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Aisha M. Alqahtani, Adnan, Umar Khan, Naveed Ahmed, Syed Tauseef Mohyud-Din, Ilyas Khan, "Numerical Investigation of Heat and Mass Transport in the Flow over a Magnetized Wedge by Incorporating the Effects of Cross-Diffusion Gradients: Applications in Multiple Engineering Systems", Mathematical Problems in Engineering, vol. 2020, Article ID 2475831, 10 pages, 2020. https://doi.org/10.1155/2020/2475831
Numerical Investigation of Heat and Mass Transport in the Flow over a Magnetized Wedge by Incorporating the Effects of Cross-Diffusion Gradients: Applications in Multiple Engineering Systems
The flow over a wedge is significant and frequently occurs in civil engineering. It is significant to investigate the heat and mass transport characteristics in the wedge flow. Therefore, the analysis is presented to examine the effects of preeminent parameters by incorporating the cross-diffusion gradients in the energy and mass constitutive relations. From the analysis, it is perceived that the temperature drops against a higher Prandtl number. Due to concentration gradients in the energy equation, the temperature rises slowly. Moreover, it is examined that the mass transfer significantly reduces due to Schmidt effects and more mass transfer is pointed against the Soret number. The shear stresses increase due to stronger magnetic field effects. The local thermal performance of the fluid enhances against more dissipative fluid, and DuFour effects reduced it. Furthermore, the mass transport rate drops due to higher Soret effects and increases against multiple Schmidt number values.
Importance of boundary layer flow cannot be ignored because of its diverse class of applications in daily life and industries as well. These comprised in civil engineering, aerodynamics, and many more.
The Newtonian flow over a stagnant wedge was firstly developed by Falkner and Skan . They transformed a dimensional model to a third-order nonlinear self-similar differential equation by applying similarity variables. Afterwards, in 1937, Hartree  explored the boundary layer model approximately. Later, researchers turned toward the study of wedge flow under different flow conditions. In 1961, Koh et al.  explored the shear stresses and quantities of practical interest such as nusselt and Sherwood numbers over a wedge in the existence of a porosity parameter. They discussed the model by encountering the impacts of inconstant wall temperature and suction property. In 1987, Lin et al.  discussed the approximate solutions for wedge flow of any Prandtl number. They analyzed the temperature regimes in the existence of forced convection.
Lately, Hussanan et al.  examined the boundary layer model in the existence of porous media by considering constant concentration gradients and resistive heating. Impacts of various physical quantities such as radiative heat flux and suction/injection on the chemically heating boundary layer model are reported in . Chamber et al.  reported the boundary layer in the existence of a chemical reaction and diffusion gradients. The Falkner Skan flow model for a static or moving wedge saturated with nanofluid was discussed by Yacob et al.  in 2011. Flow in a wavering sheet was examined in . They encountered the effects of natural convection and Ohmic heating. Su et al.  inspected the magnetohydrodynamic flow of Newtonian fluid composed by nanoparticles. Influences of applied Lorentz forces, slip flow condition, and ohmic heating were discussed in their study.
In 2016, Khan et al.  investigated the bioconvection model in the existence of a porosity parameter over a wedge. They explored the influence of viscous dissipation, resistive heating, and imposed Lorentz forces on the flow of a gyrotactic microorganism. They discussed the impacts of different self-similar physical parameter on the momentum, thermal, density motile, and concentration profiles of the microorganism. Graphical analysis for shear stresses, local mass, and heat transfer comprised in their study. They treated a particular model numerically and, for the accuracy of the results, made comparison with the prevailing literature. They investigated that velocity field increases for stronger magnetic field. Furthermore, significant analysis regarding to the characteristics of the flow behavior in various geometries under multiple flow conditions is perceived in [12–22].
In 2007, Ishak et al.  studied Falkner Skan flow through an accelerating wedge with the addition of suction/injection properties. Forced convection magnetohydrodynamic flow over a nonisothermal wedge by prevailing time-dependent viscosity was discussed by Pal et al.  in 2009. A magnetonanofluid model by considering the heat generation/absorption and the influence of a convective flow condition was inspected by Rahman et al.  in 2012.
