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This paper is dedicated to the mathematical analysis of an axisymmetric, steady Newtonian fluid flow through a horizontal pipe within the occurrence of radiation, Dufour, and Soret effects. The flow is exposed to associate outwardly functional constant suction above the pipe along Zdirection. The homotopy analysis methodology (HAM) is utilized to get semianalytical solutions for the coupled differential equations. The results of diverse rising constraints on velocities, thermal and solutal are discussed and pictured. The flow is studied through streamlines, isotherms and pressure contours area unit likewise shown as pictured. It is identified that the temperature can increase with an increase in Dufour parameter but decelerates with an improvement in the radiation parameter. For the given increase within the Soret number, the concentration decelerates.
Newtonian fluid, suction, pipe, thermal radiation, crossdiffusions, stream function, homotopy parameter
The study of flow through permeable boundaries has several important industrial applications such as biophysics, lubrication technology, spatial dynamics, blood flow through arteries and veins, and control of boundary layers. From this viewpoint, the porous channel/pipe problems of similar model type have been studied by many researchers (Berman [1], Terril and Shrestha [2], Zaturska and Banks [3]. Bujurke et al. [4], Erdogan and Imrak [5], Devakar and Iyengar [6]). The other interesting problems for porous boundaries with different geometrics can be seen in Ramana Murthy et al. [7], Nagaraju and Ramana Murthy [8], Srinivas et al. [9], and Aparna et al. [10]. It is documented that heat and mass transport are mixed as a result of the fluid density depends on thermal and solutal. In the power and chemical industries, coupling laminar heat and mass transfer can be found. In many engineering applications such as heat exchangers, solar thermal collectors, and boilers, doublediffusive convection in circular tubes in laminar flow occurs. Of explicit interest is the problem of derivation within the thermal entrance incompressible flow along a pipe with a fully improved velocity distribution; this problem is resulted to as the Graetz problem. It has drawn interest not only from engineers but also from mathematicians due to the difficulties in deriving its solution. Ganesan and Loganathan [11] have studied the doublediffusive viscous flow through a semiinfinite vertical cylinder. The Doublediffusive natural convection was examined by Mani Sankar [12] through a vertical openended cylindrical annulus. He found that the number of Sherwood is decreasing as the radii ratio increases. Brahim and Jemni [13] investigated a twodimensional Darcy–Brinkman–Forchheimer heat tube's thermal analysis. Okafor et al. [14] studied the distribution of nonsymmetric heat flux through a mixed horizontal convective tube.
The Dufour effect is the heat flux caused by a gradient of concentration that was resulted in 1873 by Dufour as a coupled effect of irreversible systems; it is referenced to as the thermodiffusion effect. The effect of Dufour is of such magnitude that it cannot be disregarded. On the other hand, the Soret effect (thermodiffusion) is a phenomenon experimental in mobile particle mixtures where different particle types react differently to the temperature gradient force. The effects of Soret and Dufour are important because the effect of Soret is used for the separation of isotopes, electrostatic precipitators, drug discovery, etc. Several researchers studied and reported results for such fluid flows, due to importance of the importance of the crossdiffusion effects for very light and medium molecular weights fluids (Nabil et al. [15], Rani and Reddy [16], Srinivas et al. [17], Odelu and Naresh kumar [18], Nagaraju et al. [19, 20].)
Yang and Ebadian [21] examined in a circular tube the mixture of axial conductivity and axial radiation. Yih [22] used a convective isothermal vertical cylinder to investigate radiation and porous medium effects. Ganesan and Loganathan [23] examined the effect of radiation on flow past a semiinfinite vertical cylinder that was impulsively started. They identified the radiation parameter increases, the fluid's speed and the temperature rise sharply near the cylinder. The other interesting studies on radiative heat and mass transfer flow by diverse geometries can be found in Govardhan et al. [24], Nagaraju et al. [25], and Abdullah AlMamun et al. [26].
