Analytical Solution for Fluid Flow and Heat Transfer in a Three-Dimensional Inclined Horizontal Channel and Under The Influence of Thermal Radiation

Abstract


I. INTRODUCTION
Numerous engineering applications, including transpiration cooling, drag reduction, thrust bearing, and design of radial diffusers, benefit from the study of heat transport. Typically, fluids are utilized to convey heat in industrial and transportation systems for heating and cooling purposes. It is also noted that academics have been interested in the stretching sheet for a long time. The physical phenomena and heat transmissions across a stretching plate have been the subject of several studies by researchers. Numerous significant industrial production processes use it, such as the manufacturing of glass filters, the extrusion of plastic sheets, and the condensation of metallic plates. The quality of the finished product depends heavily on the skin friction coefficient and the rate of surface heat transfer, thus the research of flow and heat transfer is crucial. The study of flow over a stretching sheet has recently been expanded to include many diverse scenarios, making it more intriguing [12]. Anurag, J. P. Maurya and A. K. Singh utilized finite Hankel transform to solve one-dimensional convection heat transfer problem containing the magnetic field [1]. Aziz-Ur-Rehman, Muhammad Bilal Riaz, Syed Tauseef Saeed and Shaowen Yao applied Laplace transformation to solve the magnetohydrodynamic problem [2]. E. N. Mac ̂do, R. M. Cotta and H. R. B. Orlande also utilized generalized integral transform technique to get the solution of convection and radiation problem [3]. F. A. A. Gomes, J. B. C. Silva and A. J. Diniz also used generalized integral transform technique to solve radiation heat transfer problem [4]. Gilvan do Nascimento Filho, Jakler Nichele and Leonardo Santos de Brito Alves used classical integral transform technique to solve one-dimensional time dependent heat conduction problem in no-ablation time period [5].Laplace transform technique implemented by Muhammad Iftikhar, Zubair Ahmad, Saqib Murtaza, Ibn e Ali and Ilyas Khan to solve radiation heat transfer problem that was formed by Caputo-Fabrizio fractional operator [9]. N.T. Eldabe, M. El-Shahed and M. Shawkey used the Laplace transform and generalized finite Hankel transform to solve the equation of unsteady flow through a concentric annulus [13]. In this paper we will solve three-dimensional radiation heat transfer problem in Cartesian coordinate by using quadruple Laplace transform.

II. The Model and Mathematical Method:
The governing equations and illustration of problem are [14] Take : Now after combine navier-stokes equations (2), (3)and (4) and simplicity we get [7]:

With boundary and initial conditions:
Apply quadruple Laplace transform to equation (9): Then we get: Take : From energy equation solution (8): Then:

III. Results:
By use Matlab we get results represented by the following figures: Figure (1-2) shows distribution of temperature for Figure (1-3) shows distribution of temperature for , .

IIII. Conclusions:
We notice the temperature distribution is gradual and in the form of a plate and as the value of z increases the value of the temperature increases, that is clear from the figures (1-2),(1-3) and (1-4) while temperature is maximum, when temperature is decrease and at temperature is minimum. It was also shown that the fluid flow is gradually and in the form of a plate and as the value of z increases the value of the fluid flow increases with incline angle , that is clear from the figures (1-5),(1-6)and(1-7) while fluid flow is maximum, when fluid flow is decrease and at fluid flow is minimum, That means the increases in fluid flow is related to the increases in temperature and vice versa. Figure (1-8) shows effect of radiation parameter which show that, when value of increase then temperature increase gradually.