1 Introduction
Recently, owing to their multifaceted and numerous micro- structures,non-Newtonian liquids[1-3] constantly persist an arena of actual desire for engineers and researchers. Therefore, widespread endeavors have been prepared to study their worth in numerous built-up applications. The noteworthy feature of these liquids are their advanced apparent viscosities thus, laminar flow conditions rise more habitually when allied with Newtonian liquids and Prandtl numbers are maximized. Numerous materials in the biological liquids, compound or foodstuff developments and petrochemical are non-Newtonian in nature. These materials comprise plasma, salvia, microelectronic cooling structures, firewood in aquatic and china soil, sewage slurry, greasepaints and vapor--liquid disseminations categorized as non-Newtonian liquids. The liquids follow nonlinearity properties between shear rate and shear stress named as non-Newtonian liquids. These liquids modify flow aspects of liquid, which outcomes affect the aptitude of the liquid's in transferring heat. Additionally, diverse liquid models wished-for to clarify the structure and rheological aspects of nonlinear materials. The Maxwell model is a rate type nonlinear model which forms the stress relaxation of numerous polymeric liquids. Limited studies covering Maxwell fluid under a diverse flow structure can be discussed via Refs. [4--8]. Fractional Maxwell liquid with variable thickness was reported by Liu and Liu.[9] They attained elucidations of the problem numerically via L1-scheme. Khan $et al.$[10] studied the performance of nanoparticles condition and heat sink/source in radiative Maxwell nanomaterial. They inspected that the Deborah number and radiation parameter have possess behavior on temperature field. Impact of nonlinear radiation in Maxwell nanofluid subject to rotating disk was elaborated by Ahmed $et al.$[11] They noted that the thermal radiation parameter enhances the fluid temperature.
Modern growths in nanotechnology have fetched new discussion of heat transfer liquids acknowledged as nanofluids. Nanofluids are solid-fluid, complex materials prepared of condensed nanoparticles with size (1--100) nm and possess an aptitude of advanced thermal conductivity. These particles, mostly a metal or metal oxide, intensify conduction and convection coefficients, letting for new heat transfer outside the coolant. Owing to their worth in numerous built-up and biomedical applications nanofluids have established thoughtfulness in modern years. Moreover, nanofluids and nanofluids constructed phase conversion materials have been acknowledged as prospective candidate for numerous energy storage and heat transport applications, still they are facing numerous experiments for example pressure drop, stability and extraordinary inflating power. Nano size particle shows a better role in obstructing nanofluid applications; however, the impact of nanoparticles size does not acquire the thoughtfulness as it deserves. The inconsistence and conflict of material provide an impression that the region is not well assumed, thus additional material essential to be prepared. In that instance, overview of nanofluids applications via nanotechnology may be incorporated into each of these technological ranges that are: drag diminutions, nanofluids in boring, magnetic fluids, heating and cooling of constructions, cooling of microchip technology and welding, astral and defense, high-control lasers, warm cylinders and thermal storing. Numerous authors scrutinized critically the applications of nanofluids and acknowledged investigation breakdowns for advance research. Additionally, experiments and forthcoming directions and applications of these liquids with diverse properties have been reviewed and presented in Refs. [12--20]. In addition, Irfan $et al.$[21] scrutinized the properties of MHD and heat sink/source in convective flow of Maxwell nanofluid. They inspected that the curvature and Maxwell parameter spectacle conflicting performance on the velocity field while analogous impact on temperature field is being identified. Hayat $et al.$[22] reported the behavior of thermal radiation and chemical reaction on 3D MHD flow of nanofluid. The non-linear radiative mixed convection flow with deferred nanoparticles in melting surface was studied by Mahantesh $et al.$[23] Their study indicates that the melting and moving parameters diminish the drag forces or friction. The aspects of activation energy and binary chemical reaction on Williamson nanomaterial was analyzed by Hamid $et al.$[24] Their upshots specify that the temperature of Williamson fluid intensifies for thermophoresis parameter while the heat transport amount diminishes for reaction parameter.