The survey exhibits that twodimensional viscous flow with crossdiffusion and radiation has not been studied elsewhere. Therefore, the objectives of this study are to mathematically solve the steady viscous flow problem through a horizontal pipe subjected to crossdiffusion and radiation for the fully developed region, and derive completely analytical solutions for the fluid thermal profile and concentration using HAM (Liao [27]). With the help of planned graphs, the consequence of connected parameters on the distribution of thermal and concentration was discussed thoroughly.
Fluid thought of during this study is Newtonian with fully developed flow through a horizontal pipe of radius ‘a’ in occurrence of crossdiffusion and thermal radiation. By constant wall temperature and concentration on the pipe surface, the heat and mass transfer method are studied. The fluid flow is subjected to constant external suction inside the conventional path through the wall. Backed by these assumptions, the dimensional equations of the fields of viscous liquid, thermal and solute are expressed as follows.
$\frac{\partial W}{\partial Z}+\frac{1}{R}\frac{\partial }{\partial R}\left( RU \right)=0$ (1)
$\rho \left( U\frac{\partial }{\partial R}+W\frac{\partial }{\partial Z} \right)U=\frac{\partial P}{\partial R}+\mu \left( \frac{\partial }{\partial R}\left( \frac{\partial U}{\partial R}+\frac{U}{R} \right)+\frac{{{\partial }^{2}}W}{\partial {{Z}^{2}}} \right)$ (2)
$\rho \left( U\frac{\partial }{\partial R}+W\frac{\partial }{\partial Z} \right)W=\frac{\partial P}{\partial Z}+\mu \left( \frac{{{\partial }^{2}}W}{\partial {{Z}^{2}}}+\frac{1}{R}\frac{\partial W}{\partial R}+\frac{{{\partial }^{2}}W}{\partial {{R}^{2}}} \right)$ (3)
$\begin{aligned} U \frac{\partial T}{\partial R}+W \frac{\partial T}{\partial Z}=& \frac{k}{\rho C_{p}}\left(\frac{\partial^{2} T}{\partial R^{2}}+\frac{1}{R} \frac{\partial T}{\partial R}+\frac{\partial^{2} T}{\partial Z^{2}}\right) \\ &+\frac{D K_{T}}{C_{s} C_{p}}\left(\frac{\partial^{2} C}{\partial R^{2}}+\frac{1}{R} \frac{\partial C}{\partial R}+\frac{\partial^{2} C}{\partial Z^{2}}\right) \\ &\frac{1}{\rho C_{p}} \frac{1}{R} \frac{\partial}{\partial R}\left(R q_{R}\right) \end{aligned}$ (4)
$\begin{aligned}\left(U \frac{\partial}{\partial R}+W \frac{\partial}{\partial Z}\right) C & \\ &=D\left(\frac{\partial^{2} C}{\partial Z^{2}}+\frac{1}{R} \frac{\partial C}{\partial R}+\frac{\partial^{2} C}{\partial R^{2}}\right) \\ &+\frac{D K_{T}}{T_{m}}\left(\frac{\partial^{2} T}{\partial Z^{2}}+\frac{1}{R} \frac{\partial T}{\partial R}+\frac{\partial^{2} T}{\partial R^{2}}\right) \end{aligned}$ (5)
with
$U=\frac{\partial W}{\partial R}=\frac{\partial T}{\partial R}=\frac{\partial C}{\partial R}=0$ at $\mathrm{R}=0$ (6)
$U=v_{0}, W=0, T=T_{w}$ and $C=C_{w}$ at $\mathrm{R}=a$ (7)
Introducing the dimensionless variables
$U=u v_{0}, W=w v_{0} R=r a, P=p \rho v_{0}^{2}, Z=a z, \boldsymbol{T}=$
$\theta\left(T_{w}T_{\infty}\right)+T_{\infty}, C=\phi\left(C_{w}C_{\infty}\right)+C_{\infty}$ (8)