Lately, Ullah et al.  investigated the non-Newtonian model past a wedge in the existence of imposed Lorentz forces. Ullah et al.  inspected the non-Newtonian model past a nonlinearly stretchable sheet by encountering the influences of various physical parameters. Aman et al.  studied a nanofluid model composed of gold nanoparticles. They also explored the influences of radiative heat flux and crisscross diffusion on the flow characteristics. Analysis of a ferrofluid composed of cylindrical-shaped nanoparticles was conducted in . Various flow models under certain boundary flow conditions in different channels are investigated in [30–33]. Brinkman sort of a nanofluid model was examined in . A natural convection flow model stretchable sheet was studied by Ullah et al. . A nanofluid model bounded by Riga plates and a second-grade flow model between an oblique channel were studied in [36, 37], respectively. Impacts of the effective Prandtl model and thermal radiation on the Newtonian model between a converging/diverging channel were explored in [38, 39], respectively. For further study regarding nanofluids and regular fluid from different aspects, we can analyze [40–46].
From a careful science literature review, it is pointed that, to date, no one analyzed the energy and mass transportation in MHD wedge flow by prevailing crisscross diffusion gradients. Initially, formulation of the model is carried out, and then, mathematical analysis of the model is performed by means of the RK technique. Section 3 is dedicated to highlight the impacts of varying a nondimensional parameter in the flow characteristics. A fruitful comparison has been made for reliability of the study. In the end, major results of the work are highlighted in the last section.
2. Self-Similar Analysis
Consider the time-independent Newtonian flow past a wedge positioned in the main stream. The fluid is electrically conducting, and the effects of cross diffusion are also under consideration. The main stream velocity of the fluid is . The flow is considered in the Cartesian coordinate system, and and axis are chosen in such a way that the surface of the wedge is along the , and the makes a right angle with the and perpendicular to the wedge surface. Furthermore, the wedge surface is kept at variable temperature and concentration . The ambient temperature and concentration of the fluid are denoted by and , respectively. Inconstant magnetic field imposed perpendicularly to the wedge with the assumptions of a smaller magnetic Reynolds number and inconsequential induced magnetic field. The schematic theme of the MHD flow model is demonstrated in Figure 1:
In the light of the aforementioned restrictions, a particular flow model can be described in the existence of various physical quantities in the following manner :
Equation (1) is the dimensional mass conservation law. Momentum, energy, and concentration equations by prevailing magnetic field and thermal and concentration gradients are embedded in equations (2)–(4), respectively. Furthermore, represent the velocities along coordinates. Moreover, is the main stream velocity, dynamic , density , imposed magnetic field , temperature , concentration , heat capacity , mass diffusivity , mean temperature of the fluid , thermal diffusion , and concentration susceptibility .
The feasible set of auxiliary conditions for the current wedge flow is as follows :
Here, shows the applied magnetic field which is a function of x and is in the form , where denotes the uniform magnetic field (for precedence, we can see [48, 49]), denotes the free stream velocity and described in function of x, and is , in which is an invariable quantity and () describes the Falkner Skan power law parameter. The expression , in which describes the Hartree pressure gradient. Furthermore, and describe the cases along the x and y coordinates, respectively.
The following self-similar model is attained from the dimensional system after incorporating the feasible similarity variables which comprised the influences of cross diffusion and magnetic:
The corresponding feasible boundary conditions are in the following manner:
In equations (8)–(10), with nonhomogeneous auxiliary conditions described in equations (11) and (12), self-similar quantities are the Hartmann number , Prandtl number (), Dufour number (), Soret number (), and Schmidt number ().
The following are the formulas to estimate the shear stresses, local Nusselt (), and Sherwood numbers :
Hence, the nondimensional form for shear stress, local heat, and mass transfer is defined as follows:where describes the local Reynolds number.
3. Mathematical Treatment
Usually, closed form solutions are very rare or not even exist for those models which are coupled and attain high nonlinearity. It is better to tackle such sort of models either numerically of asymptotically. A particular model is coupled and nonlinear in nature. Therefore, we adopted a numerical method called the Runge–Kutta method . To initiate the technique, feasible substitutions are as follows:
Consequently, supporting initial conditions are
After this, we performed the numerical calculation and obtained the tabulated results for the model. Table 1 presents the solutions for velocity, temperature, and concentration fields over the domain of interest.