The consideration of this study is to inspect the influence of stagnation point and variable conductivity in MHD flow of Maxwell nanofluid due to stretching cylinder. The aspects of heat sink/source and chemical reaction are also incorporated. Solutions of the formulated problem are attained via homotopic analysis method. Dynamic features of diverse inferential variables are presented graphically and discussed. Additionally, comparison tables are reported to authenticate the solutions of the problem in limiting sense.
2 Mathematical Formulation
We scrutinize steady 2D stagnation point flow of a Maxwell nanomaterial influenced by stretching cylinder of radius $R$. Additionally, variable thermal conductivity, heat sink/source, chemical reaction are incorporated for heat and mass transport phenomena. The stretching and free stream velocities of the cylinder are assumed to be $\left( {U_{0}z}/{l},{U_{\infty }z}/{l}\right) $ along $z$-direction, where $\left( U_{0},U_{\infty }\right) $ are the reference velocities and $l$ the specific length, respectively. Let us take cylindrical polar coordinates $(z,r)$ in such a way that $z$-axis goes close to axis of the cylinder and $r$-axis is restrained along the radial direction. Furthermore, a uniform magnetic field of length $B_{0}$ is applied along $r$-direction and further assumed that the features of induced magnetic field are insignificant owing to small magretic Reynolds number (as clarified in Fig. 1). Under the norm of these circumstances, the flow problem of Maxwell nanofluid is as follows
$$ \frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0\,, $$
$$ u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}+\lambda \left[ w^{2}\frac{\partial ^{2}w}{\partial z^{2}}+u^{2}\frac{\partial ^{2}w}{\partial r^{2}}+2uw\frac{\partial ^{2}w}{\partial r\partial z}\right] = +W_{e}\frac{{d}W_{e}}{{d}z}+\nu\left[ \frac{\partial ^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}\right]-\frac{\sigma B_{0}^{2}}{\rho _{f}}\left((W_{e}-w)+\lambda u\frac{\partial w}{\partial r}\right)\!,\\ $$
$$ u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\frac{1}{\left( \rho c\right) _{f}}\frac{1}{r}\frac{\partial }{\partial r}\left[K(T)\left( r\frac{\partial T}{\partial r}\right) \right] +\tau \left[ D_{B}\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}+\frac{D_{T}}{T_{\infty }}\left( \frac{\partial T}{\partial r}\right) ^{2}\right] +\frac{Q_{0}(T-T_{\infty })}{(\rho c)_{f}}, $$
$$ u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_{B}\frac{1}{r}\frac{\partial }{\partial r}\left[ \left( r\frac{\partial T}{\partial r}\right) \right] +\frac{D_{T}}{T_{\infty }}\frac{1}{r}\frac{\partial }{\partial r}\left[ \left( r\frac{\partial T}{\partial r}\right) \right]-k_{c}(C-C_{\infty }), $$
Fig. 1 Geometry of coordinates system. |
subject to the boundary conditions
$$ w(z,r)=W(z)=\frac{U_{0}z}{l}, \quad u(z,r)=0, \nonumber\\ T=T_{w},\quad C=C_{w}\quad {\rm at} \quad r=R, $$
$$ w\rightarrow W_{e}(z)=\frac{U_{\infty }z}{l},\quad T\rightarrow T_{\infty },\nonumber\\ C\rightarrow C_{\infty }\quad {\rm as}\quad r\rightarrow \infty \text{.} $$
Here $(u,w)$ denotes the velocity components in the $r$- and $z$-directions, respectively, $\nu $ the kinematic viscosity, $\lambda $ the relaxation time, $(\rho _{f},c_{f})$ the liquid density and specific heat respectively, $(T,C)$ the nanoliquid temperature and concentration, respectively, $% (D_{B},D_{T})$ the Brownian and thermophoresis diffusion coefficients, respectively, $\tau $ the ratio of effective heat capacity of nanoparticles material to the heat capacity of the base liquid, $Q_{0}$ the heat sink/source coefficient and $k_{c}$ the coefficient of chemical reaction. Moreover, the variable thermal conductivity $k(T)$ is defined as[25-27]
$ k(T)=k_{1}\left[ 1+\varepsilon \left( \frac{T-T_{\infty }}{\Delta T}\right)\right] ,$
where $k_{1}$ is the nanoliquid thermal conductivity, ε the small scale thermal conductivity parameter and $\Delta T$ the temperature difference between the liquid temperature of stretched surface and far away from the surface of cylinder.