We consider that q_{R} under the Rossel and approximation has the following form:
$q_{R}=\frac{4 \sigma^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial R}$ (9)
Substitute in Eqns. (1)(7), we get the governing dimensionless equations as:
$\frac{\partial w}{\partial z}+\frac{1}{r} \frac{\partial}{\partial r}(r u)=0$ (10)
$\operatorname{Re}\left(u \frac{\partial}{\partial r}+w \frac{\partial}{\partial z}\right) u=\operatorname{Re} \frac{\partial p}{\partial r}+\left(\frac{\partial}{\partial r}\left(\frac{\partial u}{\partial r}+\frac{u}{r}\right)+\frac{\partial^{2} w}{\partial z^{2}}\right)$ (11)
$\operatorname{Re}\left(u \frac{\partial}{\partial r}+w \frac{\partial}{\partial z}\right) w=R e \frac{\partial p}{\partial z}+\left(\frac{\partial^{2} w}{\partial z^{2}}+\frac{1}{r} \frac{\partial w}{\partial r}+\frac{\partial^{2} w}{\partial r^{2}}\right)$ (12)
$\begin{aligned} \operatorname{RePr}\left(u \frac{\partial \theta}{\partial r}+w \frac{\partial \theta}{\partial z}\right) & \\ &=\nabla^{2} \theta+\operatorname{Pr} D_{f} \nabla^{2} \phi \\ &+R d\left(\frac{\partial^{2} \theta}{\partial r^{2}}+\frac{1}{r} \frac{\partial \theta}{\partial r}\right) \end{aligned}$ (13)
$\operatorname{ReSc}\left(u \frac{\partial \phi}{\partial r}+w \frac{\partial \phi}{\partial z}\right)=\operatorname{Sc} \operatorname{Sr} \nabla^{2} \theta+\nabla^{2} \phi$ (14)
$f=D^{2} f=\frac{\partial \theta}{\partial r}=\frac{\partial \phi}{\partial r}=0$ at $\mathrm{r}=0$ (15)
$f^{\prime}=0, f=\theta=\phi=1 \quad$ at $\mathrm{r}=1$ (16)
The Laplacian operator defined as $\nabla^{2}=\frac{\partial^{2} \theta}{\partial r^{2}}+\frac{1}{r} \frac{\partial \theta}{\partial r}+\frac{\partial^{2} \theta}{\partial z^{2}}$, $\operatorname{Re}=\frac{\rho v_{0} a}{\mu}$, $P r=\frac{\mu c_{p}}{k}$, $D_{f}=\frac{D K_{T}\left(C_{w}C_{\infty}\right)}{v C_{S} C_{p}\left(T_{w}T_{\infty}\right)}$, $S c=\frac{v}{D}$, $R_{d}=\frac{16 T_{\infty}^{3} \sigma^{*}}{3 k k^{*}}$ and $S r=\frac{D K_{T}\left(T_{w}T_{\infty}\right)}{v T_{m}\left(C_{w}C_{\infty}\right)}$.
The stream function ψ is defined to be:
$u=\frac{\psi_{z}}{r}$ and $w=\frac{\psi_{r}}{r}$ (17)
From (11) and (12), the pressure (p) is removed, produces:
$\operatorname{Re}\left(\frac{2}{r^{3}} \psi_{z} E^{2} \psi+\frac{1}{r^{2}}\left(\psi_{z} \frac{\partial E^{2} \psi}{\partial r}\psi_{r} \frac{\partial E^{2} \psi}{\partial z}\right)\right)=\frac{E^{2}\left(E^{2} \psi\right)}{r}$ (18)
where, $E^{2}=\frac{\partial^{2}}{\partial r^{2}}\frac{1}{r} \frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial z^{2}}$.
The following form is supposed for y [2].
$\psi=(Nz) f(r)$ (19)
Substituting Eq. (19) in (18), results in,
$\operatorname{Re}\left(\frac{3 f f^{\prime \prime}}{r^{2}}\frac{3 f f^{\prime}}{r^{3}}\frac{f f^{\prime \prime \prime}}{r}+\frac{f^{\prime \prime}}{r}\frac{f^{\prime 2}}{r^{2}}\right)=D^{4} f$ (20)
where, $D^{2}=\frac{d^{2}}{d r^{2}}\frac{1}{r} \frac{d}{d r}$ is a differential operator.
