Abstract
The assumption of Hall current and ion slip is extremely crucial in several industrial and manufacturing processes, such as MHD (magneto hydrodynamics) accelerators, preservation coils, transmission lines, electric converters, and heating elements. Keeping this in view, the main aim of this article is to present a computational analysis of MHD ion Hall current with nonlinear thermal radiation on the sloping flow of shear thinning fluid through a porous medium on a stretching sheet that allows fluid suction and injection. The major mathematical modelling of governed problems is converted into a system of nonlinear ODEs (ordinary differential equations) by means of appropriate similarity relations. The influence of all relative physical parameters on velocity and temperature is studied through graphs and discussed in a detailed physical manner. Some beneficial mathematical quantities from the practical engineering and industrial point of view, such as skin friction factor and heat transfer rate at the porous surface, are calculated numerically and presented through graphs. It has been observed that flow may become unstable when M is small and the existence of a magnetic field and a porous ground contributes to a highly rough flow over the stretching surface. Suction is actually a resistive force which results in higher skin friction that is beneficial in controlling flow separation. Temperature of the fluid rises with stronger magnetic field and higher thermal radiation effects. The local heat flux decreases as the magnetic field strength and permeability parameter increase.
1. Introduction
Engineers, scientists, and mathematicians face an enormous challenge while dealing with nonlinear rheology of working fluids. There are number of means through which such types of nonlinearity can be confronted. One of the simplest ways in which viscoelastic fluids have been classified is the methodology given by Rivlin and Ericksen [1]. Noll and Truesdell [2] presented stress tensor as a symmetric tensor with velocity gradient and its derivatives in constitutive equations. In this modern era, researchers like [3] have made a lot of contribution in the field of non-Newtonian fluid flows, due their high-tech implication in industries. It is also noteworthy that these types of fluids exhibit very stimulating mathematical features in their governed flow equations. Oblique stagnation point non-Newtonian fluid flow studies become a more exciting challenge for researchers and investigators due to its wide applications in industries. The fluid flow over a stretched surface is highly significant in so many manufacturing practices. Lok et al. [4] studied time-independent viscid incompressible fluid flow impinging at some arbitrary angles of incidence on a stretching panel. Labropulu et al. [5] developed the study of oblique stagnation-point flow of non-Newtonian fluid towards a stretching surface. Mahapatra et al. [6] analysed a time independent 2D radiative oblique stagnation-point flow with heat transfer characteristics on a shrinking sheet. Sadiq et al. [7] described MHD features of oscillatory oblique stagnation point flow of micropolar nanofluid. Some more studies developing different physical effects on nonorthogonal stagnation point flows dealing with several non-Newtonian models may be found in [8–10].
The nonlinear radiative electrically conducting fluid flow in manifestation of magnetic field is widely beneficial into electrical control generators, cosmological flows, stellar and lunar power control machinery, planetary automobile re-entry, fissionable production plants, and many other engineering areas. At great operational temperatures, nonlinear thermal radiation becomes more vital and obvious, particularly under nonisothermal conditions. Nonlinear thermal radiation is highly significant when polymer extrusion procedure is monitored by thermally controlled environment. The influence of linear as well as nonlinear thermal radiation on Newtonian as well as non-Newtonian fluid flows in the presence as well as absence of magnetic field has already been discussed by numerous researchers and scientists [11, 12].
Hall and ion currents in influence with magnetic field is the most noteworthy phenomena in modern research due to its intensive, keen-sighted, and immense implications in abundant engineering fields such as power control originators, MHD generators, preservation coils, broadcast ranks, electrical converters, and boiler essentials. By applying Ohm’s law directly, mostly the required results are unattainable due to weak magnetic strength but it can be enhanced by adding Hall and ion slip effects in this law. When applied magnetic field is in the direction of magneto hydrodynamics force in combination with Hall ion slip currents, then it becomes tremendously noteworthy in modern research because Hall and ion currents have strong influence on size and track of existing density and subsequently on the magnetic meter.