2.1 Appropriate Conversions
Letting
$$ u\!=\!-\!\frac{R}{r}\sqrt{\frac{U_{0}\nu }{l}}f(\eta ),\quad w\!=\!\frac{U_{0}z}{l}f^{\prime }(\eta ),\quad \theta (\eta )\!=\!\frac{T\!-\!T_{\infty }}{T_{w}\!-\!T_{\infty }},\phi (\eta )=\frac{C-C_{\infty }}{C_{w}-C_{\infty }},\text{}\eta =\sqrt{\frac{U_{0}}{\nu l}}\left( \frac{r^{2}-R^{2}}{2R}\right). $$
In outlook of Eq. $(7)$ and overhead alterations $(8)$, Eq. $(1)$ is satisfied automatically and Eqs. (2)--(6) compact to
$$ (1+2\alpha \eta )f^{\prime \prime \prime }+2\alpha f^{\prime \prime}+ff^{\prime \prime }-f^{\prime 2}+2\beta ff^{\prime }f^{\prime \prime}-\beta f^{2}f^{\prime \prime \prime } \nonumber \\-\frac{\alpha \beta }{(1+2\alpha \eta )}f^{2}f^{\prime \prime}-M^{2}(f^{\prime }-A-\beta ff^{\prime \prime })+A^{2}=0, $$
$$ (1\!+\!2\alpha \eta )(\theta \theta ^{\prime \prime }\!+\!\theta ^{\prime2})\varepsilon \!+\!2\alpha \varepsilon \theta \theta ^{\prime }\!+\!(1\!+\!2\alpha \eta)\theta ^{\prime \prime }\!+\!2\alpha \theta ^{\prime }\!+\!(1\!+\!2\alpha \eta )Pr N_{b}\theta ^{\prime }\phi ^{\prime }\!+\!Pr f\theta^{\prime }\!+\!(1\!+\!2\alpha \eta )Pr N_{t}\theta ^{\prime 2}\!+\!Pr \delta \theta=0, $$
$$ (1+2\alpha \eta )\phi ^{\prime \prime }+2\alpha \phi ^{\prime }+Le Pr f\phi^{\prime }+(1+2\alpha \eta )\left( \frac{N_{t}}{N_{b}}\right) \theta^{\prime \prime }+2\alpha \left( \frac{N_{t}}{N_{b}}\right) \theta ^{\prime}-Le Pr C_{r}\phi =0, $$
$$ f(0)=0,\quad f^{\prime }(0)=1,\quad \theta \left( 0\right) =1,\quad \phi (0)=1, $$
$ f^{\prime }(\infty )=A,\quad \theta (\infty )=0,\quad \phi(\infty )=0\,.$
Here $(\alpha$ , $\beta$ ,$A,$ $M,$ $N_{b}$,$N_{t},\ \delta $ $Pr$ ,$Le$, $C_{r})$ are, respectively, the curvature parameter, Deborahnumber, velocities ratio parameter, magnetic parameter, Brownian motion parameter, thermophoresis parameter, heat sink/source parameter, Prandtl number, Lewis number and chemical reaction parameter and are defined as
$$ \!\alpha\! \left(\!=\!\frac{1}{R}\sqrt{\frac{\nu l}{U_{0}}}\right) ,\quad \!\beta\!\left(\!=\!\frac{\lambda U_{0}}{l}\right),\quad A\!=\!\frac{U_{\infty } }{U_{0}},\nonumber\\\ M\!\left(\!=\!\sqrt{\frac{\sigma B_{0}^{2}l}{U_{0}\rho _{f}}}\right), \quad N_{b}\!\left(\!=\!\frac{\tau D_{B}(C_{w}\!-\!C_{\infty })}{\nu }\right),\nonumber \\\ N_{t}\!\left(\!=\!\frac{\tau D_{T}(T_{w}\!-\!T_{\infty })}{\nu T_{\infty }}\right),\quad \!\delta \!\left(\!=\!\frac{lQ_{0}}{U_{0}(\rho c)_{f}}\right),\nonumber\\\ \!Pr \!\left(\!=\!\frac{\nu }{\alpha _{1}}\right),\quad Le\left(\!=\!\frac{\alpha _{1}}{D_{B}}\right),\quad \!C_{r}\!\left(\!=\!\frac{k_{c}l}{U_{0}}\right). $$
