Marangoni flow and mass transfer of power-law non-Newtonian fluids over a disk with suction and inje
本站小编 Free考研考试/2022-01-02
Yanhai Lin,, Meng YangFujian Province University Key Laboratory of Computation Science and School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
Abstract We scrutinize the approximate analytical solutions by the optimal homotopy analysis method (OHAM) for the flow and mass transfer within the Marangoni boundary layer of power-law fluids over a disk with suction and injection in the present paper. Concentration distribution on the surface of a disk varies in a power-law form. The non-Newtonian fluid flow is due to the surface concentration gradient without considering gravity and buoyancy. According to the conservation of mass, momentum and concentration, the governing partial differential equations are established, and the appropriate generalized Kármán transformation is found to reduce them to the nonlinear ordinary differential equations. OHAM is used to access the approximate analytical solution. The influences of Marangoni the number, suction/injection parameters and power-law exponent on the flow and mass transfer are examined. Keywords：non-Newtonian fluid;Marangoni boundary layer;optimal homotopy analysis method;disk;suction and injection;mass transfer
PDF (997KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yanhai Lin, Meng Yang. Marangoni flow and mass transfer of power-law non-Newtonian fluids over a disk with suction and injection. Communications in Theoretical Physics, 2020, 72(9): 095003- doi:10.1088/1572-9494/aba247
1. Introduction
Non-Newtonian fluids can be found in a wide range of nature and production processes. For example, concentrated solutions and suspensions of polymer such as polyethylene, petroleum and toothpaste belong to non-Newtonian fluids. Furthermore, all kinds of engineering plastics, tomato juice and jam and so on are included as well. Because of their widespread existence, there are considerable investigations [1–3] focused on them for a long time. Some models for non-Newtonian fluids were studied, such as power-law fluid model, Bingham fluid model [4, 5], third grade model [6, 7], Powell–Eyring fluid model [8], Carreau fluid model [9, 10], Williamson fluid model [11] and so on. Moreover, power-law fluid is one of the most common fluids among them which rheological law is given by $\tau =K{\left(\tfrac{\partial u}{\partial y}\right)}^{n},$ where $\tau $ is shear stress, $K$ is the consistency index, $n$ is the power-law exponent of the fluid. As a typical kind of time independent non-Newtonian fluids, power-law fluids have attracted wide attention of researchers such as their convective heat transfer under various conditions [12–14]. The methods used to solve the established mathematical and physical models are also studied, including exact and approximate solutions, numerical solutions and analytical solutions. Because of behavior of non-Newtonian fluids is complex and the transport essence is different to the Newtonian fluids, the classical Fourier’s law of heat conduction and Fick’s law of diffusion in Newtonian fluids can no longer accurately describe the heat transfer and diffusion process of non-Newtonian fluids. As a result, Pop et al [15] raised a modified Fourier’s heat conduction law that the power-law viscosity coefficient was also considered in the temperature field, and Zheng et al [16] assumed that the temperature field resembled the velocity field, and the power-law viscosity coefficient was taken into account in the thermal conductivity for heat transfer process. N-power-law diffusion equation was considered as mass transfer process. Some investigations [17, 18] showed that they approximately expressed the motion of the actual non-Newtonian fluids.
Marangoni convection is a capillary convection dominated by a surface tension gradient at two-phase interface which caused by evaporation, dissolution, migration of surface active substances or temperature effects etc. In the mid-1960s, C Marangoni founded this phenomenon and named it. It exists in many processes among aerospace, crystal growth, materials science and a great deal of studies [19–22] revealed that this phenomenon is dreadfully significant under microgravity conditions. The Marangoni convection differs by the induced factors that contribute to the surface tension gradient, divided into a temperature-driven thermal Marangoni effect (EMT) and a concentration-driven solute Marangoni effect (EMS). The initial model which illustrates a connection between surface tension and surface temperature is established by Pearson [23]. Acrivos et al [24] was the first one to apply the boundary layer theory to power-law fluids, and derived the classical governing equations for power-law fluid boundary layer. Scriven and Sterling [25] had firstly established a physical model of the flow induced by temperature gradient coupled with concentration gradient. Napolitano and Golia [26] numerically solved the flow field coupled kinematically and thermally in the Marangoni boundary layers. Pop et al [27] investigated the Marangoni problem at interface caused by thermal gradient coupled with solute gradient. The mixed convection boundary layer within Marangoni effect as well as buoyancy effect was explored by Chamkha et al [28]. Almudhaf and Chamkha [29] presented the magnetohydrodynamic (MHD) Marangoni convection flow within heat generation\absorption and chemical reaction. In addition, many scholars had also scrutinized Marangoni effect on a power-law thin liquid film. Chen [30] examined the Marangoni thin flow heat diffusion of power-law fluids film driven by a stretched sheet, and founded the thermally-induced Marangoni convection affected velocity and temperature distributions significantly.
