Abstract
The intention of the present study is to scrutinize the three-dimensional MHD mixed convection flow of Casson nanofluid over an exponentially stretching sheet using the impacts of Hall and ion slip currents. Moreover, the impacts of thermal radiation and heat source are considered in this study. The governing partial differential equations are transformed into a system of joined nonlinear ordinary differential equations using similarity transformations, and they are solved numerically employing a spectral relaxation method (SRM). The obtained results are contrasted with existing specific cases, and a reasonable harmony is established. The impacts of noteworthy physical parameters on the velocities, thermal and concentration distributions, skin friction coefficients, local Nusselt number, and local Sherwood number are investigated graphically. It is found that the rise in Casson fluid and magnetic field parameters reduce the velocity profiles along both and directions while the reverse tendency is observed with an increment in Hall, ion slip, and mixed convection parameters. Moreover, the increase in both radiation and heat source parameters enhances the temperature profile. It is also observed that both the skin friction coefficients reduced with an increase in Casson fluid, Hall, and ion slip parameters. Furthermore, the local Nusselt number enhances with an augment in radiation parameter, whereas the opposite trends of local Nusselt and Sherwood numbers are found with an increase in heat source parameter.
1. Introduction
Nanofluid is a kind of heat transport medium containing nanoparticles less than 100 nm which are consistently and steadily dispersed in a base fluid like water, oil, and ethylene glycol. These dispersed nanoparticles, mostly a metal or metal oxide, enormously improve the thermal conductivity of the nanofluid and enhance conduction and convection coefficients taking into consideration more heat transport. Reddy et al. [1] utilized finite element method to portray the impact of magnetohydrodynamic boundary layer stream and heat transport of nanofluid over a porous contracting sheet with divider mass suction and heat source/sink. They found that an increase in magnetic field and suction parameters leads to a rise in velocity profile, whereas opposite trends of the temperature and nanoparticle volume fraction profiles are observed. Also, Ramya et al. [2] numerically dissected the boundary layer viscous flow of nanofluids and heat transport over a nonlinearly extending sheet within the sight of a magnetic field utilizing Keller box method and found that the temperature profile and nanoparticle concentration increment with expanding values of the magnetic parameter. Zhao et al. [3] have explored the three-dimensional stream and heat transport of a nanofluid in the boundary layer region over a flat sheet extended constantly in two lateral directions utilizing homotopy analysis method (HAM), and they reported that the heat transport conductivity of the nanofluid is greater than that of the pure fluids. Furthermore, Khan et al. [4] examined the three-dimensional flow of nanofluid over a skin friction exponentially extending sheet utilizing Keller box method. They reported the existence of remarkable Sparrow–Gregg-type hills for temperature profile in line with some range of parametric values. Moreover, Hayat et al. [5] computed three-dimensional boundary layer stream of viscous nanofluid over a bi-directional linearly extending sheet within the sight of Cattaneo–Christov two fold diffusion and reported that temperature and concentration profiles reduced with an increment in thermal and concentration relaxation parameters.
Recently, Shah et al. [6] investigated natural convection flow of hybrid nanofuid () with as base fluid in a porous media via control volume finite element method (CVFEM). They confirmed that the Nusselt number is an increasing function of porosity parameter, whereas opposite trend is noticed for Lorentz forces. Most recently, Shah et al. [7] scrutinized the effect of thermal radiation on Darcy–Forchheimer flow of micropolar ferrofluid with as base fluid and iron oxide as electromagnetite nanoparticles in a porous and dynamic sheet exposed to both suction and injection. They found that the velocity profile enhances with the increment in microrotation and electric field strength parameters for the stretching sheet, whereas the opposite result is observed for the shrinking sheet in both suction and injection cases. Moreover, Shah et al. [8] described the flow and heat diffusion of blood that carries the micropolar nanofluid of gold in a porous channel in presence of thermal radiation and found that the temperature distribution for the micropolar nanoparticles augments when the suction/injection parameter is positive, i.e., and it reduces when for either moving or stationary walls of porous channel. Further, Alreshidi et al. [9] have discussed the time-independent and incompressible flow of MHD nanofluid past a permeable rotating disc with slip conditions. Besides, they studied the mass and heat diffusion with viscous dissipation effect and found that the fluid velocities diminish with the intensification in velocity slip, porosity, and magnetic parameters.