3 Physical Quantities of Interest
The rates of heat and mass transport $(Nu_{z},Sh_{z})$ are
$$ Nu_{z}=\frac{zq_{m}}{k(T_{f}-T_{\infty })}, \quad Sh_{z}=\frac{zj_{m}}{D_{B}(C_{w}-C_{\infty })}, $$
where $\left( q_{m},j_{m}\right) ,$ respectively, the heat and mean fluxes
$$ q_{m}=-k\left( \frac{\partial T}{\partial r}\right) _{r=R},\quad j_{m}=-D_{B}\left( \frac{\partial C}{\partial r}\right)_{r=R}. $$
Utilizing the above equations we attain
$$ Nu_{z}{Re}_{z}^{-{1}/{2}}=-\theta ^{\prime }(0),\quad Sh_{z}{Re}_{z}^{-{1}/{2}}=-\phi ^{\prime }\left( 0\right) , $$
where ${Re}_{z}={W(z)z}/{\nu }$ depicts the local Reynolds number.
4 Homotopic Solutions
The homotopy analysis method provides us excessive independence and an informal approach to regulate and control the convergence constituency of the series elucidations. To scrutinize the stimulus of inferential parameters a well-organized analytical methodology, namely homotopy analysis method (HAM) has been betrothed which solves nonlinear ODEs. The initial estimates $\left( f_{0},\text{ }\theta _{0},\text{ }\phi _{0}\right) $ and linear operators $(￡ _{f},$ $￡ _{\theta },$ $￡ _{\phi })$ for such scheme are defined by
$$ f_{0}(\eta )=1\!-\!A(1-\eta )-e^{-\eta }(1-A),\nonumber\\\ \theta _{0}=e^{-\eta },\quad \phi _{0}\!=\!e^{\!-\!\eta }\,, $$
$$ ￡ _{f}=f^{\prime \prime \prime }-f,\quad ￡ _{\theta}=\theta ^{\prime \prime }-\theta ,\quad ￡ _{\phi }=\phi^{\prime \prime }-\phi\,, $$
with the properties
$$ ￡ _{f}[B_{1}^{\ast }\!+\!B_{2}^{\ast }e^{\!-\!\eta }\!+\!B_{3}^{\ast }e^{\eta }]=0,\quad ￡ _{\theta }[B_{4}^{\ast }e^{\!-\!\eta }\!+\!B_{5}^{\ast }e^{\eta}]=0,\nonumber\\\ ￡ _{\phi }[B_{6}^{\ast }e^{\!-\!\eta }\!+\!B_{7}^{\ast}e^{\eta }]=0, $$
in which $B_{i}^{\ast }$ $(i=\text{1--7})$ are the arbitrary constants.
5 Graphical Depiction and Analysis
To explore the aspect of influential variables in MHD flow of Maxwell nanofluid near the stagnation region with the inducement of chemical reaction and variable thermal conductivity the current section is structured. The graphs are intended and conferred for the velocity, temperature, concentration and reduced Nusselt number with detail.