For the past decade, researches on Marangoni convection in power-law fluids have been developed. The exact analytical solutions of Marangoni convection within heat sources/sinks and MHD Marangoni convection with external magnetic field were revealed by Magyari and Chamkha [31, 32], respectively. Hamid et al [33] extended the model of Pop et al [27] to a permeable wall and obtained dual solutions by the shooting method. Zheng and his coworkers [34–38] discussed Marangoni convection on the power-law fluids a lot. Zheng et al [34] established a model of Marangoni flow boundary layer induced by the power-law temperature gradient and the heat transfer equation adopted Zheng’s model [17]. Lin et al [35] examined that flow and heat transfer of pseudo-plastic nanofluids in Marangoni boundary layer driven by a varied temperature. The nanofluids consisting of five different nanoparticles were compared and the radiation conditions were taken into account. Jiao et al [36] reported MHD thermosolutal Marangoni convection flow and transfer courses driven by the temperature and concentration in power-law form. Jiao et al [37] also investigated the transport characteristics of Marangoni velocity and temperature boundary layers of power-law fluids in the presence of porous medium. Zheng et al [38] further investigated the previous problem with chemical reaction using the homotopy analysis method (HAM). Mahdy and Ahmed [39] analyzed the influences of various parameters including Soret and Dufour parameters on MHD thermosolutal Marangoni boundary layer past a vertical flat plate due to non-uniform temperature and concentration. Sheikholeslami and Ganji [40] reported the flow and heat diffusion of Marangoni boundary layer in CuO–H2O nanofluid with applied magnetic field, the results showed that the velocity and temperature fields enhance as the volume fraction of CuO augments. The Marangoni convective heat transfer of two-phase nanofluids was also deliberated numerically by Sheikholeslami and Chamkha [41]. Refer to problems of Marangoni boundary layer flow on a disk, the transport characteristics of the Marangoni boundary layer on the porous media disk was considered by Lin and Zheng [42]. The numerical solutions as well as approximate analytical solution of the established two-dimensional boundary layer model were obtained. Mahanthesh et al [43] addressed the Marangoni convection due to an exponential space dependent heat source of SWCNT and MWCNT nanofluids with magnetic force and radiation in a disk, and the numerical simulation solutions exhibited that the heat transfer rate increases via Marangoni convection, but decreases due to the applied magnetic force. Under the same conditions, Mahanthesh et al [44] analyzed the impacts of nanoparticle shapes and Marangoni convection on the flow fields in an infinite disk, and the results showed that the stronger the volume fraction of nanofluids and Marangoni convection are, the faster the heat transfer rate is. Mahanthesh et al [45] showed the Marangoni boundary layer flow and transport behavior of Casson fluids on an infinite disk.
Because non-Newtonian power-law fluids have strong nonlinear properties, there are few related investigations on Marangoni boundary flow over a disk [43–46]. However, HAM [47] is an analytical approximation method which is suit for solving strongly nonlinear problems. It was proposed by Liao and its principle based on application of the homotopy theory to transform a nonlinear problem into a series of linear problems. It solved many problems [48, 49] about strongly nonlinear problems till now. However, in the process of calculation, the slow convergence of series solution may lead to inefficient calculation. For this reason, Liao [50] raised an improvement of HAM named optimal homotopy analysis method (OHAM). The Marangoni flow and heat transfer of power-law non-Newtonian fluids over a disk using OHAM is scrutinized in this paper. Effects of suction and injection are considered. A serial of approximate analytical solutions of velocity and concentration are given. The effects of some important physical parameters such as Marangoni number and power-law exponent on the flow and mass transfer are illustrated with graphs and explained.