Ibrahim and Anbessa [13] scrutinized the 3D nanofluid flow of Casson fluid in the presence of applied magnetic field with ion Hall currents and mixed convection over an exponentially stretching surface. Krishna et al. [14] investigated combined effects of Hall and ion slips on MHD spinning stream of ciliary momentum of miniscule bacterium over absorbent intermediate. Rajakumar et al. [15] deliberated the flow of Casson fluid in their research, and they explored the influences of free convection with effects of radiation and viscid indulgence in existence with magnetic fields and Hall ion effects. Kumar and Vishwanath [16] established a scientific arrangement of non-Newtonian fluid flow over a permeable surface with a uniform distribution of magnetic field with Hall current and ion slip effects. Shah et al. [17] defined the flow of micropolar nanofluid in presence of thermally radiative rotating disks for investigation of mass flux and heat flux. Few more related studies on the said topic can be found from the references [18–22].
For continuity of fluid flow, suction/injection is highly recommended, particularly in boundary layer flows. Mainly these types of flows have applications in field of aerodynamics and planetary fields where the use of minimum drag forces is ensured. Suction is used for improvement in efficiency of diffusers. Shojaefard et al. [23] investigated the control flow of fluid on the surface of a subsonic aircraft by using suction/injection. Braslow [24] showed that fuel ingesting and pollution caused by subsonic aircraft as well as price of commercial aircrafts can be reduced to a good extent only with the help of suction/injection.
Stagnation point flows under influence of suction injection have become one of the great interests for modern researchers. Zeeshan and Majeed [25] inspected the characteristics of Jeffery fluid past a stretched plate under influence of attractive dipole with suction/injection. Similarly, El-Arabawy [26] studied the impact of radiative heat transfer with suction and injection on a constant rotating sheet for a micropolar fluid. Chamkha et al. [27] examined the properties of chemical species and heat and mass transmission on a stretched surface in a permeable medium. Pandey and Kumar [28] categorized the viscid dissipation with the presence of suction/injection on MHD flow of a nanofluid in a porous medium. Rundora and Makinde [29] discussed third-grade fluid with assumptions of time- and temperature-dependent variable viscosity findings in presence of suction/injection in a porous station. Similar type of studies may be seen in [30–36].
The all cited works of numerous researchers and scientists depicted that inclined flow of non-Newtonian fluid in existence with a strong magnetic field with ion and Hall currents and nonlinear thermal radiation in porous medium with suction injection influence is highly suitable in many engineering problems and found to be new in this combination, although many researchers in this modern era of research have explored these types of problem but not yet this one. The novelty of governed fluid problem is stated as follows:(i)A picture of the inclined Casson fluid stagnation point flow on a stretched horizontal plate is captured(ii)Suction injection phenomenon is taken into consideration, and the horizontal stretched plate is supposed to be permeable.(iii)The body force on this bioconvective nanofluid flow is the magneto hydrodynamic force with ion slip and Hall currents(iv)Nonlinear thermal radiation is supposed to be added with convective boundary conditions
The current findings and implications are presented by including graphs of fluid distributions that reveal all new impacts of various parameters. Morevoer, validation of current results with previously existing literature for Newtonian case is provided.
2. Mathematical Scheme
The mathematical model is constructed by using assumptions of two-dimensional, steady oblique flow of MHD Casson fluid along with Hall and ion slip conditions with suction injection and nonlinear thermal radiation. To keep surface stretched, two equal balanced forces are applied in opposite directions along the axis, and the origin is maintained fixed as shown in Figure 1. The basic fundamental laws in component forms as per stated assumptions are [37–39]
The consistent boundary conditions are [39]where and are the velocity components in and directions, and , , , , , , , , , and are the kinematic viscosity, pressure, density, temperature, nonlinear radiative heat flux, specific heat, ambient fluid temperature, thermal coefficient, the constants, and Casson fluid parameter, respectively. is defined as generalised Ohm’s law, where , , and are the current density, electrical conductivity, and electric field intensity, respectively. Equations (1)–(8) are transformed into nondimensional form [37–39]:where , , , , , , , , and represent the Hall parameter, ion parameter, Prandtl number, suction /injection parameter, temperature ratio parameter, Biot number, magnetic field parameter, permeability parameter, stretching ratio parameter, and obliqueness of the flow, respectively.