Figure 2 portrays the aspect of stagnation point $A$ on the velocity component $f^{\prime }(\eta )$. Remarkably, we noted that $f^{\prime }(\eta ) $ intensified when the value of $A$ exaggerates. The thickness of boundary layer progresses for $A＜1$ as stretching rate exceeds the free stream amount. Moreover, the straining motion close to the stagnation area rises which intensifies the acceleration of the external stream and declines the thickness of the boundary layer with rise in $A$. However, it falloffs when $% A＞1,$ i.e.\ the amount of free stream velocity is higher when correlated with amount of stretching velocity. There is no development of boundary layer when $A=1$ as the stretching and free stream velocities are equivalent as noted via Fig. 2. Furthermore, we concluded that the conflicting performance is being acknowledged for both $A＞1$ and $A＜1$.
Fig. 2 (Color online) Influence of $A$ on $f^{\prime }(\eta)$. |
An escalation in thermal conductivity parameter $\varepsilon $ and Deborah number $\beta $ have conflicting behavior on $\theta (\eta )$ as portrayed in Figs. 3(a), 3(b). Caused by the enormous quantity of heat transfer from surface to material the liquid thermal conductivity of Maxwell liquid intensifies with the temperature field boost up. Moreover, enrichment in $% \beta $ spectacles divergent performance in temperature and decays the temperature of Maxwell liquid as shown in Fig. 3(b). As collision between liquid components boost up when $\beta $ intensifies resulting in a decay the liquid temperature. To picture the stimulus of heat sink/source for heat transfer feature, Figs. 4(a), 4(b) are strategized. It is remarked that the temperature field falloff for $\delta ＞0$ and enriched for $\delta ＞0$. Furthermore, in instance of heat source parameter results in heat to be delivered to the liquid owing to which the liquid temperature rise while heat is fascinated in instance of heat sink parameter due to which the temperature of liquid decline. Thus, a differing behavior is being commented for $\delta ＜0$ and $\delta ＞0.$ Figures 5(a), 5(b) are delineated to picture the influence of nanoparticles $N_{b}$ and $N_{t}$ on $\theta (\eta )$. An augmenting enactment of temperature field is observed for both $N_{b}$ and $% N_{t}$. Physically, higher $N_{b}$ has advanced $D_{B}$ which improves the temperature of nanoliquid because $N_{b}$ has direct relation with $D_{B}$. Additionally, the theromophoretic force intensifies for $N_{t}$. Thus, the liquid particles transport rapidly from hot to cold area. Therefore, the nanoliquid temperature heightens. Figures 6(a), 6(b) report the graphical sketch of magnetic parameter $M$ and Prandtl number $Pr $ on
Fig. 3 (Color online) Influence of $\varepsilon $ (a) and $\beta $ (b) on $\theta (\eta )$. |
Fig. 4 (Color online) Influence of $\left( \delta ＜0\right) $ (a) and $\left( \delta＞0\right) $ (b) on $\theta (\eta )$. |
temperature field $\theta (\eta )$. These displays portray that the temperature rises when $M$ intensifies while it declines for $Pr $. The thermal thickness of thermal layer and temperature rise for boosting the value of $M$. The conflicting enactment is being acknowledged for $Pr $. The progressive value of $Pr $ indicates to the lesser thermal diffusivity and decreases the aptitude of energy transfer, which spectacles the reduction of temperature of Maxwell liquid. Furthermore, the relative thickness of velocity and thermal thickness of boundary layers is controlled by $Pr $.
The aspects of diverse values of Brownian $N_{b}$ and thermophrotic $N_{t}$ nanoparticles on concentration field $\phi (\eta )$ are described via Figs. 7(a), 7(b). The $N_{b}$ is diminishing function of concentration of Maxwell liquid, but its augmenting function of $N_{t}.$ The behavior of $ N_{b}$ occurred because $N_{b}$ intensifies the unsystematic gesture of the liquid particles which generates extra heat and diminish the concentration of Maxwell liquid. Moreover, the liquid thermal conductivity enhances in the existence of nanoparticles. Higher $N_{t}$ contributes intensification in the conductivity which displays higher concentration and allied thickness. Hence, we established that both parameters $N_{b}$ and $ N_{t}$ have opposite trend on $\phi (\eta )$. Figures 8(a), 8(b) illustrate the features of chemical reaction $C_{r}( C_{r}＜0$ and $C_{r}＞0)$ on concentration field $\phi (\eta )$. These plots specify a reverse propensity on concentration field. The larger destructive chemical reaction parameter $C_{r}＞0$ decreases the concentration field and an opposite behavior is reported for generative chemical reaction $C_{r}＜0$ and for $C_{r}=0$ no reaction occurs. Physically, the higher value of $C_{r}＞0$ reasons to decline in the chemical molecular diffusivity, which decays the concentration field.Hence, conflicting behavior is noted for both cases on concentration field.