2. Problem model and basic equations
Consider the steady, two-dimensional, laminar, boundary layer Marangoni convection flow of a non-Newtonian power-law fluid over an infinite disk. Gravity and buoyancy are not considered. Concentration changes in a power-law form at the surface. Assuming that the fluid is incompressible and the flow is axi-symmetric. Establishing a cylindrical coordinate system on such a physical model which is offered in figure 1, and the role of Marangoni convection is reflected in the boundary layer condition of this model. On the base of the conservation of mass, momentum and mass transfer, the model is written as follows [42]$ \begin{eqnarray}\displaystyle \frac{\partial U}{\partial R}+\displaystyle \frac{U}{R}+\displaystyle \frac{\partial W}{\partial Z}=0,\end{eqnarray}$$ \begin{eqnarray}U\displaystyle \frac{\partial U}{\partial R}+W\displaystyle \frac{\partial U}{\partial Z}=\displaystyle \frac{\partial }{\partial Z}\left(\nu {\left|\displaystyle \frac{\partial U}{\partial Z}\right|}^{n-1}\displaystyle \frac{\partial U}{\partial Z}\right),\end{eqnarray}$$ \begin{eqnarray}U\displaystyle \frac{\partial C}{\partial R}+W\displaystyle \frac{\partial C}{\partial Z}=\displaystyle \frac{\partial }{\partial Z}\left(\lambda {\left|\displaystyle \frac{\partial C}{\partial Z}\right|}^{n-1}\displaystyle \frac{\partial C}{\partial Z}\right),\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}Z=0:\rho \nu {\left|\displaystyle \frac{\partial U}{\partial Z}\right|}^{n-1}\displaystyle \frac{\partial U}{\partial Z}=-\displaystyle \frac{\partial \sigma }{\partial R},W=k{R}^{\displaystyle \frac{n-1}{2-n}},\\ C={C}_{\infty }+A{R}^{\displaystyle \frac{1}{2-n}},\end{array}\end{eqnarray}$$ \begin{eqnarray}Z\to \infty :U=0,C={C}_{\infty },\end{eqnarray}$where $U$ and $W$ are radial and axial velocities, respectively. $C$ is the concentration. The kinematic viscosity coefficient is denoted by $\nu {\left|\tfrac{\partial U}{\partial Z}\right|}^{n-1}$ and $\nu $ is a constant. $\rho $ is the fluid density. The mass diffusivity is denoted by $\lambda {\left|\tfrac{\partial C}{\partial Z}\right|}^{n-1}$ and $\lambda $ is a constant. $k$ is the surface suction/injection constant. When $k\gt 0,$ it indicates that there is injection at the interface, and when $k\lt 0,$ there is suction. $n$ is the power-law exponent [51]. $A$ represents a positive constant. $\sigma $ represents the surface tension, the expression of the surface tension and concentration can be written as$ \begin{eqnarray}\sigma ={\sigma }_{0}+\displaystyle \frac{{\gamma }_{C}}{2}{\left(C-{C}_{\infty }\right)}^{2},\end{eqnarray}$and$ \begin{eqnarray}{\gamma }_{C}={\left.\displaystyle \frac{{\partial }^{2}\sigma }{\partial {C}^{2}}\right|}_{C={C}_{\infty }}.\end{eqnarray}$
The dimensionless variables can be written as$ \begin{eqnarray}u=\displaystyle \frac{U}{{U}_{{\rm{\Delta }}}},w=\displaystyle \frac{W}{{U}_{{\rm{\Delta }}}}R{e}^{\displaystyle \frac{1}{n+1}},r=\displaystyle \frac{R}{{R}_{{\rm{\Delta }}}},z=\displaystyle \frac{Z}{{R}_{{\rm{\Delta }}}}R{e}^{\displaystyle \frac{1}{n+1}},c=\displaystyle \frac{C}{{C}_{\infty }},\end{eqnarray}$and$ \begin{eqnarray}Re=\displaystyle \frac{{{U}_{{\rm{\Delta }}}}^{2-n}{{R}_{{\rm{\Delta }}}}^{n}}{\rho \nu },Sc=\displaystyle \frac{\rho \nu {{U}_{{\rm{\Delta }}}}^{n-1}}{\lambda {{C}_{\infty }}^{n-1}}.\end{eqnarray}$In these variables, ${U}_{{\rm{\Delta }}}$ is unit of the velocity and ${R}_{{\rm{\Delta }}}$ is unit of the length. $Re$ is Reynolds number and $Sc$ is Schmidt number.
