# Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equations of Motion

_{0}is the constant magnetic field applied in the z-direction, φ is the porosity, k

_{0}is the permeability of the porous medium, c

_{p}is the specific heat, T is the temperature and λ is the thermal diffusivity.

_{b}and the temperature of the inner cylinder at T

_{s}, and considering the temperature T of the following form:

## 3. Basics of the Optimal Auxiliary Functions Method (OAFM)

_{f}(η) is an unknown function, H is the domain of interest, B is a boundary operator and F(η) is an unknown function at this stage. In most of the cases, an exact solution for strongly nonlinear equations of types (1) and (2) is hard to find [23].

_{0}(η) and the first approximation F

_{1}(η,C

_{i}) are determined as follows. We chosethe initial approximation F

_{0}in order to satisfy the initial/boundary conditions

_{i}, i = 1,2,…, p are unknown parameters at this moment.

_{i}are arbitrary unknown parameters and f

_{i}(η) are functions depending on the initial approximation F

_{0}(η), on the functions which appear in N[F

_{0}(η)], or are combinations of such expressions. These auxiliary functions f

_{i}are very important and are not unique, and it should be emphasized that we have much freedom to choose such auxiliary functions. Using the previous considerations, for instance if F

_{0}(η) and N[F

_{0}(η)] are polynomial functions, then f

_{i}are sums of polynomial functions; if F

_{0}(η) and N[F

_{0}(η)] are exponential (logarithmic) functions, then f

_{i}are sums of exponential (logarithmic) functions, respectively; if F

_{0}(η) are trigonometric functions and N[F

_{0}(η)] are polynomial functions, then f

_{i}are sums of combinations of trigonometric and polynomial functions, and so on. In conclusion, the auxiliary functions are of the same form like F

_{0}(η) and N[F

_{0}(η)]. In other words, F

_{0}(η) and N[F

_{0}(η)] are “source” for the auxiliary functions f

_{i}.

_{i}can be optimally identified via rigorous methods, and we have much freedom to choose between the last square method, Ritz method, collocation method, Galerkin method, or Kantorowich method or by minimizing the square residual error:

_{i}are obtained from the following system:

_{i}.

## 4. Application of the Optimal Auxiliary Functions Method

_{0}given from Equation (22) are not unique. In the following, we present only three possibilities to choose the linear operators and the initial known function F

_{f}(η) for Equations (9)–(11).

_{1}(η) from Equation (32) are obtained from Equations (41) and (59):

_{i}to the first approximations g

_{1}from Equation (33) are the following:

_{1}is obtained from Equation (28):

_{1}can be obtained from the following:

_{0}, q

_{0}and s

_{0}can be obtained from the following equations:

_{i}can be written as follows:

_{1}, q

_{1}and s

_{1}are obtained from the following equations:

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Geometry of the problem: R is the radius of the inner cylinder, bR is the radius of the outer cylinder, Ω is the angular velocity of the inner cylinder and w is the velocity in the axial direction.

**Figure 3.**Comparison between the approximate solution (73) and numerical results for the velocity profile f’.

**Figure 5.**Comparison between the approximate solution (74) and numerical results for the velocity profile g’.

**Figure 7.**Comparison between the approximate solution (75) and numerical results for the velocity profile h’.

**Figure 10.**Comparison between the approximate solution (100) and numerical integration results for s.

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**MDPI and ACS Style**

Marinca, V.; Herisanu, N.
Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder. *Symmetry* **2020**, *12*, 1335.
https://doi.org/10.3390/sym12081335

**AMA Style**

Marinca V, Herisanu N.
Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder. *Symmetry*. 2020; 12(8):1335.
https://doi.org/10.3390/sym12081335

**Chicago/Turabian Style**

Marinca, Vasile, and Nicolae Herisanu.
2020. "Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder" *Symmetry* 12, no. 8: 1335.
https://doi.org/10.3390/sym12081335