Fig. 5 (Color online) Influence of $N_{b}$ (a) and $N_{t}$ (b) on $\theta (\eta )$. |
Fig. 6 (Color online) Influence of $M$ (a) and $Pr $ (b) on $\theta (\eta )$. |
Fig. 7 (Color online) Influence of $N_{b}$ (a) and $N_{t}$ (b) on $\phi (\eta )$. |
To pinpoint the performance of $N_{t},$ $N_{b},$ $Pr $ and $\varepsilon $ on heat transfer amount, Figs. 9(a), 9(b) and 10(a), 10(b) are drafted. These plans exhibit similar behavior for $N_{t},$ $N_{b}$ and $\varepsilon $ whereas conflicting trend is being reported for $Pr $. The heat transfer amount declines for $N_{t},$ $N_{b}$ and $\varepsilon $ but enhances for $% Pr $.
Fig. 8 (Color online) Influence of $\left( C_{r}＜0\right) $ (a) and $\left(C_{r}＞0\right) $ (b) on $\phi (\eta )$. |
Fig. 9 (Color online) Behavior of $N_{t}$ (a) and $N_{b}$ (b) on $-\theta ^{\prime }(0)$. |
Fig. 10 (Color online) Behavior of $Pr $ (a) and $\varepsilon $ (b) on $-\theta ^{\prime}(0)$. |
Table 1 establishes the numerical value of $-f^{\prime \prime }(0)$ associated with that of Refs. [28--29]. Additionally, Tables 2 and 3 spectacle the comparison values of $Pr $ for $-\theta ^{\prime }$(0) with previous studies. A remarkable settlement between these upshots is being detected in these tables. Hence, we are assured that the latest consequences are very precise.
Table 1 Numerical values of $-f^{\prime \prime }(0)$ for$\ \beta $ in limiting cases when $\alpha =M=A=0$. |
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Table 2 Numerical values of $-\theta ^{\prime }(0)$ when $\alpha=\beta =\delta = A =M=N_{t}=\varepsilon =C_{r}=0$ and$\ N_{b}\rightarrow 0.$ |
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Table 3 Numerical values of $-\theta ^{\prime }(0)$ when $\alpha=\beta =\delta =A =M=N_{t}=\varepsilon =C_{r}=0$ and$\ N_{b}\rightarrow 0.$ |
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6 Closing Remarks
To disclose the aspects of the stagnation point flow in MHD Maxwell nanofluid caused by stretching cylinder this section has been established. For the solution of the problem homotpy analysis method (HAM) has been consumed. The crucial outlooks of this study are as follows;
$\bullet$ For $A＜1$ the thickness of the momentum boundary layer enriched;however, for $n＞1$ it diminished when $A$ increases.
$\bullet$ The larger magnetic parameter $M$ enhanced the temperature of Maxwell fluid while the Deborah number $\beta$ decayed the temperature field.
On temperature field quite conflicted behaviors were being detected for heat sink/source parameter $\delta$.
$\bullet$ The variable thermal conductivity $\varepsilon $ increased nanoliquid temperature field.
$\bullet$ A similar trend was being acknowledged for thermophrotic parameter $N_{t}$ on temperature and concentration of Maxwell nanofluid.
$\bullet$ Local Nusselt number enhanced for increasing Prandtl number $Pr $ and declined for increasing thermophrotic parameter $N_{t}$ and variable conductivity $\varepsilon$ .
Conflict of interest
Authors have no conflict of interest.