By stream-function transformation as defined [38] in equations (9)–(14) with
Over consuming stream-function transformation as defined in [38] into equations (15)–(18) and after integration once,
The consistent boundary conditions (17) and (18) convertwhere and can be obtained from B.Cs (23) as
Defining the transformationwe get
3. Physical Quantities of Interest
Skin friction coefficients at surface and the local heat flux [37] are the physical quantities of interest that have extensive use in numerous engineering and manufacturing productions.
The stagnation points are
4. Numerical Scheme
The mathematical form of governed problems (26)–(30) is the system of coupled highly nonlinear set of ordinary differential equations. To solve such system of equations by shooting technique, first of all, make them into set of first-order initial value problem by using following transformation.
We getwhere , are the shooting factors with assumption of three decimal places tolerance level.
5. Results and Physical Discussion
Comprehensive computational calculations have been conducted and demonstrated by graphs herein segment. The numerical investigation of oblique stagnation point flow of MHD ion Hall current with suction injection of non-Newtonian fluid along with nonlinear thermal radiation in porous medium is presented in this segment. Figures 2–10 are settled to attain the norms and standards of this theoretical research.
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Figure 2 is constructed to note the influence of permeability constraint on fluid’s normal and tangential velocity , , and temperature of fluid with suction/injection. Figure 2(a) presents normal velocity shrinkages for rising values of permeability parameter , and it is quite evident that the existence of permeable surface becomes the reason of strong restriction to flowing of fluid, so the velocity becomes decelerate. Also, it is worth mentioning here that the magnitude of suction is greater than the magnitude of injection for this case, higher suction becomes more effective in porous surface as it delays the boundary layer separation and flow becomes more and more stable. Figure 2(b) indicates other component of velocity upswings nearby surface since a high permeability allows fluids to pass through more freely, and after inflection point, it reverses its behaviour and comes to decline far off from sheet because of higher inspiration of porousness parameter and behaves extremely resistive. It is noted in this graph that close to wall injection is higher than suction but away from the wall effect of suction/injection reversed. Figure 2(c) describes the effect of permeability parameter on temperature of fluid, when permeability parameter rises so that the temperature distribution with thermal boundary layer increases. This happens because permeable surface slows down the motion of fluid flow, and this restriction in flowing of fluid becomes responsible to enhance the temperature of governed fluid. Figure 2(c) also depicts the stronger influence of suction .
Figure 3 describes the impact of ion slip constraint on fluid’s velocities and as well as on temperature of fluid with suction/injection. Conductivity of fluid increases when values of ion slip constraint escalates and as a reaction, restraining energy comes down and fluid’s molecules freely moves and fluid’s velocity raises as noted in Figure 3(a). Figure 3(b) is plotted to show similar kind of increasing behaviour for away from the surface but at surface velocity reverses its behaviour and declines at wall because at wall fluids have resistance which opposes the flow. Figure 3(c) displays that temperature of fluid declines; also, the thermal boundary layer becomes thinner for ion slip parameter due to dropping of damping energy in the direction of flow. It is worth mentioning here that velocity profile for suction is greater than injection . Strong influence of suction is highly useful to reduce the drag in boundary layer flow.
Figure 4 is planned to recognize the enactment of both velocities , , and temperature of fluid for Hall parameter . Since the resistance is produced by magnetic field when Lorentz force is strong enough, but due to the presence of Hall parameter , the resistive force becomes weak due to decline in conductivity, so fluid’s normal velocity proliferates with the rise in Hall parameter as mentioned in 4(a), and for suction , velocity is greater, but for injection , velocity of fluid is smaller. Figure 4(b) expresses the significance of Hall constraint , and it governs when fluid is far away from surface and fluid’s tangent component of velocity, intensificates away from wall, but near to surface, it takes differing conduct and contracts. Given that the Hall parameter is calculated as the sum of the frequency and the time of electron collisions. An increase in this parameter indicates an increase in the frequency of electrons, the duration of electron collisions, or both. Figure 4(c) shows that temperature drops down when Hall parameter increases; because of weak resistive force, there is decline in thermal conductivity and as a result the temperature profile declined. The effect of suction is stronger in these graphs, and it is more applicable in practical world problems/models and useful for situation where to increase output of diffusers of governed fluid through reducing separation drag. Boundary layer suction particular in porous media close to trailing edge is useful to maximize the lift and minimize the drag force of automobiles, aerofoils, and jet planes.
