Thermal characteristics of longitudinal fin with Fourier and non-Fourier heat transfer by Fourier sine transforms

The quest for high-performance of heat transfer components on the basis of accommodating shapes, smaller weights, lower costs and little volume has significantly diverted the industries for the enhancement of heat dissipation with variable thermal properties of fins. This manuscript proposes the fractional modeling of Fourier and non-Fourier heat transfer of longitudinal fin via non-singular fractional approach. The configuration of longitudinal fin in terms of one dimension is developed for the mathematical model of parabolic and hyperbolic heat transfer equations. By considering the Fourier and non-Fourier heat transfer from longitudinal fin, the mathematical techniques of Fourier sine and Laplace transforms have been invoked. An analytic approach is tackled for handling the governing equation through special functions for the fractionalized parabolic and hyperbolic heat transfer equations in longitudinal fin. For the sake of comparative analysis of parabolic verses hyperbolic heat conduction of fin temperature, we depicted the distinct graphical illustrations; for instance, 2-dimensional graph, bar chart, contour graphs, heat graph, 3-dimensional graphs and column graphs on for the variants of different rheological impacts of longitudinal fin.

A Note on a Fourier Sine Transform

This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results are new.

Modeling electro-osmotic flow and thermal transport of Caputo fractional Burgers fluid through a micro-channel

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

Natural frequency and critical velocities of heated inclined pinned PP-R pipe conveying fluid

Purpose: The flow velocity and pressure of fluid flowing through a pipeline can cause the vibration of pipes, and consequently result in the modification in natural frequency via fluid-structure interaction. The value of the natural frequency of a component when approaches the excitation force to a certain degree, a severe resonance failure may occur. Hence, avoiding the resonance failure of a pipe subjected to complex conditions is an essential issue that requires to be solved urgently in the engineering field. This work treats the transverse vibration for flexible inclined heated pipe, made of polypropylene randomcopolymer (PP-R), conveying fluid assuming pinned connections at the ends. The pipe was placed at different support angles and subjected to variant temperatures. Design/methodology/approach: The inclined pipe is modelled as Euler-Bernoulli beam taking into account its self-weight, temperature variation, inclination angle, aspect ratio, and internal fluid velocity. The integral transforms method, which includes the finite Fourier sine and the Laplace transforms, was used to develop an analytic solution to the modified equation of motion and the analytical expressions for dual natural frequencies of the pipefluid interaction system were computed. Findings: The proposed solution technique via finite Fourier sine and Laplace transforms offers a more convenient alternative to calculate the dynamic characteristic of pipes conveying fluid. The obtained results showed that the dynamical behaviour of pipe–fluid system is strongly affected by fluid flow velocity, degree of inclination, temperature variation, and aspect ratio of the pipe in transverse modes. Research limitations/implications: This work focuses on fundamental (first) mode in the most discussions. Practical implications: It was revealed that the thermal effects in the pipe are a very important factor and more significant in comparison with the internal fluid velocity and the inclination angle has a larger impact on vibration characteristics at a higher aspect ratio. The findings can be useful for the design of engineering components. Originality/value: Determining the combining effect of inclination angle, aspect ratio, and thermal loading on vibration characteristic of the pipes conveying fluid by using an improved analytic solution to the modified equation of motion via mixed of finite Fourier sine and Laplace transforms.

Tích chập suy rộng với hàm trọng g(y) = sin ay đối với các phép biến đổi tích phân Fourier cosine và Fourier sine

Tích chập suy rộng mới với hàm trọng đối với hai phép biến đổi tích phân Fourier cosine và Fourier sine được chúng tôi xây dựng và nghiên cứu trong bài báo này. Chúng tôi chứng minh sự tồn tại của tích chập suy rộng mới này trong không gian L(R¬+). Đẳng thức nhân tử hóa cốt yếu với sự có mặt của hai phép biến đổi tích phân khác biệt là Fourier cosine, Fourier sine và hàm trọng cùng một số tính chất khác như tính không giao hoán, tính không kết hợp khác với các tích chập của một phép biến đổi tích phân được phát biểu và chứng minh. Cuối cùng là áp dụng tích chập suy rộng mới được xây dựng để giải hệ phương trình tích phân kiểu Toeplitz-Hankel và nhận được nghiệm dưới dạng đóng. Trong ba thập niên trở lại đây, tích chập suy rộng được các nhà toán học quốc tế và trong nước quan tâm nghiên cứu. Đồng thời các nhà toán học cũng ứng dụng chúng trong việc giải các bài toán về phương trình tích phân, phương trình vi tích phân,… Vì vậy, việc nghiên cứu tích chập suy rộng là vấn đề thời sự. Do đó, nhóm tác giả chúng tôi đã viết bài báo này.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.