4. Graphical Results and Discussion
The impacts of ingrained self-similar quantities on the momentum, temperature, and mass of the fluid are incorporated in this section. These physical quantities are , , , , and . The impacts of the magnetic field and thermal and concentration gradients are also under consideration. It is very important to mention that, in the current nonlinear flow model, if , then it represents the wedge flow. If and , it represents the flow of the horizontal plate and stagnation point flow, respectively. For an example, we can refer .
The flow behavior against an imposed magnetic field is given in Figure 2. It is explored that, against a stronger magnetic field, fluid motion increases. For horizontal plate flow, the velocity rises promptly comparative to the stagnation and wedge flow case. Physically, over a horizontal surface, the fluid particles move freely due to which the velocity upturns. Near the surface, these alterations are almost inconsequential. The physical reason behind this behavior is the force of friction between the surface and fluid layer adjacent to the surface. Due to the force of friction, the motion of fluid particles near the surface declines and rest of the fluid layer flow abruptly. Furthermore, for wedge and stagnation cases, increment in the velocity is observed quite slowly. Figure 2(b) presents a 3D image of the velocity against multiple values of .
The behavior of thermal performance against Prandtl and Dufour parameters are given in Figures 3(a) and 3(b), respectively. The declines in are noticed against stringer Prandtl values. For the stagnation point, these alterations are prompt, and maximum declines are perceived in the region . The temperature is against the horizontal plate flow case. Physically, the fluid flows abruptly over the horizontal surface due to which the collision between the particles increases; consequently, the temperature drops slowly. The significant increasing variations in are pointed against Dufour effects. Due to the Dufour number, the temperature upturns. However, maximum increment in the temperature is pointed against horizontal plate flow. Figures 4(a) and 4(b) show the 3D image of against Prandtl and Dufour parameters, respectively.
The mass transfer for multiple values of Schmidt and Soret parameters is given in Figures 5(a) and 5(b), respectively. It is perceived that the mass transfer is in inverse proportion to the Schmidt number. Due to higher Schmidt effects, less mass transfer at the surface is noted. However, maximum decrement is pointed against the stagnation point case. Near the surface, an almost inconsequential behavior of is noted for wedge, stagnation, and horizontal flat plate cases. The Soret number which appears due to cross-diffusion gradients favor the mass transfer . The mass transfer profile rises against stronger Soret effects. For a horizontal plate, maximum alterations in the mass transfer trends are detected comparative to wedge and stagnation cases. The 3D behavior of the fluid concentration is shown against Sc and Sr in Figures 6(a) and 6(b), respectively.
The behavior of shear stress against the magnetic field and is shown in Figures 7(a) and 7(b), respectively. It is pointed out that the shear stresses upturn against both the parameters. Due to stronger Hartmann effects, the fluid motion reduces; consequently, the shear stresses increase.
The local thermal performance and mass transfer rate against multiple parameters are shown in Figures 8 and 9, respectively. From inspection of the plotted results, it is noticed that the local thermal performance declines against the Dufour number. For the stagnation case, a drop in the local heat transfer rate is slow comparative to wedge and horizontal plate cases. On the other hand, viscous dissipative effects favor the local thermal performance rate. It is pointed out that, against more dissipative fluid, the local heat transfer rate increases. The mass transfer rate against Soret and Schmidt parameters is accessed in Figures 9(a) and 9(b), respectively. It is examined that the mass transfer rate enhances due to a stronger Schmidt number, and inverse variations are perceived against the Soret number.
Table 2 shows the comparative study against restricted flow parameters. The value of is fixed at one and the computation is carried out. From the inspection of Table 2, it is perceived that the presented results and adopted technique are reliable and acceptable.
The presented work encountered MHD flow over a wedge. The stimuli of cross diffusion are taken under consideration. The model is effectively treated by using a numeric technique called the RK technique. The impacts of ingrained quantities on the flow behavior are presented, and lastly, the following key findings are noted:(i)The fluid velocity is accelerating for increasing , and in the case of stagnation point flow, these variations are rapid(ii)The Dufour parameter favors the temperature , whereas the Prandtl number is against the fluid temperature(iii)increasing Schmidt number leads to a decrease in the concentration of the fluid, and for a varying Soret number, these variations are reverse(iv)Less heat and mass transfer are investigated for Dufour and Soret parameters(v)More heat and mass transfer are noted for Prandtl and Schmidt parameters in the wedge, stagnation point, and horizontal plate flows
No data were used in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.
