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We examined how injection/suction impacts the flow characteristics for mixed convection across a vertical cone saturated porous medium in the presence of double dispersion and chemical reaction effects. We perform suitable transformations to convert the nonlinear system of partial differential expressions into a system of non-dimensional form and received dimensionless equations solved numerically by the bivariate Chebyshev spectral collocation quasi-linearization method. We explain the outcomes of the flow characteristics over various variables through diagrams and numerical benchmarks. We also establish precision verification of the chosen numerical technique through a comparison with prior published computations and found to be in great assent. The residual analysis section also illustrated which unblocks convergence of the present results.
mixed convection, vertical cone, injection/suction effect, double dispersion effect, chemical reaction effect, bivariate Chebyshev spectral collocation quasi-linearization method (BSCQLM)
Flow across cone-shaped frames ordinarily encountered which has brought significant attention in many engineering applications, the physical phenomenon over a vertical cone takes place in heat exchangers, geothermal reservoirs, nuclear reactor cooling systems and style of canisters for nuclear waste disposal, etc. Mixed convection flow through saturated porous media is another essential field that attracted in the eyes of the researcher's many researchers who analyzed it on various geometries. Magnetohydrodynamic study over a vertical plated was proposed by Aldoss et al. [1]. Khanafer and Chamkha [2] introduce flow in a lid-driven filled with fluid. Flow over a vertical wedge offered by Kumari et al. [3]. Aydın and Kaya [4] introduced MHD flow over a permeable vertical plate. A fully developed flow between inclined parallel plates presented by Cimpean et al. [5]. Mukhopadhyay [6] offered flow along with a stretching cylinder and we can view more appropriate work as [7-11].
Mass transfer through a wall slot into the boundary layer is enthusiastic because of several prospective applications including thermal protection, energizing the inner section of the boundary layer region in adverse pressure gradient and skin friction reduction in control surfaces. Mass transfer into a slot completely impacts the progress of the boundary layer along the surface and can inhibit or at least limit the separation of the viscous region. Some analysts offered it on various bodies. Flow in a non-Darcian fluid-saturated porous enclosure under multiple suction effects was offered by Kumar and Murthy [12]. Daba and Devaraj [13] analyzed over a vertical stretching sheet of unsteady double-diffusive flow. Unsteady flow through a moving vertical slender cylinder with the viscous dissipation offered by Patil et al. [14]. Murthy et al. [15] presented a parallel finite element study of 3D flow in a cubic porous enclosure. The unsteady flow over a vertical cone introduced by Roy et al. [16]. Ravindran and Ganapathirao [17] presented non-uniform slot suction/injection over a vertical cone. Ganapathirao et al. [18] introduced a study of unsteady flow over a vertical cone with non-uniform slot suction (injection).
The impact of chemical reactions on the flow is also an imperative field because of its key role in the various sections like the design of equipment of hydrometallurgical and chemical processing industries, food processing plants, cooling towers, damage of crops for freezing, combustion processes and other engineering processes. Several researchers analyzed it on different bodies. We can see a few of which here. An experimental study of miscible viscous fingering with viscosity changes induced by variations was examined by Nagatsu et al. [19]. Patil and Kulkarni [20] introduced free convection flow through porous media of a polar fluid with internal heat generation. Flow over a vertical truncated cone with heat generation/absorption and variable viscosity effects on the double-diffusive convection analyzed by Mahdy [21]. Ravindran et al. [22] proposed unsteady MHD flow over a vertical cone with heat generation/absorption and non-uniform slot mass transfer. Unsteady flow over a wedge with suction/injection and heat generation/absorption introduced by Ganapathirao et al. [23]. We can view more relevant work on the chemical reaction effect [24-31].
The study of the combined impact of injection/suction and chemical reaction effects on the mixed convection flow across a vertical cone saturated porous medium in the presentation of double dispersion effect did not recognize so far and through this, we aim it. We code the set of equations of the flow model in MATLAB via the bivariate Chebyshev spectral collocation quasi-linearization method and the computations stated in the results and discussions section with the significant applications. It establishes the precision verification of the chosen numerical technique through a comparison with previously published Kumari et al. [32] computations and found to be in great assent. The residual analysis section also illustrated which unblocks convergence of the present results.