A mixed convection flow is the method of heat transport happened due to the consolidated impacts of free and forced convection flows. As of late, investigation of mixed convection boundary layer flow past a plate has increased striking consideration as it expects a vital part in numerous industrial and technological applications in nature, for example, streams in the sea and in the ambiance, sun oriented recipients exposed to wind currents, atomic reactors cooled during emergency shutdown, electronic devices cooled by fans, heat exchangers put in a low-velocity condition, etc. Izadi et al. [10] numerically contemplated the mixed convection heat transport and entropy generation of a nanofluid containing carbon nanotube flowing in a three-dimensional rectangular channel exposed to contradicted buoyant forces utilizing finite volume technique and found that with an expansion in the contradicted buoyancy parameter the nanofluid velocity close to the channel divider definitely lessens and, in this way, causes a decrease in the Nusselt number. Also, numerical simulation of mixed convection heat transport in a lid driven triangular hole filled with power law nanofluid and with an opening was performed under the impact of a slanted magnetic field by Selimefendigil and Chamkha [11]. They found that average heat transport reduces with Hartmann number, and in the company of magnetic field heat transfer rate is superior for dilatant fluid, while without the magnetic field a pseudoplastic fluid provides the maximum value of average heat transport. Most recently, numerous scholars concentrated on the examination of mixed convection flow of fluids by taking different angles and geometries [12–14].
Non-Newtonian fluids has obtained significant attention because of its wide range of applications in different industries, for example, structure of strong lattice heat, atomic waste transfer, synthetic synergist reactors, geothermal energy creation, ground water hydrology, transpiration cooling, oil supplies, and so forth. These fluids are progressively confounded when contrasted with Newtonian fluids because of nonlinear connection among stress and strain rate. Numerous models have been proposed for the investigation of non-Newtonian liquids, however, yet not a solitary model is built up that displays all properties of non-Newtonian fluids. Among various non-Newtonian fluids, Casson fluid is the most famous fluid which has many applications in nourishment handling, metallurgy, drilling activities, and bio-engineering tasks. Casson fluid is a shear thinning fluid which is accepted to have a limitless viscosity at zero rate of shear, a yield stress underneath which no flow happens and a zero viscosity at a boundless rate of shear. Some common examples of Casson fluid are honey, tomato sauce, jelly, soup, concentrated fruit juices, blood, and so on. Casson fluid model is a non-Newtonian fluid with yield stress which is extensively used for modeling blood flow in narrow arteries. Furthermore, Casson fluid possesses yield stress and has great importance in polymer processing industries and biomechanics. Hayat et al. [15] considered the mixed convection stream of Casson nanofluid over an extending surface in nearness of thermal radiation, heat source/sink, and first order chemical reaction. They reported that thermal boundary layer thickness is an increasing function of thermal radiation and internal heat generation. Moreover, Concentration distribution and associated boundary layer thickness increase with the increment in generative chemical reaction while reverse tendency is observed for destructive chemical reaction. Also, Kamran et al. [16] numerically examined Casson nanofluid past flat extending surface with magnetic impact and Joule heating considering slip and thermal convective boundary conditions utilizing Keller box method and they established that the effect of expanding Hartmann number resulted in the decline of both Sherwood and Nusselt number. Afify [17] numerically researched the effects of multiple slips with viscous dissipation on the boundary layer stream and heat transfer of a Casson nanofluid over an extending surface utilizing a shooting strategy with fourth-order Runge–Kutta integration scheme and they found inverse impact with generative chemical reaction and concentration slip parameter. Recently, boundary layer flow of Casson nanofluids stream over various geometries was considered by numerous authors in references [18–24].