The concentration gradient at the interface induced flow as $\tfrac{\partial \sigma }{\partial R}=\tfrac{\partial \sigma }{\partial C}\cdot \tfrac{\partial C}{\partial R}.$ Then, the governing boundary layer equations and the corresponding boundary conditions (1)–(5) are converted into$ \begin{eqnarray}\displaystyle \frac{\partial u}{\partial r}+\displaystyle \frac{u}{r}+\displaystyle \frac{\partial w}{\partial z}=0,\end{eqnarray}$$ \begin{eqnarray}u\displaystyle \frac{\partial u}{\partial r}+w\displaystyle \frac{\partial u}{\partial z}=\displaystyle \frac{\partial }{\partial z}\left({\left|\displaystyle \frac{\partial u}{\partial z}\right|}^{n-1}\displaystyle \frac{\partial u}{\partial z}\right),\end{eqnarray}$$ \begin{eqnarray}u\displaystyle \frac{\partial c}{\partial r}+w\displaystyle \frac{\partial c}{\partial z}=\displaystyle \frac{{\rm{1}}}{Sc}\displaystyle \frac{\partial }{\partial z}\left({\left|\displaystyle \frac{\partial c}{\partial z}\right|}^{n-1}\displaystyle \frac{\partial c}{\partial z}\right),\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}z=0:{{\left|\displaystyle \frac{\partial u}{\partial z}\right|}^{n-1}\displaystyle \frac{\partial u}{\partial z}| }_{z=0}=-\displaystyle \frac{1}{2-n}\displaystyle \frac{M{a}_{C}}{Sc}R{e}^{-\displaystyle \frac{3}{n+1}}{r}^{\displaystyle \frac{n}{2-n}}\,,\\ \,w(0)=k{{R}_{{\rm{\Delta }}}}^{\displaystyle \frac{n-1}{2-n}}{r}^{\displaystyle \frac{n-1}{2-n}},c(0)=1+\displaystyle \frac{A{{R}_{{\rm{\Delta }}}}^{\displaystyle \frac{1}{2-n}}}{{C}_{\infty }}{r}^{\displaystyle \frac{1}{2-n}},\end{array}\end{eqnarray}$$ \begin{eqnarray}z\to \infty :u=0,c=1,\end{eqnarray}$where $M{a}_{C}$ is the solutal Marangoni number and $M{a}_{C}=\tfrac{{\gamma }_{C}{A}^{2}}{\lambda }\tfrac{{{R}_{{\rm{\Delta }}}}^{\displaystyle \tfrac{n}{2-n}+n}}{{U}_{{\rm{\Delta }}}{{C}_{\infty }}^{n-1}R{e}^{\displaystyle \tfrac{3-n}{n+1}}}.$
Then, the equations (10)–(14) simplified to the equations (18)–(22)$ \begin{eqnarray}\displaystyle \frac{3-n}{2-n}f\left(\eta \right)+h^{\prime} \left(\eta \right)=0,\end{eqnarray}$$ \begin{eqnarray}\left({\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right)\right)^{\prime} =\displaystyle \frac{1}{2-n}{f}^{2}\left(\eta \right)+h\left(\eta \right)f^{\prime} \left(\eta \right),\end{eqnarray}$$ \begin{eqnarray}\left({\left|\phi ^{\prime} \left(\eta \right)\right|}^{n-1}\phi ^{\prime} \left(\eta \right)\right)^{\prime} =Sc\left[\displaystyle \frac{1}{2-n}f\left(\eta \right)\phi \left(\eta \right)+h\left(\eta \right)\phi ^{\prime} \left(\eta \right)\right],\end{eqnarray}$$ \begin{eqnarray}f^{\prime} \left({\rm{0}}\right)=-Ma,h\left({\rm{0}}\right)={h}_{0},\phi \left({\rm{0}}\right)=1,\end{eqnarray}$$ \begin{eqnarray}f\left(\infty \right)=0,\phi \left(\infty \right)=0,\end{eqnarray}$where, ${h}_{0}=\tfrac{{C}_{2}}{{C}_{1}}k{{R}_{{\rm{\Delta }}}}^{\tfrac{n-1}{2-n}}$ denotes the suction or injection parameter such that ${h}_{0}\gt 0$ and ${h}_{0}\lt 0$ mean injection and suction respectively. $Ma$ is the Marangoni number and it satisfies as $ \begin{eqnarray*}M{a}^{n}=\displaystyle \frac{1}{2-n}\displaystyle \frac{M{a}_{C}}{Sc}R{e}^{-\displaystyle \frac{3}{n+1}}{({C}_{1}{C}_{2})}^{-n}.