This work is organized in the following pattern: Section 2 presents the mathematical formulation of the model, section 3 proposes the numerical simulation of the problem, while section 4 unfolds the numerical computations and we concluded in section 5.
Consider a vertical cone with half-angle δ saturated with Darcy-Brinkman porous model in a steady, laminar, mixed convection flow of an incompressible viscous fluid. The cone surface with its generator is aligned with x-axis and y-axis is normal to the surface; the flow moves in an upward direction, parallel to the axis of the cone. The walls of the cone are supposed to be permeable so that the fluid can inject/sucted with velocity v_{w}(x). A schematic representation of the flow model and coordinates system is sketched in Figure 1.
Figure 1. Flow model and coordinates system
The governing equations of the model are formulated as follows, by considering the work [17, 22, 32].
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
$\begin{align} & u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U\frac{dU}{dx}+\frac{{{\mu }_{e}}}{\rho }\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \\ & -\frac{\mu }{\rho K}u+g\cos \delta \left( {{\beta }_{T}}(T-{{T}_{\infty }})-{{\beta }_{C}}(C-{{C}_{\infty }}) \right) \\ \end{align}$ (2)
$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{\partial }{\partial y}\left( {{\alpha }_{e}}\frac{\partial T}{\partial y} \right)$ (3)
$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{\partial }{\partial y}\left( {{D}_{e}}\frac{\partial C}{\partial y} \right)-{{K}_{c}}\left( C-{{C}_{\infty }} \right)$ (4)
The boundary conditions are:
$u=0, \quad v=v_{w}, T(\mathrm{x}, \mathrm{y})=T_{w}, \quad C(\mathrm{x}, \mathrm{y})=C_{w}$ at $y=0$
$u \rightarrow U=u_{\infty} x^{m}, T(\mathrm{x}, \mathrm{y}) \rightarrow T_{\infty}, \quad C(\mathrm{x}, \mathrm{y}) \rightarrow C_{\infty}, \mathrm{as}$
$\mathrm{y} \rightarrow \infty$ (5)
Introducing the following non-dimensional variables:where, u and v are velocity components in x and y directions. δ is the half-angle of the cone, T is the fluid temperature, C is the concentration, v is the kinematic viscosity, g is the gravitational acceleration, K is the permeability coefficient of the porous medium; K_{c} is the chemical reaction rate, β_{T} and β_{C} are volumetric coefficients of the thermal and concentration expansions, respectively. α_{e} and D_{e} are the effective thermal and Solutal diffusivities which can be write as $\alpha_{e}=\alpha+\gamma d u$ and $D_{e}=D+\zeta d u$, where α and D are the molecular thermal and Solutal diffusivities, respectively.
$\xi(\mathrm{x})=\left(\frac{2}{m+1} \frac{v x}{U}\right)^{1 / 2}$
$\eta(\mathrm{x}, \mathrm{y})=y\left(\frac{m+1}{2} \frac{U}{v x}\right)^{1 / 2}$,
$\psi(\mathrm{x}, \mathrm{y})=\left(\frac{2}{m+1} v x U\right)^{1 / 2} f(\xi, \eta)$,
$T(\mathrm{x}, \mathrm{y})=\left(T_{w}-T_{\infty}\right) \theta(\xi, \eta)+T_{\infty}$,
$C(\mathrm{x}, \mathrm{y})=\left(C_{w}-C_{\infty}\right) \phi(\xi, \eta)+C_{\infty}$ (6)
By the definition of stream function, we introduce as:
$u=\frac{\partial \psi}{\partial y}$ and $v=-\frac{\partial \psi}{\partial x}$ (7)
Thus, we have:
$\begin{aligned} u=U f_{\eta}, & \begin{array}{l}v=-\frac{1}{2}\left(\frac{2}{m+1} \frac{v U}{x}\right)^{1 / 2} \\ {\left[(\mathrm{~m}+1) \mathrm{f}+(\mathrm{m}-1)\left(\eta \mathrm{f}_{\eta}-\xi \mathrm{f}_{\xi}\right)\right]}\end{array} \end{aligned}$ (8)
Enforcing Eqns. (6), (7) and (8) into Eqns. (2)-(4), then got the following forms, respectively.