The investigation of MHD flows, the Hall current and ion slip relations in Ohms law have been disregarded in order to effortlessly lead scientific examination of the flow. Nevertheless, the result of the Hall current and ion slip is significant within the sight of a high magnetic field. Thus, in various characteristic conditions, it is fundamental to involve the impact of Hall current and ion slip terms of the magnetohydrodynamics articulations. Attributable to these realities, numerous investigations have been accounted for MHD streams within the sight of Hall and ion slip currents. Accordingly, Nawaz et al. [25] have investigated the Hall and ion slip impacts on three-dimensional combined free and forced convection flow of a Maxwell liquid over an extending vertical surface and they reported that the Hall parameter has similar impacts on both tangential and lateral velocities whereas the ion slip parameter has opposite impacts on both velocities. In addition, Nawaz and Zubair [26] studied the Hall and ion slip effects on three-dimensional flow equations of nano-plasma fluid in the company of homogeneous applied magnetic field and found that the inclusion of copper (Cu) and silver (Ag) nanoparticles greatly influences the velocity components and temperature of the nano-plasma. Some more investigations related to Hall and ion slip currents can be found in references [27–29].
Thermal radiation assumes a significant role in manufacturing industries for the design of atomic power plants and a few designing applications. Because of its essential applications various scientists have given their consideration to thermal radiation impact. Hayat et al. [30] scrutinized the impact of thermal radiation on three-dimensional mixed convection stream of viscoelastic fluid and reported that mixed convection parameter has opposite effect on velocity and temperature boundary layers. Makanda et al. [31] investigated the impacts of radiation on MHD natural convection flow of Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy permeable medium and they established that both velocity and temperature profiles are increasing functions of the radiation parameter. Ullah et al. [32] numerically examined the results of chemical reaction on hydromagnetic free convection flow of Casson nanofluid induced as a result of nonlinearly extending sheet immersed in a permeable medium under the impact of convective boundary condition and thermal radiation. They found that Casson fluids are superior to manage the temperature and nanoparticle concentration as contrasted to Newtonian fluid for nonlinearly extending sheet. The inclusive references and in detail understanding on thermal radiation can be read in recent articles [33–37].
The aim of the present study is to investigate the Hall and ion slip impacts on the flow of MHD mixed convection flow of Casson nanofluid in presence of thermal radiation and heat source. Up to the authors’ knowledge, there is no work is reported like the investigation of three-dimensional mixed convection flow of Casson nanofluid over an exponentially stretching sheet under the effect of Hall and ion slip currents. Along these lines, inspiration of the present examination is to direct the Hall and ion slip effects for mixed convection flow of Casson nanofluid over an exponentially stretching sheet employing spectral relaxation method. In particular, the study of the effects of mixed convection, Hall, ion slip, thermal radiation, and heat source parameters makes this work a novel one.
2. Mathematical Formulation
Consider the steady three-dimensional incompressible mixed convection flow of Casson nanofluid over an exponentially extending sheet in two lateral directions. The sheet is situated at and the flow is restricted to . The fluid is electrically conducted by a consistent applied magnetic field in the direction orthogonal to the plane. The induced magnetic field is ignored under the supposition of small magnetic Reynolds number. Suppose the velocities of the sheet along and directions be and respectively, where and are constants. The sheet is kept up at temperature and the concentration where and are constants, and and are the ambient values of temperature and concentration, respectively, as shown in Figure 1.
The rheological equation of state for an isotropic and incompressible flow of Casson nanofluid can be composed as [23, 36, 38]where is the component of the deformation rate, π is the product of the component of deformation rate with itself, is a critical value of this product based on the non-Newtonian model, is the plastic dynamic viscosity of the Casson fluid, and is the yield stress of the fluid.
The generalized Ohm’s law with Hall and ion slip consequences is given by [39–41]where is the current density vector, is the intensity vector of the electric field, is the velocity vector, is the magnetic field, is the cyclotron frequency, and is the electrical collision time.
Thus, with the above assumptions and under the standard boundary layer suppositions, the equations governing the conservations of mass, momentum, energy, and nanoparticles mass are [26, 40–43]where are the velocity components along the directions, respectively.
