The steady three-dimensional flow of condensation or spraying on inclined spinning disk is studied analytically. The governing nonlinear equations and their associated boundary conditions are transformed into the system of nonlinear ordinary differential equations. The series solution of the problem is obtained by utilizing the homotopy perturbation method (HPM). The velocity and temperature profiles are shown and the influence of Prandtl number on the heat transfer and Nusselt number is discussed in detail. The validity of our solutions is verified by the numerical results. Unlike free surface flows on an incline, this through flow is highly affected by the spray rate and the rotation of the disk.
The removal of a condensate liquid from a cooled, saturated vapor is important in engineering processes. Sparrow and Gregg [
Most of the scientific problems and phenomena are modeled by nonlinear ordinary or partial differential equations. In recent years, many powerful methods have been developed to construct explicit analytical solution of nonlinear differential equations. Among them, two analytical methods have drawn special attention, namely, the homotopy perturbation method (HPM) [
We know all perturbation methods require small parameter in nonlinear equation and the approximate solutions of equation containing this parameter are expressed as series expansions in the small parameter. Selection of small parameter requires a special skill. A proper choice of small parameter gives acceptable results, while an improper choice may result in incorrect solutions. The homotopy perturbation method, which is a coupling of the traditional perturbation method and homotopy in topology, does not require a small parameter in equation modeling phenomena. In recent years, the HPM has been successfully employed to solve many types of linear and nonlinear problems such as the quadratic Ricatti differential equation [
With the above discussion in mind, the purpose of the present paper is to examine analytically the problem of condensation or spraying on an inclined rotating disk. The governing equations here are highly nonlinear coupled differential equations, which are solved by using the homotopy perturbation method. In this way, the letter has been organized as follows. In Section
Figure
The schematic diagram of steady three-dimensional problem of condensation film on inclined rotating disk.
Now, for convenience, consider the following general nonlinear differential equation:
The operator
The convergence of series (
In a homotopy equation, what we are mainly concerned about are the auxiliary linear operator
According to the steps of the homotopy perturbation method,
We mean that it must be chosen in such a way that one has no difficulty in subsequently solving systems of resulting equations. It should be noted that this condition does not restrict
Strictly speaking, in constructing
There is no unique universal technique for choosing the initial approximation in iterative methods, but from previous works done on HPM [
For example, it can be chosen to be the solution to some part of the original equation, or it can be chosen from initial/boundary conditions.
Although this condition only can be checked after solving some of the first few equations of the resulting system, this is the criteria that has been used by many authors when they encountered different choices as an initial approximation.
An important point to note is that, whenever we apply an auxiliary parameter as a factor for second term of homotopy equation (
To investigate the explicit and totally analytic solutions of present problem by using HPM, we first define homotopy
Figures
The normalized radial velocity profiles for the rotating flow obtained by the 8th-order approximation of the HPM in comparison with the numerical solution, when
The normalized velocity profiles for the draining flow (
The normalized draining flow rate (
The normalized shear on the disk in the
The normalized temperature profiles (
The temperature gradient on the disk (
The Nusselt number (
In this paper, the homotopy perturbation method (HPM) was used for finding the totally analytic solutions of the system of nonlinear ordinary differential equations derived from similarity transform for the steady three-dimensional problem of fluid deposition on an inclined rotating disk. The analytical results depicted by the graphs are consistent with the graphs produced by the fourth-order Runge-Kutta method, and, therefore, further establish the reliability and effectiveness of the HPM solution. This method provides an analytical approximate solution without any assumption of linearization. This character is very important for systems with strong nonlinearities which could be extremely sensitive to small changes in parameters. The solution obtained by means of HPM is an infinite power series for appropriate initial approximation, which can be, in turn, expressed in a closed form. In this regard, the homotopy perturbation method is found to be a very useful analytic technique to get highly accurate and purely analytic solution to such kind of nonlinear problems.