$\begin{align} & F''+\frac{2m}{m+1}\left( 1-{{F}^{2}} \right)+fF'-\frac{{{\xi }^{2}}}{Da}F \\ & +\frac{2}{m+1}\frac{\lambda }{\Pr }\left( \theta +N\phi \right) \\ & =\left( \frac{m-1}{m+1} \right)\xi \left( F'\frac{\partial f}{\partial \xi }-F\frac{\partial F}{\partial \xi } \right) \\ \end{align}$ (9)
$\begin{align} & \theta ''+\operatorname{Prf}\theta '+P{{e}_{\gamma }}\left( F\theta ''+F'\theta ' \right) \\ & =\Pr \left( \frac{m-1}{m+1} \right)\xi \left( \theta '\frac{\partial f}{\partial \xi }-F\frac{\partial \theta }{\partial \xi } \right) \\ \end{align}$ (10)
$\begin{align} & \phi ''+Scf\phi '+P{{e}_{\zeta }}\left( F\phi ''+F'\phi ' \right)-{{\xi }^{2}}Sc\Delta \phi \\ & =Sc\left( \frac{m-1}{m+1} \right)\xi \left( \phi '\frac{\partial f}{\partial \xi }-F\frac{\partial \phi }{\partial \xi } \right) \\ \end{align}$ (11)
Boundary conditions Eq. (5) reduced as:
$F(\xi, \eta)=0, \quad \theta(\xi, \eta)=1, \quad \phi(\xi, \eta)=1$ at $\eta=0$
$F(\xi, \eta) \rightarrow 1, \quad \theta(\xi, \eta) \rightarrow 0 \phi(\xi, \eta) \rightarrow 0$ as $\eta \rightarrow \infty$ (12)
here primes represent derivatives with respect to η and subscript ξ represent partial derivatives, and $f=\int_{0}^{\eta} F d \eta+f_{w}$, f_{w} is given as: $f_{w}+\left(\frac{1-m}{m+1}\right) \xi\left(f_{\xi}\right)_{w}=-\left(\frac{v_{w}}{v}\right) \xi=A \xi$;
where, $A=-\frac{v_{w}}{v}$ is the surface mass transfer parameter A<0 and A>0 are injection and suction, respectively; $\lambda=\frac{G r}{R e_{x}^{2}}$ is the buoyancy parameter, $N=\frac{\beta_{C}}{\beta_{T}}\left(\frac{C_{w}-C_{\infty}}{T_{W}-T_{\infty}}\right)$ is the buoyancy ratio, $R a_{x}=\frac{g K \beta_{T} x^{2}\left(T_{W}-T_{\infty}\right) \cos \gamma}{v \alpha}$ is the modified local Rayleigh number, $G r=\frac{R a_{x}}{P r}$ is the local Grashof number, $R e_{x}=\frac{U x}{v}$ is the local Reynold number, $P r=\frac{v}{\alpha_{m}}$ is the Prandtl number, $P e_{\gamma}=\frac{\gamma d U}{\alpha}$ is the thermal dispersion parameter, $P e_{\zeta}=\frac{\zeta d U}{D}$ is the thermal dispersion parameter, $S c=\frac{v}{D_{m}}$ is the Schmidt number, $\Delta=\frac{K_{c}}{v}$ is the chemical reaction parameter.