- V. M. Falkner and S. W. Skan, “Some approximate solutions of the boundary layer equations,” Philosophical Magazine, vol. 12, pp. 865–896, 1931.
- D. R. Hartree, “On an equation occurring in Falkner and skan’s approximate treatment of the equations of the boundary layer,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 33, no. 2, pp. 223–239, 1937.
- J. C. Y. Koh and J. P. Hartnett, “Skin friction and heat transfer for incompressible laminar flow over porous wedges with suction and variable wall temperature,” International Journal of Heat and Mass Transfer, vol. 2, no. 3, pp. 185–198, 1961.
- H.-T. Lin and L.-K. Lin, “Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl number,” International Journal of Heat and Mass Transfer, vol. 30, no. 6, pp. 1111–1118, 1987.
- A. Hussanan, Z. Ismail, I. Khan, A. G. Hussein, and S. Shafie, “Unsteady boundary layer MHD free convection flow in a porous medium with constant mass diffusion and newtonian heating,” European Physical Journal Plus, vol. 129, no. 46, 2014.
- R. Kandasamy, W. Abd, and A. B. Khamis, “Effects of chemical reaction, heat and mass transfer on boundary layer flow over a porous wedge with heat radiation in the presence of suction or injection,” Theoretical and Applied Mechanics, vol. 33, no. 2, pp. 123–148, 2006.
- P. L. Chambre and A. Acrivos, “Difiusion of a chemically reactive species in a laminar boundary layer flow,” Indian Engineering Chemical, vol. 49, 1957.
- N. A. Yacob, A. Ishak, and I. Pop, “Falkner-Skan problem for a static or moving wedge in nanoluids,” International Journal of Thermal Sciences, vol. 50, 2011.
- A. Hussanan, M. I. Anwar, F. Ali, I. Khan, and S. Shafie, “Natural convection flow past an oscillating plate with Newtonian heating,” Heat Transfer Research, vol. 45, no. 2, pp. 119–135, 2014.
- X. Su and L. Xiaohong, “Hall effect on MHD flow and heat transfer of nanofluids over a stretching wedge in the presence of velocity slip and Joule heating,” Central European Journal of Physics, vol. 11, no. 12, pp. 1694–1703, 2013.
- U. Khan, N. Ahmed, and S. T. Mohyud-Din, Inﬂuence of Viscous Dissipation and Joule Heating on MHD Bio-Convection ﬂow over a Porous Wedge in the Presence of Nanoparticles and Gyrotactic Microorganisms, Springer, Berlin, Germany, 2016.
- M. K. Nayak, J. Prakash, D. Tripathi, V. S. Pandey, S. Shaw, and O. D. Makinde, “3D Bioconvective multiple slip flow of chemically reactive Casson nanofluid with gyrotactic micro‐organisms,” Heat Transfer, vol. 49, no. 1, pp. 135–153, 2019.
- M. K. Nayak, J. Prakash, D. Tripathi, and V. S. Pandey, “3D radiative convective flow of ZnO-SAE50nano-lubricant in presence of varying magnetic field and heterogeneous reactions,” Propulsion and Power Research, vol. 8, no. 4, pp. 339–350, 2019.
- M. K. Nayak, N. S. Akbar, D. Tripathi, and V. S. Pandey, “Three dimensional MHD flow of nanofluid over an exponential porous stretching sheet with convective boundary conditions,” Thermal Science and Engineering Progress, vol. 3, pp. 133–140, 2017.
- M. K. Nayak, N. S. Akbar, D. Tripathi, Z. H. Khan, and V. S. Pandey, “MHD 3D free convective flow of nanofluid over an exponentially stretching sheet with chemical reaction,” Advanced Powder Technology, vol. 28, no. 9, pp. 2159–2166, 2017.
- M. K. Nayak, N. S. Akbar, V. S. Pandey, Z. H. Khan, and D. Tripathi, “3d free convective mhd flow of nanofluid over permeable linear stretching sheet with thermal radiation,” Powder Technology, vol. 315, pp. 205–215, 2017.