Figure 5 is intended to show the effect of magnetic field on fluid’s velocity and fluid’s temperature, every time. The existence of magnetic field means that there is birth of Lorentz strength. Lorentz force is defined as a resistive drag force, so normal velocity of fluid descents for as seen in Figure 5(a). But for tangential velocity, case is opposite and grows up close to surface but reverses its behaviour when it moves away from the surface for , see Figure 5(b). This happens due to the presence of magnetic field. Figure 5(c) contrives to implement the connection of temperature with magnetic field. It shows that temperature increases for higher values of magnetic field because of frictional stress which arise because of Lorentz force, so there occurs increment in thermal conductivity, so in temperature of fluid.
Figure 6 is plotted for inspecting the performance of radiation parameter , Biot number , and Prandtl number on temperature distribution. Fluid’s temperature enlarged for rising numbers of radiation parameter because larger radiation parameter implies more heat is provided to the fluid, so thermal boundary layer becomes thick and temperature of fluid rises as illustrated in Figure 6(a). In addition to being utilised to produce power, radiation is also used in academia, industry, and medical. Radiation is also useful in many other fields, including mining, law enforcement, space exploration, agriculture, archaeology (carbon dating), and many others. Figure 6(b) shows that temperature of fluid becomes higher with growth in Biot number . Because when convective heat conversation at the surface rises , then there is enhancement in thermal boundary layer thickness as with a higher heat transfer coefficient, and more heat is transferred from the surface to the fluid. The rate of heat transmission increases with greater estimates of . can, therefore, be used as a cooling operator in complex operations. Figure 6(c) shows the thermal boundary layer thicknesses shrinkage extremely when there is rise in Prandtl number ; so, there is escalation in the wall temperature gradient. This phenomenon occurs because of higher values Prandtl number, and then, fluid has moderately little thermal conductivity that lessens the occurrence of conduction and reduces the thickness of thermal boundary layer; hence, temperature of fluid declines. Small Prandtl values are a suitable choice for heat-transmitting liquids since they are free-flowing liquids with strong thermal conductivity. Prandtl number specifies fluids with huge thermal conductivity which crops denser thermal boundary layer as compared to the thermal boundary layer for higher Prandtl number . Suction is more prominent than injection in all these plots, and it is an efficient source for laminar boundary layer flow, it reduces the contact losses at surface and suction becomes more stable in laminar boundary layer, and it becomes thin and remains laminar throughout. These physical quantities are of great worth due to its large and high scales applications in many industrial and engineering arenas, specifically areas of aerodynamics and astronomical, and highly beneficial in controlling flow separation.
Figure 7(a) launches that skin friction coefficient at surface increases when permeability parameter rises with suction and injection , and also, it upsurges when the values of magnetic field raised for both cases. It is worth noting in this plot that suction is smaller than injection. Figure 7(b) develops the decreasing influence of local heat transfer rate rises for permeability parameter and for magnetic field parameter for both cases of suction as well as for injection . Also, suction is smaller than injection .
Figure 8(a) indicates skin friction coefficient at wall shrinkages when both Hall parameter and ion slip parameter increase. Figure 8(b) displays that local heat flux grows up for increasing values of Hall parameter ; on the other hand, it remained fixed for ion slip parameter on local heat flux for both cases of suction as well as for injection but influence of suction is smaller than injection in these two plots.
In Figure 9, it is found that effect of radiation parameter is downward for heat transfer rate at surface - but have opposite behaviour for Biot number . Also, this figure exhibits that injection is stronger and enhancing than suction. The fluid flow in the channel is controlled by suction or injection phenomenon. Figure 10 shows flow pattern through stream lines for suction and injection in the presence and absence of permeability parameter . Figure 10(a) reveals the flow pattern with and without permeability for injection and for suction in Figure 10(b). Figure 10(c) simultaneously shows the stream lines pattern for both suction as well as injection . The stream contour touches the partition , at stagnation point , and zero skin friction.