The principal bodily quantities of interest which can no longer be skipped are skin friction, heat and mass transfer rates in terms of local skin friction coefficient C_{f}, local Nusselt number Nu_{x}, and local Sherwood number Sh_{x} can be written as: $C_{f}=\frac{2 \tau_{w}}{\rho U^{2}}, \quad N u_{x}=\frac{q_{w}}{\left(T_{w}-T_{\infty}\right)} \frac{x}{k}$ and $S h_{x}=\frac{q_{m}}{\left(C_{w}-C_{\infty}\right)} \frac{x}{D_{f}}$ here, $\tau_{w}=\mu\left(\frac{\partial u}{\partial y}\right)_{y=0}, q_{w}=-k_{e}\left(\frac{\partial T}{\partial y}\right)_{y=0}$ and $q_{m}=-D_{e}\left(\frac{\partial C}{\partial y}\right)_{y=0}$are defined as wall skin friction, heat flux and mass flux, respectively.
Thus, we have
${{C}_{f}}\operatorname{Re}_{x}^{1/2}=2{{\left( \frac{m+1}{2} \right)}^{1/2}}F'(\xi ,0)$ (13)
$\begin{align} & N{{u}_{x}}\operatorname{Re}_{x}^{-1/2}=-{{\left( \frac{m+1}{2} \right)}^{1/2}} \\ & \left( 1+P{{e}_{\gamma }}F(\xi ,0) \right)\theta '(\xi ,0) \\ \end{align}$ (14)
$\begin{align} & S{{h}_{x}}\operatorname{Re}_{x}^{-1/2}=-{{\left( \frac{m+1}{2} \right)}^{1/2}} \\ & \left( 1+P{{e}_{\zeta }}F(\xi ,0) \right)\phi '(\xi ,0) \\ \end{align}$ (15)
The non-similar equations are solved numerically through the well explained numerical technique call the bivariate Chebyshev spectral collocation quasi-linearization method by Motsa et al. [33], Meena et al. [34, 35]. To test the efficiency of the numerical technique; the residual analysis study is computed, which converges to 10^{-12}.
In this section, the outcomes for non-similarity solution of Eqns. (9-11) with boundary conditions Eq. (12) are displayed graphically and numerically via the above explained technique. The governing parameters are considered in the following ranges 0.66≤Sc≤2.57, -1.0≤N≤1.0, 0≤m≤1/2, 0.733≤Pr≤6.7, 1.0≤λ≤7.0, 0.5≤Pe_{γ}≤2.0, -1.0≤Δ≤1.0, 0.5≤Pe_{ζ}≤2.0, -1.0≤A≤1.0 and 0.0≤ξ≤1.0. We carry the boundary layer region between 4.0 to 7.0.
Figure 2 shows the residual analysis over iterations of F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles for Da variation with m=1/2, Pr= 0.733, λ=1.0, Pe_{γ}=0.5, Pe_{ζ}=0.5, N=1.0 and Δ=1.0, respectively. It establishes the convergence of present adopted numerical technique. In Figure 2 (a), we observe that the method converges after eleven iterations for the various values of Da to 10^{-11}, while in Figure 2 (b) the method converges after ten iterations for the various values of Da to 10^{-11} and in Figure 2 (c) the method converges after twelve iterations for the various values of Da to 10^{-12}. This in turn validates the stability and convergence of the methods, and hence the results are accurate to approximately eleven digits.
(a)
(b)
(c)
Figure 2. Residual analysis over iterations with m=1/2, Pr=0.733, λ=1.0, Pe_{γ}=0.5, Pe_{ζ}=0.5, A=1.0, N=1.0 and Δ=1.0 are represented by (a), (b) and (c), respectively
Figure 3(a) exhibits a comparison of prior published computation with Kumari et al. [32] which authenticated validation of the chosen numerical technique. Figure 3(b, c) introduces the impact of Pr and Pe_{γ} together on the F(ξ, η) and θ(ξ, η) profiles for m=1/2, Sc=2.57, λ=1.0, A=1.0, Pe_{ζ}=0.5, N=1.0 and Δ=0.5. Both the F(ξ, η) and θ(ξ, η) profiles intensify with Pe_{γ} and clashing with Pr. The intensification rate is greater with a small Prandtl number because of lower viscosity fluid. Therefore, thermal dispersion impact is more prevalent on low viscosity fluid. We perceive it from the velocity profile that the enhancement is less near to the surface of the cone because of the no-slip condition and increases far away and the similar trend concerned in the temperature profile.
