Nonsimilar solutions for double diffusive convection near a frustum of a wavy cone in porous media
Introduction
Double diffusive natural convection flow in a fluid-saturated porous medium may be met in geophysical, geothermal and industrial applications, such as the migration of moisture through air contained in fibrous insulations and the underground spreading of chemical contaminants through water-saturated soil.
Bejan and Khair [1] examined the free convection boundary layer flow driven by both temperature and concentration gradients. Lai and Kulacki [2] studied the natural convection boundary layer along a vertical surface with constant heat and mass flux and the effect of wall injection. Yih [3] studied the heat and mass transfer driven by natural convection from a truncated cone embedded in a porous medium with variable wall temperature and concentration or with variable heat and mass flux.
The study of heat or mass transfer near irregular surfaces is of fundamental importance; that is because it is often met in many practical applications. Yao [4] studied the natural convection heat transfer from isothermal vertical wavy surfaces, such as sinusoidal surfaces, in Newtonian fluids. Chiu and Chou [5] examined the transient free convection heat transfer from a vertical wavy surface in micropolar fluids. Rees and Pop [6] studied the natural convection flow over a vertical wavy surface with constant wall temperature in porous media. Pop and Na [7] studied the natural convection flow along a vertical wavy cone in porous media. Pop and Na [8] studied natural convection over a frustum of a wavy cone with constant wall temperature in a porous medium Hossain and Rees [9] studied the heat and mass transfer in natural convection flow along a vertical wavy surface with constant wall temperature and concentration in Newtonian fluids. Cheng [10] studied the problem of free convection heat and mass transfer near a wavy cone in a porous medium. Wang [11] studied the effect of thermophoresis on particle deposition rate from a natural convection flow onto a vertical wavy plate Wang and Chen [12] examined the thermophoretic deposition of particle from a boundary layer flow onto a continuously moving wavy surface.
In this paper, we want to extend the works of Pop and Na [8] and Cheng [10] to study the heat and mass transfer along the surface of a frustum of a wavy cone with constant wall temperature and concentration. This work use a coordinate transformation to transform a wavy conical surface to a smooth conical surface, and the transformed nonsimilar boundary layer governing equations are then solved by the cubic spline collocation method [13], [14]. The effects of the Lewis number, the buoyancy ratio, the half cone angle and the amplitude–wavelength ratio on the local Sherwood number and the local Nusselt number for natural convection heat and mass transfer near a vertical frustum of a wavy cone in a fluid-saturated porous medium are carefully examined.
Section snippets
Analysis
Consider the boundary layer flow near a vertical frustum of a wavy cone in a porous medium saturated with a Newtonian fluid as shown in Fig. 1. The wavy surface profile is given bywhere is the amplitude of the wavy surface, is the slant height at the lower end of the cone, and 2l is the characteristic length of the wavy surface. The origin of the coordinate system is placed at the apex of the wavy frustum cone. The surface of the wavy frustum cone is maintained
Numerical method
Here we first use the cubic spline collocation method [13], [14] to solve the governing equations (25), (26) and their boundary conditions (27), (28). Note that the steady-state solution can be calculated by using a pseudo-transient formulation in which a false transient term is added to Eqs. (25), (26). The Simpson’s rule for variable grids is then used to obtain the value of f at every position from Eq. (24) and its corresponding boundary conditions (27), (28). At every grid point, this cycle
Results and discussion
To assess the accuracy of the solution, the present results are compared with the results obtained by other researchers. Table 2 shows the local Nusselt number for ξ0 = 1, a = 0 and N = 0, the conditions for heat transfer of an isothermal truncated cone with smooth surfaces, obtained by the present method and the results reported by Yih [3] and Cheng et al. [15]. It is shown that the present results are in excellent agreement with the results obtained by Yih [3] and Cheng et al. [15].
Conclusions
The coupled heat and mass transfer by natural convection of a Darcian fluid flow along a vertical frustum of a wavy cone has been studied. Here a coordinate transformation is employed to transform a complex wavy conical surface to a smooth conical surface, and the transformed nonsimilar boundary layer equations are then solved by the cubic spline collocation method. The effects of the half cone angle, the amplitude–wavelength ratio, the buoyancy ratio, and the Lewis number on the Sherwood and
Acknowledgement
This work was supported by National Science Council of Republic of China under the grant no. NSC 94-2212-E-218-016.
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