\end{eqnarray*}$
3. Optimal homotopy analysis solution
The power-law index $n$ is a positive real number. For $0\lt n\lt 1,$ the equations (1)–(5) describe shear thinning fluids, and they describe dilatants fluids for $n\gt 1.$ As $n=1,$ the equations (1)–(5) degenerate into the case of Newtonian fluids. We convert the equation (18) into the equation (23) firstly$ \begin{eqnarray}f\left(\eta \right)=-\displaystyle \frac{2-n}{3-n}h^{\prime} \left(\eta \right).\end{eqnarray}$
Next we assume that $h^{\prime\prime} \left(\eta \right)\gt 0$ and $\phi ^{\prime} \left(\eta \right)\lt 0$ due to the boundary conditions (21), (22) and substitute (23) into (19)–(22), the equations (19)–(22) then become$ \begin{eqnarray}\begin{array}{l}{\left(\displaystyle \frac{2-n}{3-n}\right)}^{n-1}n{h^{\prime\prime} }^{n-1}\left(\eta \right)h\prime\prime\prime \left(\eta \right)+\displaystyle \frac{1}{3-n}{h^{\prime} }^{2}\left(\eta \right)\\ \,-h\left(\eta \right)h^{\prime\prime} \left(\eta \right)=0,\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}n{\left(-\phi ^{\prime} \left(\eta \right)\right)}^{n-1}\phi ^{\prime\prime} \left(\eta \right)+Sc\left[\displaystyle \frac{1}{3-n}h^{\prime} \left(\eta \right)\phi \left(\eta \right)\right.\\ \,-\left.h\left(\eta \right)\phi ^{\prime} \left(\eta \right)\right]=0,\end{array}\end{eqnarray}$$ \begin{eqnarray}h^{\prime\prime} \left({\rm{0}}\right)=\displaystyle \frac{3-n}{2-n}Ma,h\left({\rm{0}}\right)={h}_{0},\phi \left({\rm{0}}\right)=1,\end{eqnarray}$$ \begin{eqnarray}h^{\prime} \left(\infty \right)=0,\phi \left(\infty \right)=0.\end{eqnarray}$
Solving the above equations with OHAM, the following initial guess solutions are selected first.$ \begin{eqnarray}\begin{array}{l}{h}_{0}\left(\eta \right)={h}_{0}-\displaystyle \frac{3-n}{2-n}Ma+\displaystyle \frac{3-n}{2-n}Ma{{\rm{e}}}^{-\eta },\\ {\phi }_{0}\left(\eta \right)={{\rm{e}}}^{-\eta }.\end{array}\end{eqnarray}$
Then, the suitable auxiliary linear operators are selected$ \begin{eqnarray}{L}_{h}=h\prime\prime\prime -h^{\prime} ,\end{eqnarray}$$ \begin{eqnarray}{L}_{\phi }=\phi ^{\prime\prime} -\phi ,\end{eqnarray}$with$ \begin{eqnarray}{L}_{h}\left({C}_{1}+{C}_{2}{{\rm{e}}}^{\eta }+{C}_{3}{{\rm{e}}}^{-\eta }\right)=0,{L}_{\phi }\left({C}_{4}+{C}_{5}{{\rm{e}}}^{-\eta }\right)=0,\end{eqnarray}$where ${C}_{i}(i=1,2,3,4,5)$ are the integral constants.