(a)
(b)
(c)
Figure 3. Comparison of the present results with Kumari 1989 of the F(ξ, η) and θ(ξ, η) profiles for m=0, Sc=0, λ=0, Pe_{γ}=0, A=0, Pe_{ζ}=0, N=0 and Δ=0 represented by (a). Impression of Pr and Pe_{γ} together on the F(ξ, η) and θ(ξ, η) with m=1/2, Sc=2.57, λ=1.0, Da=1.0, A=1.0, Pe_{ζ}=0.5, N=1.0 and Δ=0.5 are represented by (b) and (c), respectively
Figure 4 portrays the effect of Sc and Δ together on the F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles with m=1/2, Pr= 0.733, Pe_{γ}=0.5, λ=1.0, A=1.0, Pe_{ζ}=0.5, N=1.0 and Δ=0.5. All the F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles diminishes with Δ and it has more variation with small Schmidt number Sc=0.22 which refers to the less viscosity of the fluid. The θ(ξ, η) profile does not have enormous variation with Δ, therefore, the chemical reaction does not affect much on the temperature distribution of the flow.
(a)
(b)
(c)
Figure 4. Impression of Sc and Δ together on F(ξ, η), $\theta(\xi, \eta)$ and ϕ(ξ, η) with m=1/2, Pr=0.733, λ=1.0, Da=1.0, A=1.0, Pe_{γ}=0.5, Pe_{ζ}=0.5, N=1.0 and Δ=0.5 are represented by (a), (b) and (c), respectively
Figure 5 shows F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles profiles for variation of the Sc and A together with m=1/2, Pr= 0.733, Pe_{γ}=0.5, λ=1.0, Pe_{ζ}=0.5, N=1.0 and Δ=0.5. All the F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles enhance with A<0 (injection) but converse with A>0 (suction), therefore, decreases. The physical cause of this can count as in the injection case the fluid injected from the cone which enhances the velocity of flow and the heat and concentration profiles also enhance but with the suction surface of the cone suck the fluid that reduces the velocity and the temperature and concentration profiles also reduces. It is also noted that if the viscosity of fluid increases then the impact of injection suction reduces, and it is greater with small Sc=0.66.
(a)
(b)
(c)
Figure 5. Impression of Sc and A together on F(ξ, η), θ(ξ, η) and ϕ(ξ, η) with m=1/2, Pr= 0.733, λ=1.0, Da=1.0, A=1.0, Pe_{γ}=0.5, Pe_{ζ}=0.5, N=1.0 and $\Delta=0.5$ are represented by (a), (b) and (c), respectively
Figure 6 (a, b) offers the impression of Pe_{ζ} and Sc jointly on the F(ξ, η) and ϕ(ξ, η) profiles regarding m=1/2, Pr= 0.733, Pe_{γ}=0.5, λ=1.0, A=1.0, N=1.0 and Δ=0.5. Both the F(ξ, η) and ϕ(ξ, η) profiles enhances with Pe_{ζ}, away the surface of the cone, and not much dominant on the F(ξ, η) profile near the surface because of the non-slip condition and similar followed by θ(ξ, η) profile. Figure 6 (c) reflects the impact of axial distance $\xi$ on the F(ξ, η), θ(ξ, η) and ϕ(ξ, η) profiles and observed decrement in all the profiles with ξ.