At the surface of the pipe the shear stress, thermal, and solutal fluxes and can be derived from:
$C_{f}=\frac{2 \bar{\tau}_{r z}}{\rho v_{0}^{2}}, N u=\frac{a q_{h}}{k\left(T_{w}T_{\infty}\right)}$ and $S h=\frac{a q_{m}}{D\left(C_{w}c_{\infty}\right)}$ at $R=a$ (21)
where,
$\left\{\begin{array}{c}\bar{\tau}_{r z}=\frac{\mu v_{0}}{a} \frac{(Nz)}{r} D^{2} f \\ q_{h}=k \frac{\partial T}{\partial R}=\frac{k\left(T_{w}T_{\infty}\right)}{a} \frac{\partial \theta}{\partial r} \\ q_{m}=D \frac{\partial C}{\partial R}=\frac{D\left(C_{W}c_{\infty}\right)}{a} \frac{\partial \phi}{\partial r}\end{array}\right.$ at r = 1
The dimensionless form of (21) is given by:
$\left\{\begin{array}{c}C_{f}=\frac{2(Nz)}{R e} D^{2} f \\ N u=\frac{\partial \theta}{\partial r} \\ S h=\frac{\partial \phi}{\partial r}\end{array}\right.$ at r = 1 (22)
The given initial approximations of f_{0} (r), $\theta_{0}(r)$ and ϕ_{0}are derived for HAM solutions as:
$f_{0}(r)=2 r^{2}r^{4}, \theta_{0}(r)=1$ and $\phi_{0}(r)=1$ (23)
with the auxiliary operators are:
$L_{2}=L_{3}=\nabla^{2}$ and $L_{1}=D^{4}$
with
$L_{1}\left[c_{1} r^{4}+c_{2}\left(2 r^{2} \log rr^{2}\right)+c_{3} r^{2}+c_{4}\right]=0$
$L_{2}\left[(Nz)^{2}\left(c_{5}+c_{6} \log r\right)+c_{7} r^{2}\right.$
$\left.\quad+c_{8} r^{2}(\log r1)\right]=0$
$L_{3}\left[(Nz)^{2}\left(c_{9}+c_{10} \log r\right)+c_{11} r^{2}\right.$
$\left.+c_{12} r^{2}(\log r1)\right]=0$ (24)
where, $C_{i}$ (i = 1 to 12) are constants. h_{i }(i=1, 2, 3) (regulator parameters) are used in zerothorder deformation equations.
The zerothorder deformations; nonlinear operators N_{i}(i=1 to 3) are considered as described in the effort of Nagaraju et al. [28, 29].
The flow Eqns. (11)(14) subject to the border line conditions (15) to (16) are coupled, hence the system of equations is solved semianalytically using the Homotopy method. For the convergent series solutions, we have picked the values of h_{1}=0.4, h_{2 }= 0.5 and h_{3}= 0.8 through the permissible range of h values based on h curves as shown in Figures 1(a)1(c).
Figure 1. hcurves for (a) velocity, (b) thermal and (c) concentration
Figures 2(a)–2(f) present the nature of f, $f^{\prime}$, θ and ϕ under the influence of Re. Out of Figure 2(a) as Re enhances, the f increases. It is observed that f' enhances up to the point r = 0.6 then there is a decrement in the axial velocity as Re increases from Figure 2(b). Figures 2(c)–2(f) shows that an increase in Re results in a rise in radial θ levels, ϕ, and decreases in axial θ.
Figure 2. Variation of Reynold number on (a) f, (b) polar velocity, (c) radial θ, (d) axial θ, (e) radial ϕ, and (f) axial ϕ
Figure 3. Variation of Dufour number on (a) radial θ, (b) axial θ
From Figures 3(a)3(b) we can see that radial and axial temperature increases as the Dufour number increases. From Figures 4(a)4(b), we can observe that radial and axial temperature decreases as the radiation parameter rises. These results show that the radiation has significant dominance in the flow field. It can be shown in Figures 5(a)5(b) that radial and axial θ enhances as the Prandtl number increases.