Table 1 provides the comparison of numerical values of local heat flux with previously published results in literature, so that to authenticate the current computational results. For this purpose, the findings of Makinde and Aziz [40], Khan and Pop [41], and Wang [42] are compared with present values of heat transfer rate in Table 1. Here, the assumptions that are made for comparison are fixed temperature with very large Biot number in BCs also with negligence of permeability parameter and suction/injection effects. These values depicted in the table that the current results of heat flux at surface against several numerical figures of Prandtl number took upto 3 decimal places with those values of heat flux presented in [40–42].
6. Concluding Remarks
The major presentation of this type of existing research is particularly in the field of aerodynamics and astral, planetary, cosmological, and astrophysical disciplines so that drag may minimize to reduce the loss of energy. So, in this respect, the prevailing article inspects the blend suction injection in permeable surface for non-Newtonian fluid with MHD Hall and ion slip effects over a nonlinear thermally radiative stretched surface. The nonlinear radiative electrically conducting fluid flow in manifestation of magnetic field is widely bump into electrical control generators, cosmological flows, stellar and lunar power control machinery, planetary automobile re-entry, fissionable production plants, and many other engineering areas.(i)Permeability developed the cause to decline in both velocities but enhances temperature of fluid as this happens in fluids due to high permeability so it allows fluids to pass through more freely. This can be helpful in materials such as aquifers, petroleum reservoirs, cements, and ceramics.(ii)Ion and Hall slip parameters and are the causes for rise in velocities. Several engineering issues including those involving power generators, magneto hydrodynamic accelerators, refrigeration coils, transmission lines, electric transformers, and heating used these types of currents.(iii)Velocities for magnetic field parameter falls down for . But for temperature distribution, it rises. Also, both velocities for magnetic field parameter with suction is recognized more superior than occurrence of injection . The discovery that the interaction of a plasma with a magnetic field could take place at far greater temperatures than were feasible in a spinning mechanical turbine served as the initial catalyst for interest in MHD power generation.(iv)Influence of radiation parameter and Biot number on temperature of fluid is more dominant, but for Prandtl number , it became subservient. Several different applications, such as thermal management, spectroscopy, optoelectronics, and energy-conversion devices, depend on the capacity to control heat radiation.(v)Local heat flux is enormous for ion and Hall slip parameter and with injection as compared to suction(vi)Heat transfer rate at surface drops down in the presence of radiation parameter , while it flourishes against different values of Biot number Bi, injection in this case is more prominent than suction. Suction/injection is a mechanical phenomenon that is used to control the fluid flow in the channel and reduce surface drag in order to reduce energy losses in the boundary layer region.(vii)A solid confirmation is obtained in tabular format of numerical figures with present existing literature. An outstanding agreement is attained for restrictive case.(viii)A strong convective boundary condition indicated that for numerous figures of Prandtl number, local heat flux at surface upturns
Nomenclature
| : | x–components of velocity |
| : | Viscosity |
| : | Density |
| : | Permeability parameter |
| : | Free stream temperature |
| : | Specific heat of fluid |
| : | Nonlinear radiative heat flux |
| : | Radiation parameter |
| : | Hall parameter, ion parameter |
| : | Hall parameter, ion parameter |
| : | Prandtl number |
| BCs: | Boundary conditions |
| : | Obliqueness of fluid flow |
| ODEs: | Ordinary differential equations |
| : | y–components of velocity |
| : | Pressure |
| : | Temperature |
| : | Heat transfer coefficient |
| : | Constants |
| : | Thermal conductivity |
| : | Casson fluid parameter |
| : | Temperature ratio parameter |
| : | Magnetic field constraint |
| : | Suction /injection parameter |
| : | Stretching ratio constraint |
| : | Biot number |
| : | Boundary layer displacement constant |
| PDEs: | Partial differential equations. |
Data Availability
No underlying data were collected or produced in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest with any individual or organization regarding publication of this research.
Acknowledgments
The authors were thankful for technical and financial support from University of Wah, HITEC University, and Shandong University of Science and Technology, Qingdao, China.