The boundary conditions for the considered flow problem arewhere , and
The radiative heat flux expressed in relation to Rosseland approximation is set aswhere is the Stefan-Boltzmann constant and is the mean absorption coefficient. can be conveyed as linear function of temperature. By expanding in a Taylor series about and disregarding higher terms, we can write
Substituting equations (9) and (10) into equation (6), we obtain
Use the following dimensionless variables [44, 45]:
Equation (3) is identically satisfied and Equations (4)−(8) and (11) take the following forms:where
The skin friction coefficients and , the local Nusselt number , and Sherwood number are defined as follows:where and are the wall shear stress along the and directions, respectively, is the wall heat flux and is the wall mass flux. These are as under:
From equations (12) and (18) and (19), we obtainwhere .
3. Method of Solution
Equations (13)–(16) depending on the boundary conditions (17) are solved employing the spectral relaxation method [42, 45–50]. This method is chosen as it has been exposed to be accurate and in general easier to employ compared to other ordinary numerical methods, for instance finite difference method. The spectral relaxation algorithm utilizes the notion of Gauss–Seidel method to decouple the system of governing Equations (13)–(16). The method is developed by evaluating the linear terms at the present iteration level and nonlinear terms at the preceding iteration level . The Chebyshev pseudospectral strategy is utilized to solve the decoupled equations. In this method, we present a differentiation matrix D which is roughly the derivative of the unknown variables, for instance, at collocation points as the matrix vector product iswhere is the number of collocation points (or grid points), and is the vector function at the collocation points. is a finite length which is sufficiently large so that we can easily include the condition at infinity in this point. A variable is used to map the truncated interval to the interval on which the spectral method can be executed.
The algorithm for the SRM may be summarized as follows:(1)Introduce the transformation and convey the original equation in terms of to reduce the order of the momentum equation (13) for .(2)Assuming that is identified from a prior iteration (denoted by ), make an iteration scheme for by assuming that only linear terms in are to be calculated at the recent iteration level (denoted by ), and all other linear and nonlinear terms are understood to be known from the prior iteration. Besides, nonlinear terms in are calculated at the prior iteration.(3)Introduce the transformation and convey the original equation in terms of to reduce the order of the momentum equation (14) for .(4)Assuming that is identified from a prior iteration (denoted by ), make an iteration scheme for by assuming that only linear terms in are to be calculated at the recent iteration level (denoted by ) and all other linear and nonlinear terms are understood to be known from the prior iteration. Besides, nonlinear terms in are calculated at the prior iteration.(5)The iteration schemes for the remaining governing dependent variables are developed correspondingly but at the present using the updated solutions of the variables determined in the preceding equation.(6)Chose suitable initial guesses which satisfy the given boundary conditions.
Thus, to employ the SRM, we begin by reducing the order of the momentum, equations (13) and (14), from third to second order introducing the transformation so that . Thus, equations (13)–(16) becomeand the boundary conditions are written as
Implementing the SRM to equations (22)–(27), we get the subsequent iteration scheme:and the boundary conditions are written as
The system of the equations along with the boundary conditions are written in a matrix form asand the matrices are defined aswhere and diag [·] are the identity and diagonal matrices of order , respectively, and , and are, respectively, the values of f, p, , h, θ, and , when evaluated at the collocation (or grid) points. equations (31)–(36) constitute the SRM scheme. Since they are decoupled, they may be solved separately. This is preceded by applying boundary conditions as shown follows:
Selected initial guesses that gratify the given boundary conditions (17) are
4. Results and Discussions
Equations (13)–(16) subject to the boundary conditions (17) are solved numerically employing SRM. The numerical solutions for the given flow problem are obtained for velocities, thermal and concentration distributions, skin friction coefficients, local Nusselt, and Sherwood numbers for different values of physical parameters through graphs shown in Figures 2–18. Unless otherwise stated, the default numerical values for the parameters are taken to be fixed as , , and . The ranges of parameters used in Figures 2–18 are , and The numerical values for the physical parameters are calculated in the chosen interval in an attempt to understand the flow behavior in a better way and to be sufficient in providing accurate solution.
For the substantiation of the numerical method used, the results are compared with the previously obtained results for various values of parameters, and it indicates an excellent agreement as depicted in Tables 1 and 2.