- A. Shahid, Z. Zhou, M. M. Bhatti, and D. Tripathi, “Magnetohydrodynamics nanofluid flow containing gyrotactic microorganisms propagating over a stretching surface by successive taylor series linearization method,” Microgravity-Science and Technology, vol. 30, pp. 445–455, 2018.
- N. S. Akbar, D. Tripathi, and Z. H. Khan, “Numerical simulation of nanoparticles with variable viscosity over a stretching sheet,” in Numerical Simulations in Engineering and Science, IntechOpen, London, UK, 2018.
- N. S. Akbar, D. Tripathi, and Z. H. Khan, “Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection,” Discrete & Continuous Dynamical Systems, vol. 11, no. 4, pp. 583–594, 2018.
- N. S. Akbar, D. Tripathi, Z. H. Khan, and O. A. Bég, “A numerical study of magnetohydrodynamic transport of nanofluids over a vertical stretching sheet with exponential temperature-dependent viscosity and buoyancy effects,” Chemical Physics Letters, vol. 661, pp. 20–30, 2016.
- M. Turkyilmazoglu, “Extending the traditional Jeffery-Hamel flow to stretchable convergent/divergent channels,” Computers & Fluids, vol. 100, pp. 196–203, 2014.
- M. Turkyilmazoglu, “Slip flow and heat transfer over a specific wedge: an exactly solvable falkner-skan equation,” Journal of Engineering Mathematics, vol. 92, no. 1, pp. 73–81, 2015.
- A. Ishak, R. Nazar, and I. Pop, “Falkner-Skan equation for flow past a moving wedge with suction or injection,” Journal of Applied Mathematics and Computing, vol. 25, no. 1-2, pp. 67–83, 2007.
- D. Pal and H. Mondal, “Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 194–208, 2009.
- M. M. Rahman, M. A. Al-Lawatia, I. A. Eltayeb, and N. Al-Salti, “Hydromagnetic slip flow of water based nanofluids past a wedge with convective surface in the presence of heat generation (or) absorption,” International Journal of Thermal Sciences, vol. 57, pp. 172–182, 2012.
- I. Ullah, I. Khan, and S. Shafie, “Hydromagnetic Falkner-Skan flow of Casson fluid past a moving wedge with heat transfer,” Alexandria Engineering Journal, vol. 55, no. 3, pp. 2139–2148, 2016.
- I. Ullah, K. Bhattacharyya, S. Shafie, and I. Khan, “Unsteady MHD mixed convection slip flow of casson fluid over nonlinearly stretching sheet embedded in a porous medium with chemical reaction, thermal radiation, heat generation/absorption and convective boundary conditions,” PLoS ONE, vol. 10, no. 11, 2016.
- S. Aman, I. Khan, Z. Ismail, and M. Z. Salleh, “Impacts of gold nanoparticles on MHD mixed convection Poiseuille flow of nanofluid passing through a porous medium in the presence of thermal radiation, thermal diffusion and chemical reaction,” Neural Computing & Applications, vol. 30, no. 3, pp. 789–797, 2016.
- A. Khalid, I. Khan, and S. Shafie, “Heat transfer in ferrofluid with cylindrical shape nanoparticles past a vertical plate with ramped wall temperature embedded in a porous medium,” Journal of Molecular Liquids, vol. 221, pp. 1175–1183, 2016.
- A. Gul, I. Khan, and S. Shafie, “Energy transfer in mixed convection MHD flow of nanofluid containing different shapes of nanoparticles in a channel filled with saturated porous medium,” Nanoscale Research Letters, vol. 10, 2015.
- I. Ullah, S. Shafie, and I. Khan, “Effects of slip condition and Newtonian heating on MHD flow of Casson fluid over a nonlinearly stretching sheet saturated in a porous medium,” Journal of King Saud University, vol. 29, no. 2, pp. 250–259, 2016.
- A. Gul, I. Khan, S. Shafie, A. Khalid, and A. Khan, “Heat transfer in MHD mixed convection flow of a ferrofluid along a vertical channel,” PLOS One, vol. 11, no. 10, pp. 1–14, 2015.
- M. Zin, N. Athirah, I. Khan, and S. Shafie, “The impact silver nanoparticles on MHD free convection flow of Jeffery fluid over an oscillating vertical plate embedded in a porous medium,” Journal of Molecular Liquids, vol. 222, pp. 138–150, 2016.