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(b)
(c)
Figure 6. Impression of Sc and Pe_{ζ} together on F(ξ, η) and ϕ(ξ, η) with m=1/2, Pr= 0.733, λ=1.0, Da=1.0, A=1.0, Pe_{γ}=0.5, A=1.0, N=1.0 and Δ=0.5 are represented by (a) and (b), respectively. Impression of ξ on F(ξ, η), θ(ξ, η) and ϕ(ξ, η) represented by (c)
Figure 7 introduces the $C_{f} R e_{x}^{1 / 2}$, $N u_{x} R e_{x}^{-1 / 2}$ and $\operatorname{Sh}_{x} R e_{x}^{-1 / 2}$ for mutation in Sc and A together over the axial distance ξ with m=1/2, Pr=0.733, Da=1.0, Δ=0.5, Pe_{γ}=0.5, Pe_{ζ}=0.5, N=1.0 and λ=1.0. All properties fall with injection over axial distance, but contrary to suction therefore enhances. The impact of A is more dominant on the $C_{f} R e_{x}^{1 / 2}$ with low viscosity fluid, hence with low Schmidt number (Sc= 0.66) but the $S h_{x} R e_{x}^{-1 / 2}$ is inimical, hence more dominant with the high Schmidt number (Sc= 2.57).
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(b)
(c)
Figure 7. Impression of Sc and A together on $C_{f} R e_{x}^{1 / 2}$, $N u_{x} R e_{x}^{-1 / 2}$ and $\operatorname{Sh}_{x} R e_{x}^{-1 / 2}$ with m=1/2, Pr= 0.733, λ=1.0, Da=1.0, Pe_{ζ}=0.5, Pe_{γ}=0.5, N=1.0 and $\Delta=0.5$ are represented by (a), (b) and (c), respectively
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(b)
Figure 8. Impression of Pr and Pe_{γ} together on $C_{f} R e_{x}^{1 / 2}$ and $N u_{x} R e_{x}^{-1 / 2}$ with m=1/2, Sc=2.57, λ=1.0, Da=1.0, Pe_{ζ}=0.5, A=1.0, N=1.0 and Δ=0.5 are represented by (a) and (b), respectively
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(b)
Figure 9. Impression of Δ on $C_{f} R e_{x}^{1 / 2}$ and $S h_{x} R e_{x}^{-1 / 2}$ with m=1/2, Sc=2.57, λ=1.0, Da=1.0, Pe_{γ}=0.5, Pe_{ζ}=0.5, A=1.0 and N=1.0 are represented by (a) and (b), respectively
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(b)
Figure 10. Impression of Sc and Pe_{ζ} together on $C_{f} R e_{x}^{1 / 2}$ and $\operatorname{Sh}_{x} R e_{x}^{-1 / 2}$ with m=1/2, Pr= 0.733, λ=1.0, Da=1.0, Pe_{γ}=0.5, Δ=0.5, A=1.0 and N=1.0 are represented by (a) and (b), respectively
Figure 8 presents the $C_{f} R e_{x}^{1 / 2}$ and $N u_{x} R e_{x}^{-1 / 2}$ for variation of Pr and Pe_{γ} together over axial distance ξ with m=1/2, Sc=2.57, Da=1.0, Δ=0.5, A=1.0, Pe_{ζ}=0.5, N=1.0 and λ=1.0. Both the $C_{f} R e_{x}^{1 / 2}$ and $N u_{x} R e_{x}^{-1 / 2}$ enhances with Pe_{γ} and hold similar tendency with the axial distance ξ. Figures 9 and 10 reflects $C_{f} R e_{x}^{1 / 2}$ and $\operatorname{Sh}_{x} R e_{x}^{-1 / 2}$ for variation in Δ and Pe_{ζ} with m=1/2, Pr=0.733, Da=1.0, Δ=0.5, A=1.0, Pe_{γ}=0.5, N=1.0 and λ=1.0, respectively. The $C_{f} R e_{x}^{1 / 2}$ diminishes with Δ over ξ but $S h_{x} R e_{x}^{-1 / 2}$ is contradictory. Both the $C_{f} R e_{x}^{1 / 2}$ and $S h_{x} R e_{x}^{-1 / 2}$ enhances with Pe_{ζ} over ξ.
