solved for velocity field and shear stress by using the Laplace transform technique. The inverse Laplace transform is alternately calculated by Stehfest's algorithm using “MATHCAD” numerically. The numerically obtained solutions are in the form of modified Bessel's equations of first and second kind and satisfying all the imposed physical conditions. Finally, there is a comparison between exact and obtained solutions. It is observed that semi analytical technique and exact technique are approximately the same and satisfy imposed boundary conditions. Through graphs, the impact of physical parameters (relaxation time, retardation time kinematic viscosity, and dynamic viscosity) and fractional parameters on both velocity and shear stress is observed.

A Semianalytical Gradient Model for Characterization of Conductors With Surface Roughness

In this paper, a novel semianalytical gradient model (SAGM) for characterizing conductors with surface roughness is presented. Based on the linear function of conductivity, the governing equation of the magnetic field is reduced to the modified Bessel equation, which has an analytical solution. With the piecewise linear approximation technique, the SAGM is derived to deal with arbitrary functions of conductivity. Three typical distributions, including the uniform, normal, and Rayleigh distributions, are investigated. The proposed SAGM is employed to evaluate the propagation properties of typical transmission line structures with different surface roughness profiles. The simulation results agree well with those from measurements, which confirm the validity and accuracy of the proposed method.

Approximate analytical treatment of annular fins of rectangular profile for teaching fin heat transfer: Utilization of the mean value theorem for integrals

In the teaching of annular fins of rectangular profile, the main obstacle lies in solving the quasi one-dimensional heat conduction equation, the modified Bessel equation of first kind. In the modified Bessel equation, the variable coefficient [Formula: see text] multiplying the first order derivative of temperature [Formula: see text] is problematic. The principal objective of the present paper on engineering education is to solve the modified Bessel equation of first kind in approximate analytical form. Specifically, we seek to apply the mean value theorem for integrals to the variable coefficient [Formula: see text], viewed as an auxiliary function in the domain of the annular fin extending from the inner radius [Formula: see text] to the outer radius [Formula: see text]. It is demonstrated in a convincing manner that approximate analytical temperature distributions having exponential functions are easy to obtain without resorting to the exact analytical temperature distribution embodying four modified Bessel functions of first kind. Furthermore, the easiness in calculating heat transfer rates in annular fins of rectangular profile for realistic combinations of the two controlling parameters: the normalized radius ratio and the dimensionless thermo-geometrical parameter is verifiable.

Approximate, analytical procedure for rectangular annular fins by accommodating the Cauchy–Euler equation

An approximate, analytical treatment is presented for the rectangular annular fin by transforming the complicated modified Bessel equation of zero order into a rudimentary Cauchy-Euler equation. The essential step in the computational procedure revolves around a simple manipulation of the radial coordinate that sets up a variable coefficient in the third term of the modified Bessel equation of zero order. In the third term, the radial variable will be replaced by the mean radius of the inner and outer radius, whereas the radial variable prevails in the first and second terms. This action paves the way to the easier Cauchy-Euler equation. For a collection of rectangular annular fins of interest in engineering applications, approximate, analytical temperature distributions and heat transfer rates (via the fin efficiency) written in terms of two binomials demonstrate excellent quality levels in all cases. Additionally, relative error distributions are presented in detailed manner using as the baseline cases the classical exact, analytical temperature distributions and heat transfer rates expressed in terms of the complicated modified Bessel functions of first and second kind.

On nabla discrete fractional calculus operator for a modified Bessel equation

In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the nabla-discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method

Semi-analytical technique for the solution of fractional Maxwell fluid

In this work, the flow of a fractional Maxwell fluid is discussed. The velocity function and time-dependent shear stress of a Maxwell fluid with fractional derivatives are calculated. It is considered that the fluid in the infinitely long circular cylinder is moving with a velocity ft. The fluid in the infinitely long circular cylinder of radius R is initially at rest and at t = 0+, because of shear, it instantly starts to move longitudinally. To obtain the solutions, we have employed Laplace transformation and modified Bessel equation. The solutions are in series form, which are expressed in terms of modified Bessel functions [Formula: see text] and [Formula: see text], and satisfy all given conditions. In this paper, Laplace inverse transformation has been calculated numerically by using MATLAB. The behavior of the following physical parameters on the flow are investigated: relaxation time, dynamic viscosity, kinematics viscosity, similarity parameters of fractional derivatives and radius of the circular cylinder. Finally, the impact of the fractional parameter and material elements is shown by graphical demonstration.

Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress

The velocity field and shear stress for unsteady flow of a second grade fluid with fractional derivatives through an infinite long circular cylinder are evaluated. The fluid is initially at rest and at t=0+, the fluid inside the cylinder starts to move longitudinally due to tangential shear stress. Semi analytical solutions are obtained with the help of Laplace transformation and modified Bessel equation. This hybrid technique, which we have used has less computational efforts and time cost as compared to other schemes that are commonly used. The solutions in the transformed domain, are presented in terms of modified Bessel functions I0(·) and I1(·) and satisfy all given conditions. Inverse Laplace transformations have been numerically calculated by using MATLAB. The semi analytical solutions for the motion of second grade fluid with fractional derivatives are reduced to the similar solutions for ordinary second grade and Newtonian fluid. Finally, the effect of different parameters of the flow is graphically examined.

Self-localized states for electron transfer in nonlocal continuum deformable media

We consider the problem of electron transport in a deformable continuum medium subjected to an external harmonic substrate potential. We then consider the quasi-stationary state of the full problem to find a Gross–Pitaevskii type equation with a nonlocal external potential, which is solved by variational and numerical means (considered as the exact solution) to find the parameter conditions for the existence of self-localized solutions. The variational approach predicts a threshold on the on-site or nonlocality parameter where localized solutions cease to exist from the Non-Linear Schrödinger soliton limit. A numerical continuation of stationary state solutions in the corresponding discrete system is used to confirm the prediction of the turning value in the on-site term. We finally study the full stationary state and make use of an approximation, proposed by Briedis et al. [17], for the nonlocal term, corresponding to strong nonlocalities, to find analytic expressions for self-localized states in terms of the series solutions of a nonlinear modified Bessel equation.

An analytical investigation for oscillations in a harbor of a parabolic bottom

Based on the linear shallow water approximation, longitudinal and transverse oscillations in a rectangular harbor with a parabolic bottom are analyzed. The longitudinal ones are combinations of the Legendre functions of the first and second kinds and the transverse ones are expressed with modified Bessel equations. Analytic results for longitudinal oscillations show that the augmentation of rapidity of variation of the water depth shifts the resonant wave frequencies to larger values and slightly changes the positions of the nodes for the resonant modes. For the transverse oscillations trapped within the harbor which are typically standing edge waves, the dispersion relationship is derived and the spatial structures of the first four modes are presented. The solutions illustrate that all the trapped modes are affected by the varying water depth parameters, especially for the higher modes whose profiles extend farther and the distribution of the energy of transverse oscillations is influenced by the rapidity of variation of the bottom within the harbor.