Figure 2 shows the impact of Casson fluid parameter on the velocity distributions . An expansion in Casson fluid parameter leads to a decrease in the yield stress and momentum boundary layer thickness. Owing to this fact, velocity distributions reduce with an increment in the values of . Physically, the fluid turns out to be more viscous with mounting Casson fluid parameter Figure 3 outlines the impact of magnetic parameter M on the velocity profiles . As the value of M expands, velocity diminishes because of Lorentz forces which slow down the velocity of the fluid. Along these lines both boundary layer thickness and the magnitude of the velocity profiles diminish. The impact of Hall parameter on velocity profiles is shown in Figure 4. It is noticed that velocity profiles have expanding tendency when Hall parameter is expanded. Moreover, the momentum boundary layer thickness is expanded by expanding the Hall parameter . This is because of the fact that magnetic field and Lorentz force decline when Hall parameter is expanded.
The impact of ion slip parameter on the velocity profiles is shown in Figure 5. It is clear from Figure 5 that both boundary layer thickness and the magnitude of the velocity profiles increment when ion slip parameter is expanded in light of the fact that magnetic field impacts because ion slip is inverse to applied magnetic field. This opposes the impacts of Lorentz force because of the applied magnetic field. It implies Lorentz force diminishes when ion slip parameter is expanded and therefore increment when ion slip parameter is expanded. Figure 6 exhibits the impact of the mixed convection parameter on the velocity profile . It is seen that the velocity distribution increments as the value of rises because of buoyancy impact. Figure 7 uncovers the alterations that are seen in nanofluid temperature profiles because of increment in the values of radiation parameter . It merits seeing that the nanofluid temperature increments as thermal radiation increment because of the way that the conduction effect of the nanofluid improves within the sight of thermal radiation. Henceforth, higher values of radiation parameter imply higher surface heat flux and thus upgrade the temperature inside the boundary layer area. Figure 8 portrays the impact of velocity ratio parameter on . An expansion in ratio parameter diminishes the boundary layer thickness for and increments . Physically, when enhances, the stretching rate augments in the direction. Subsequently, the velocity amplifies in the direction. Here, denotes two-dimensional cases. In the event that , the characteristics of the stream along both directions are alike.
The impact of Brownian motion parameter on temperature and concentration is exhibited in Figure 9. Brownian motion is the arbitrary movement of small colloidal particles suspended in a fluid, brought about by the collision of the fluid atoms with the particles. An improvement in the Brownian motion yields noteworthy movement of the nanoparticles that offers an increment in the fluid kinetic energy, and as a result the fluid temperature increases. In addition, thermal boundary layer thickens when is expanded. In contrast, it is seen that the concentration and related boundary layer thickness lessens with an improvement in . Figure 10 explains the impact of thermophoresis parameter on the temperature and nanoparticle volume fraction . Escalating values of shows stronger thermophoretic force because of temperature gradient which moves the nanoparticles from the hot sheet to the quiescent fluid accordingly mounting the temperature and nanoparticle volume fraction boundary layer.
Figure 11 indicates that an augment in temperature exponent parameter caused a decrease in both temperature and concentration profiles. Besides, the thermal and concentration boundary layer thickness lessen with increment in temperature exponent parameter . Figure 12 explains the impact of heat source parameter on the temperature distribution . Overall, heat generation parameter in the fluid expands the temperature. Accordingly, an expansion in Q upgrades the temperature and thermal boundary layer thickness.
Figure 13 demonstrates the skin friction coefficient for different values of ion slip parameter mixed convection parameter . The skin friction coefficient is a decreasing function of Hall parameter . Moreover, the increase in and decreases the skin friction coefficient . Similarly, the skin friction coefficient is a diminishing function of Casson fluid parameter and the augment in and reduces the skin friction coefficient as shown in Figure 14. Furthermore, Figure 15 depicts that the local Nusselt number is an escalating function of radiation parameter and the rise in temperature exponent parameter enhances the local Nusselt number . On the contrary, Figure 16 shows that the local Nusselt number is a declining function of thermophoresis parameter and the increase in Brownian motion parameter and heat source parameter decreases the local Nusselt number . Moreover, Figure 17 represents that the local Sherwood number is an increasing function of radiation parameter and the increase in temperature exponent parameter amplifies the local Sherwood number . In contrast, the local Sherwood number is a decreasing function of thermophoresis parameter and the augment in heat source parameter Q boosts the local Sherwood number as depicted in Figure 18.
