The problem of the numerical analysis of natural convection from a spinning cone with variable wall temperature, viscous dissipation and pressure work effect is studied. The numerical method used is based on the spectral analysis. The method used to solve the system of partial differential equations is the multidomain bivariate spectral quasilinearization method (MDBSQLM). The numerical method is compared with other methods in the literature, and the results show that the MDBSQLM is robust and accurate. The method is also stable for large parameters. The numerical errors do not deteriorate with increasing iterations for different values of all parameters. The numerical error size is of the order of
The study of numerical methods in fluid flow has been limited to a very small number of methods used in the past several decades. The common methods that have been used are the finite difference based methods such as Kellerbox, finite element and finite volume. Other finitebased methods are commercially developed methods such as computational fluid dynamics (CFD) [1]. Other methods that are common are analytical methods, and these methods are only applied in simple cases arising in fluid flow models.
Analytical methods have been used in many studies, these include the work of Makinde [2], in which a simple model of entropy generation analysis was considered. Rees et al. [3] worked on an analytical solution for boundary layer flows for Bingham fluids. Other analytical solutions appear in the works involving unsteady fluid flow of third grade fluid [4] and in other nonNewtonian fluids over twodimensional porous media [5]. Silva et al. [6] used analytical solution to study power law fluids in porous media. With the increase in complexity of fluid mathematical models, analytical solutions are becoming rarely used. The use of numerical methods has been on the increase.
Numerical methods have been used in many studies in fluid flow, which include the work of Cheng [7] in which the cubic spline method was used. Altun et al. [8] used the finite difference approach with discretization using the central difference scheme. The work of Ashari and Tafreshi [9] made use of FlexPDE, which is a commercial program based on finite element method. Other works in which finite difference methods were used include Ece [10] who used the Thomas algorithm on the flow around spinning cone, Haddout et al. [11] in fluid flow involving viscous dissipation and pressure work, Javaherdeh et al. [12] in free convection fluid flow on a moving vertical plate, Kefayati [13] used the finite difference lattice Boltzmann method, Rosali et al. [14] used the Kellerbox method in the study of flow of micropolar fluids and Teymourtash et al. [15] used the finite difference method in free convection in supercritical fluids. The homotopy analysis method was used in the study of fluid flow by Dinarvand et al. [16], Tylimazoglu [17], Abd Elmaboud et al. [18] and others. Haque et al. [19] used the Nachtshim–Swigert iteration technique in the study of MHD fluid flow in micropolar fluid.
Some studies in fluid flow are based on results from actual laboratory experiments. These studies include among others the work of Markicevic et al. [20] who studied capillary force driven flows in porous medium and Pinar et al. [21] who investigated flow structure around shallow waters.
In all the aforementioned studies, none of them used spectralbased methods. There are some recently developed spectralbased methods such as the spectral quasilinearization method (SQLM), BQLM, bivariate local linearization methods studies, among others [22,23,24]. These methods are based on approximating the derivative of functions at collocation points by the Lagrange or Chebyshev polynomials. The procedure is less rigorous than in the case of finite difference methods. In this study, the system of partial differential equations (PDEs) is solved using the multidomain bivariate quasilinearization method (MDBSQLM).
This article is organized as follows: in Section 2, mathematical formulation of the problem is given. In Section 3, the method of solution in which the solution method is described is given. In Section 4, results and discussion are given, and finally in Section 5 the conclusion is given.
A spinning cone in a Newtonian fluid is considered and maintained at a variable temperature
The governing equations in this buoyantdriven flow are given as:
We introduce the nondimensional variables
Substituting equations (7)–(8) in (1)–(4) gives the following equations:
In this section, we describe the implementation of the MDBSQLM. The method considers the use of several domains as compared to the traditional single domain system. In general, the system of nonlinear PDEs is given as
The operators are of the form:
The primes refer to the derivative with respect to
To apply the MDBSQLM, we decompose the interval for
The full description of this method is described in the study by Magagula et al. [22]. Applying the method with
The system of PDEs (9)–(12) is written in the form:
Using the matrix system described in Magagula et al. [22], the solution to the systems (9)–(12) is obtained.
The problem considered in this article is a coupled system of PDEs (9)–(12) describing fluid flow around a spinning cone. In this section, the effect of changing temperature exponent
Ece [10]  Makanda [26] (SQLM)  MDBSQLM  








1  0.681483  0.638855  0.68148334  0.63885473  0.68148022  0.63885087 
10  0.68148334  0.63885473 
In Table 1, it is shown that using more than one domain resulted in obtaining accurate results.
Figure 2 depicts the effect of increasing
Figure 3 shows the effect of increasing
Figure 4 depicts the effect of increasing
In this study, it is of interest to investigate the effect of varying the suction parameter
In Figures 5–7, it is shown that increasing
In this study, the effect of varying the temperature exponent
Figures 8 and 9 show that the errors for
Figures 10 and 11 depict that the errors for
In Figures 12–13, the error graphs for
A spinning cone with variable surface temperature, suction and pressure work effect was considered. A system of PDEs describing the fluid flow around a spinning cone was solved using the MDBSQLM. The numerical method was shown to be robust and accurate by comparison with other results obtained in the literature. Considering more domains makes this method unique and accurate. The method can be preferred in solving PDEs arising in fluid flow with large parameters.
The effects of increasing the suction parameter involving large values resulted in the decrease of velocity, spin velocity and temperature profiles. The errors for all the solutions for
Conflict of interest: Authors state no conflict of interest.
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