The package bvph2.0 is used for convergent series solutions by solving deformation equations.
4. Results and discussion
The appropriate values of the control convergence parameters ${h}_{h}$ and ${h}_{\phi }$ are obtained by minimizing the error of the approximate analytical solutions. Here we quantify it by means of the average squared residual errors [50]:$ \begin{eqnarray}{\varepsilon }_{m}^{h}=\displaystyle \frac{1}{k+1}\displaystyle \sum _{j=0}^{k}{\left[{N}_{h}{\left(\displaystyle \sum _{i=0}^{m}h\left(\eta \right)\right)}_{\eta =j\delta \eta }\right]}^{2},\end{eqnarray}$$ \begin{eqnarray}{\varepsilon }_{m}^{\phi }=\displaystyle \frac{1}{k+1}\displaystyle \sum _{j=0}^{k}{\left[{N}_{\phi }{\left(\displaystyle \sum _{i=0}^{m}h\left(\eta \right),\displaystyle \sum _{i=0}^{m}\phi \left(\eta \right)\right)}_{\eta =j\delta \eta }\right]}^{2},\end{eqnarray}$and the total average squared residual error is defined as$ \begin{eqnarray}{\varepsilon }_{m}^{t}={\varepsilon }_{m}^{h}+{\varepsilon }_{m}^{\phi }.\end{eqnarray}$
In the light of bvph2.0, The values of control convergence parameters we get are ${h}_{h}=-1.0642$ and ${h}_{\phi }\,=-0.878\,236$ at 3rd order approximation when $n\,=1.6,Sc=0.6,Ma=0.5,{h}_{0}=0.$ And the individual average squared residual error and the total average squared residual error are list in table 1. To verify the accuracy and effectiveness of the present OHAM method, numerical results for $f(0)$ and $-\phi (0)$ given by shooting method are compared in table 2. Furthermore, considering different types of fluids (as $n=1.0$ is Newtonian fluid and $n=0.7$ is non-Newtonian fluid), comparison of analytical OHAM and numerical results for distributions of $f(\eta )$ and $\phi (\eta )$ for $Sc=1.0,Ma=0.5,{h}_{0}=0$ are presented in figures 2, 3. From table 2 and figures 2, 3, we can be seen that the analytical results obtained by OHAM agree well with the numerical results.
Table 1. Table 1.Individual averaged squared residual errors and total averaged squared errors at different order of approximations for $n=1.6,Sc=0.6,Ma=0.5,{h}_{0}=0.$
New window|Download| PPT slide Figure 2.Comparison of analytical and numerical results for Newtonian fluid ($n=1.0$) when $Sc=1.0,Ma=0.5,{h}_{0}=0.$
Figure 3.
New window|Download| PPT slide Figure 3.Comparison of analytical and numerical results for non-Newtonian fluid ($n=0.7$) when $Sc=1.0,Ma=0.5,{h}_{0}=0.$
Figures 4–6 present the role of power-law exponent reflects in the radial dimensionless velocity $f\left(\eta \right)$ (the velocity component along the surface of disk), the shear stress ${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right)$ and the concentration $\phi \left(\eta \right)$ in the case ${S}_{c}=0.6,{h}_{0}=0,Ma=0.5.$ As we can see from figure 4, the velocity field $f\left(\eta \right)$ decreases with the add in the location similarity variable $\eta .$ Moreover, the increase in power-law exponent reduces $f\left(\eta \right).$ The figure represents that the velocity boundary layer becomes thinner with the increment in power-law exponent. As for figure 5, the absolute value of dimensionless shear stress ${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right)$ declines as variable $\eta $ grows and the rise in power-law index has an effect to weaken the dimensionless shear stress. It is evident that increasing power-law exponent helps to reduce concentration field in figure 6.