5. Conclusions
A numerical study of three-dimensional MHD mixed convection flow of Casson nanofluid over an exponentially stretching sheet with Hall and ion slip effects has been investigated. The numerical solutions for the considered flow problem are obtained employing spectral relaxation method (SRM). The impacts of different significant parameters on the velocities, temperature, and nanoparticle concentration distributions have been considered graphically. Moreover, the numerical results for the skin friction coefficients, local Nusselt, and Sherwood numbers have been presented graphically. Accordingly, the major results of the present study are(a)The velocity profiles along both and directions enhance with an increment in the Hall parameter and ion slip parameter while the opposite inclination is observed with a rise in Casson fluid parameter and magnetic field parameter .(b)The velocity distribution along direction boosts with a rise in mixed convection parameter .(c)The temperature profile enhances with increasing values of the Brownian motion parameter , thermophoresis parameter , thermal radiation parameter and heat source parameter , and it declines for increasing values of stretching ratio parameter and temperature exponent parameter .(d)The concentration distribution rises with an increment in the thermophoresis parameter and it diminishes as the Brownian motion parameter , stretching ratio parameter , and temperature exponent parameter enlarge.(e)Both the skin friction coefficients reduce as Casson fluid parameter Hall parameter , and ion slip parameter rise.(f)Both the local Nusselt and Sherwood numbers are increasing functions of the thermal radiation parameter and temperature exponent parameter Moreover, opposite conditions are noticed for the local Nusselt and Sherwood numbers as Brownian motion parameter and heat source parameter increase.
5.1. Recommendations
It is important to emphasize few limitations about this research work. Such limitations will help researchers to analyze and provide an extension of the present work. Thus, the following assumptions and limitations can be considered:(i)The same problem can be calculated under the deliberation of different nanoparticles, viscous dissipation, ohmic heating, Newtonian heating, etc.(ii)To extend present results using convective boundary condition or/and slip condition.
Nomenclature
| : | Cartesian coordinates |
| : | Velocity components |
| : | Velocities of the stretching sheet |
| : | Constant magnetic field |
| : | Acceleration due to gravity |
| : | Fluid temperature |
| : | Surface temperature |
| : | Ambient temperature |
| : | Concentration of fluid |
| : | Surface concentration |
| : | Ambient concentration |
| : | Magnetic field parameter |
| : | Local grashof number |
| : | Local reynolds number |
| : | Buoyancy ratio |
| : | Prandtl number |
| : | Brownian diffusion coefficient |
| : | Thermophoresis diffusion coefficient |
| : | Brownian motion parameter |
| : | Thermophoresis parameter |
| : | Schmidt number |
| : | Skin friction coefficient in direction |
| : | Skin friction coefficient in direction |
| : | Local nusselt number |
| : | Local sherwood number |
| : | Velocity ratio parameter |
| : | Reference length |
| : | Temperature exponent parameter |
| : | Radiation parameter |
| : | The dimensional heat generation |
| : | Heat source parameter |
| : | Dimensionless stream functions |
| : | Density of fluid |
| : | Kinematic viscosity of the fluid |
| : | Casson fluid parameter |
| : | Mixed convection parameter |
| : | Thermal diffusivity |
| : | Thermal conductivity |
| : | Electrical conductivity |
| : | Hall parameter |
| : | Ion slip parameter |
| : | Coefficient of thermal expansion |
| : | Coefficient of concentration expansion |
| : | Dimensionless similarity variable |
| : | Dimensionless temperature |
| : | Dimensionless concentration |
| : | Effective heat capacity of a nanoparticle |
| : | Heat capacity of the fluid |
| : | Fluid |
| : | Nanoparticle |
| : | Condition at the surface |
| : | Ambient condition |
| : | Differentiation w. r. t. |
Data Availability
No data were generated or analyzed during the contact study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.