- F. Ali, M. Gohar, and I. Khan, “MHD flow of water-based Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable surface velocity, temperature and concentration,” Journal of Molecular Liquids, vol. 223, pp. 412–419, 2016.
- I. Ullah, I. Khan, and S. Shafie, “MHD natural convection flow of casson nanofluid over nonlinearly stretching sheet through porous medium with chemical reaction and thermal radiation,” Nanoscale Research Letters, vol. 11, p. 527, 2016.
- U. Khan, N. Ahmed, and S. T. Mohyud-Din, “Influence of thermal radiation and viscous dissipation on squeezed flow of water between Riga plates saturated with carbon nanotubes,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 522, pp. 389–398, 2017.
- U. Adnan, U. Khan, N. Ahmed, and S. T. Mohyud-Din, “Thermo-diffusion and diffusion-thermo effects on flow of second grade fluid between two inclined plane walls,” Journal of Molecular Liquids, vol. 224, pp. 1074–1082, 2016.
- N. Ahmed, U. Khan, and S. T. Mohyud-Din, “Influence of an effective Prandtl number model on squeezed flow of γAl 2 O 3 -H 2 O and γAl 2 O 3 -C 2 H 6 O 2 nanofluids,” Journal of Molecular Liquids, vol. 238, pp. 447–454, 2017.
- M. Adnan, M. Asadullah, U. Khan, N. Ahmed, and S. T. Mohyud-Din, “Analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels,” Journal of Molecular Liquids, vol. 224, pp. 768–775, 2016.
- M. Sheikholeslami and S. A. Shehzad, “Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition,” International Journal of Heat and Mass Transfer, vol. 106, pp. 1261–1269, 2017.
- M. Sheikholeslami, T. Hayat, and A. Alsaedi, “Numerical study for external magnetic source influence on water based nanofluid convective heat transfer,” International Journal of Heat and Mass Transfer, vol. 106, pp. 745–755, 2017.
- M. Sheikholeslami, “CuO-water nanofluid free convection in a porous cavity considering Darcy law,” The European Physical Journal Plus, vol. 132, 2017.
- M. Sheikholeslami and K. Vajravelu, “Nanofluid flow and heat transfer in a cavity with variable magnetic field,” Applied Mathematics and Computation, vol. 298, pp. 272–282, 2017.
- B. B. Mohsin, N. Ahmed, Adnan, U. Khan, and S. T. Mohyud-Din, “A Bioconvection model for squeezing flow of nanofluid between parallel plates in the presence gyrotactic microorganisms,” European Physical Journal Plus, vol. 132, 2017.
- U. Khan, N. Ahmed, and S. T. Mohyud-Din, “Influence of viscous dissipation on copper oxide nanofluid in an oblique channel: implementation of KKL model,” The European Physical Journal Plus, vol. 132, 2017.
- N. Ahmed, Adnan, U. Khan, S. T. Mohyud-Din, and A. Waheed, “Shape effects of nanoparticles on Squeezed flow between two Riga Plates in the presence of thermal radiation,” The European Physical Journal Plus, 2017.
- D. Srinivasacharya, U. Mendu, and K. Venumadhav, “MHD boundary layer flow of a nanofluid past a wedge,” Procedia Engineering, vol. 127, pp. 1064–1070, 2015.
- M. H. Cobble, “Magneto fluid dynamic flow with a pressure gradient and fluid injection,” Journal of Engineering, vol. 11, no. 2, pp. 49–56, 1977.
- Z. Zhang and J. Wang, “Exact self-similar solutions of the magnetohydrodynamic boundary layer system for power-law fluids,” Zeitschrift für angewandte Mathematik und Physik, vol. 58, no. 5, pp. 805–817, 2007.
- U. Khan, N. Ahmed, and S. T. Mohyud-Din, “Numerical investigation for three dimensional squeezing flow of nanofluid in a rotating channel with lower stretching wall suspended by carbon nanotubes,” Applied Thermal Engineering, vol. 113, pp. 1107–1117, 2017.
- P. D. Ariel, “Hiemenz flow in hydromagnetics,” Acta Mechanica, vol. 103, no. 1-4, pp. 31–43, 1994.
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