New window|Download| PPT slide Figure 5.Effects of the power-law exponent on ${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right).$
Figures 7–9 clarify the effects of Marangoni number $Ma$ on $f\left(\eta \right),$${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right)$ and $\phi \left(\eta \right)$ when $n={\rm{1.6}},{Sc}_{}=0.6,{h}_{0}=0$ meaning that there is no suction or injection for the dilatant fluids. From figure 7, We can observe that the dimensionless velocity curve $f\left(\eta \right)$ descends to convergence to 0 as the location similarity variable $\eta $ grows, and $f\left(\eta \right)$ decreases and converges slower as the Marangoni number $Ma$ decreases. In figure 8, the magnitude of dimensionless shear stress ${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right)$ decreases as the variable $\eta $ increases, and it increases as the Marangoni number decreases. As for figure 9, as $\eta $ increases, the dimensionless concentration $\phi \left(\eta \right)$ decreases, furthermore, $\phi \left(\eta \right)$ decreases with the Marangoni number increases. In other word, the concentration boundary layer becomes thinner with the increment in Marangoni number.
New window|Download| PPT slide Figure 8.Effects of the Marangoni number on ${\left|f^{\prime} \left(\eta \right)\right|}^{n-1}f^{\prime} \left(\eta \right).$
Figures 10 and 11 revel the relevance of the suction/injection parameters ${h}_{0}$ and the velocity and the concentration fields when $n=1.6,{S}_{c}=0.6,Ma=0.5.$${h}_{0}\gt 0$ means injection, and ${h}_{0}\lt 0$ means injection. The figures show that $f\left(\eta \right)$ and $\phi \left(\eta \right)$ decrease when $\eta $ increases, and the dimensionless velocity and the dimensionless concentration increase with the increase in ${h}_{0},$ which indicates that the imposition of suction or injection tends to enhance the velocity and concentration gradients.
The approximation analytical study for Marangoni convective mass transfer of power-law fluids over a disk with suction and injection is presented in this paper. A new model is proposed under the conditions that the mass diffusion caused by nonlinear concentration gradient. The generalized Kármán transformation is used to transfer the mathematical model into ordinary differential equations which work out using OHAM. The effects of various parameters consist of power-law exponent, Marangoni number and suction/injection number on the dimensionless velocity and concentration are demonstrated graphically and analyzed. Some important findings are(a) The dimensionless velocity, concentration and the dimensionless shear stress gradually decrease to zero as the location similarity variable $\eta $ grows. It means that there is no change in velocity and shear force and concentration at a distance from the disk surface. (b)The increasing power-law exponent contributes to reduce the shear stress and the velocity and concentration boundary layers. (c)The increase in Marangoni number is conducive to increase the velocity field and the shear stress while reduce the concentration field. (d)The imposition of suction or injection conduces to raise the velocity and concentration fields.
In this study, we do not consider the combined effect of the temperature and we will consider this in future work. Also, we will continue our work on non-Newtonian nanofluids with traditional Maxwell model or two-phase Buongiorno model within Marangoni boundary layer problem. Indeed, we have done a little work in Newtonian fluid and nanofluid for Marangoni boundary layer over a disk [42, 46] and found that the combined effects of temperature and concentration are strongly nonlinear and complex. In this study, OHAM greatly improves the convergence speed compared with the standard HAM by the fitting iteration technique, but at the same time, it also loses some certain accuracy. So, we will continue our work on improving OHAM to solve the related problems.
Acknowledgments
Yanhai Lin was supported by the National Natural Science Foundation of China (No. 11702101), the Fundamental Research Funds for the Central Universities and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (No. ZQN-PY502), the Natural Science Foundation of Fujian Province (No. 2019J05093) and Quanzhou High-Level Talents Support Plan. Meng Yang was supported by Subsidized Project for Postgraduates’ Innovative Fund in Scientific Research of Huaqiao University.
,, Meng YangFujian Province University Key Laboratory of Computation Science and School of Mathematical Sciences, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