Table 1 produces numerical computations of the skin friction coefficient C_{f}, local Nusselt number Nu_{x} and Sherwood number Sh_{x} for mutation of the Prandtl number Pr, Schmidt number Sc, buoyancy parameters λ, buoyancy ratio N, chemical reaction parameter Δ, surface mass transfer A, thermal dispersion parameter Pe_{γ} and Solutal dispersion parameter Pe_{ζ}, respectively. An increment in Pr decrease C_{f} and Sh_{x} but Nu_{x} is contradictory. Likewise, Sc diminishes C_{f} and Nu_{x} but Sh_{x} is inconsistent. λ and N both enhance all the C_{f}, Nu_{x} and Sh_{x} properties. C_{f} and Nu_{x} diminishes with $\Delta$ but Sh_{x} is in contrary trend, i.e. enhances. All the C_{f}, Nu_{x} and Sh_{x} increases with A, Pe_{γ} and Pe_{ζ}, respectively.
Table 1. Influence of Pr, Sc, λ, N, Δ, A and Pe_{γ} on C_{f}, Nu_{x} and Sh_{x} with respect to Pr=0.733, λ=1.0, Pe_{γ}=0.5, Sc=2.57, A=1.0, N=1.0 and Δ=0.5
Pr Sc λ N Δ A Pe_{γ} Pe_{ζ} |
C_{f} |
Nu_{x} |
Sh_{x} |
0.733 2.57 1.0 1.0 0.5 1.0 0.5 0.5 6.7 |
5.34699853 2.49358539 |
0.70571959 3.10539008 |
1.64807540 1.58992410 |
0.66 2.57 |
5.99628942 5.34699853 |
0.71848098 0.70571959 |
0.74086333 1.64807540 |
1.0 3.0 5.0 |
5.34699853 11.20234647 16.67238428 |
0.70571959 0.76470108 0.81241888 |
1.64807540 1.75659674 1.84689391 |
-1.0 -0.5 0.5 1.0 |
3.01247913 3.59273497 4.76034572 5.34699853 |
0.68574963 0.69079561 0.70078467 0.70571959 |
1.60909515 1.61886853 1.63836864 1.64807540 |
-1.0 -0.5 0.5 1.0 |
5.50363272 5.44360491 5.34699853 5.30728051 |
0.70796474 0.70709352 0.70571959 0.70516651 |
1.25389934 1.39873759 1.64807540 1.75772303 |
-1.0 -0.5 0.5 1.0 |
4.68020358 4.87394627 5.19792984 5.34699853 |
0.35246798 0.42793296 0.60482311 0.70571959 |
0.45856369 0.66610023 1.26553822 1.64807540 |
0.5 1.0 1.5 2.0 |
5.34699853 5.43034406 5.49448953 5.54607471 |
0.70571959 0.79782014 0.88245526 0.96060263 |
1.64807540 1.65138736 1.65395021 1.65601667 |
0.5 1.0 1.5 2.0 |
5.34699853 5.42477313 5.48900447 5.54346196 |
0.70571959 0.70738186 0.70883745 0.71012732 |
1.64807540 1.78200226 1.90921124 2.02922572 |
This manuscript aspires to investigate the influence of the injection/suction effects because of mixed convection flow in the presence of double dispersion and chemical reaction effects across a vertical cone saturated porous medium of a steady two-dimensional viscous incompressible fluid. The numerical computations figured and tabulated through the described technique, and we describe characteristics in the above section. The buoyancy parameter and buoyancy ratio both upgrade the C_{f}, Nu_{x} and Sh_{x}, respectively. Likewise, all the C_{f}, Nu_{x} and Sh_{x} increases with A, Pe_{γ} and Pe_{ζ}, respectively. We perceive it from the velocity profile that the enhancement is less near to the surface of the cone because of the no-slip condition and increases far away and the similar trend concerned in the temperature and concentration profiles. C_{f} and Nu_{x} diminishes with Δ but Sh_{x} enhances and noted that Δ is prevailing on the concentration profile and mass transfer and does not interrupt temperature distribution in the flow. An increment in Pr decrease C_{f} and Sh_{x} but Nu_{x} increases. Likewise, Sc diminishes C_{f} and Nu_{x} but Sh_{x} enhances. All injection/suction, double dispersion, and chemical reaction impacts are more dominant in less viscosity fluid (Pr=0.733, Sc=0.66) comparing to high (Pr=6.7, Sc=2.57).
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