It is clear from Figures 6 and 7, it can be resulted that radial and axial concentration decreases as Sc and Sr increases. It is identified that Nusselt and Sherwood number raises with Prandtl and Schmidt number raises in Figures 8 and 9 respectively. It is noticed from Figure 10, that velocity streamlines are nonnegative at z≤N and negative at z>N. For z=N, the Streamlines are numerically symmetric. It can be shown from Figure 11 that θ is symmetrical about the line z=N. From Figure 12, it is observed that concentration is symmetric about z=N and increments as r<0.8.
Figure 4. Variation of Radiation on (a) radial θ, (b) axial θ
Figure 5. Variation of Prandtl number on (a) radial θ, (b) axial θ
Figure 6. Impact of Sc on (a) radial ϕ, (b) axial ϕ
Figure 7. Impact of Sr on (a) radial ϕ, (b) axial ϕ
Figure 8. Changes of Pr on Nu
Figure 9. Changes of Sc on Sh
Figure 10. Velocity streamlines
Figure 11. Contour graph for Temperature isotherms
Table 1. Nature of skin friction coefficient, heat and mass transfer rates for various values of Re, Sr, and Df
Re 
z 
Pr 
Sc 
Sr 
Df 
Rd 
C_{f} 
Nu 
Sh 
0.75 
1 
0.71 
0.5 
1 
0.06 
1.5 
17.2233 
3.85944 
4.52758 
1.5 
5.98985 
24.722 
14.2953 

2.25 
1.81603 
53.0241 
27.0234 

1.5 
1 
0.71 
0.5 
1 
0.5 
1.5 
 
27.0956 
48.0487 
1 
29.7929 
61.5916 

1.5 
32.4902 
75.2606 

1.5 
1 
0.71 
0.5 
0.5 
0.5 
1.5 
 
37.607 
38.5889 
1 
37.9793 
50.872 

1.5 
38.3515 
63.1551 
Figure 12. Contour graph for concentration
The nature of the C_{f}, Nu, and Sh with the flow parameters can be seen in Table 1. The results from Table 1 that the Nu and Sh increase as Re, Sr and Df increase.
This paper addresses the mathematical analysis of an axisymmetric, steady Newtonian fluid flow through a horizontal pipe in the occurrence of radiation and Crossdiffusion effects. Based on results through the HAM, summarily the findings as follows:
The Reynolds number breaks the axial flow and increases the radial flow.
An increment in Pr and Df causes an increase in the temperature profiles whereas $\theta$ decreases as Rd increases.
The concentration profile rises upon increasing the Re, and decreases as Sr, and Scincreases.
The Skinfriction (C_{f}), Nusselt number (Nu), and Sherwood number (Sh) increase as Re raise.
The Nusselt number (Nu), and Sherwood number (Sh) increases when Pr and Sc number raises.
The authors are grateful to the anonymous referees for remarks which improved the work considerably.
w, W 
Nondimensional and dimensional axial velocity, m/s 
u,U 
nondimensional and dimensional radial velocity, m/s 
Cp 
specific heat, J. kg^{1}. K^{1} 
k 
thermal conductivity, W.m^{1}. K^{1} 
Tw 
surface/wall temperature 
Nu 
local Nusselt number along the heat source 
D 
Diffusion coefficient 
Tm 
mean temperature 
CS 
Concentration susceptibility 
v0 
suction/injection velocity 
Re 
Reynolds number 
Pr 
Prandtl number 
E^{2} 
Stoke’s operator 
T, C 
The fluid temperature and concentration 
N= U_{0}/v_{0} 
Constant velocity 
U0 
inlet velocity 
Df 
Dufour parameter 
Sc 
Schmidt number 
Rd 
radiation parameter 
Sr 
soret number 
T_{∞} 
temperature far away from the center 
C_{w} 
concentration at the wall 
C_{∞} 
concentration far away from the center. 
k* 
mean absorptioncoefficient. 
K_{T} 
Thermaldiffusion ratio 
Greek symbols 

$\rho$ 
fluid density, kgm^{−3} 
$\mu$ 
viscosity 
σ* 
StefanBoltzmann constant 
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