Abstract
Mounting temperatures in electronic devices during operation may damage sensitive internal components if too much thermal energy accumulates inside the system. The advent of an innovative ultrahigh-performance thermal management technology known as nanofluid has provided a veritable platform to improve the system performance and reliability by removing the high heat flux generated in the engineering and industrial devices. This paper examines the combined effects of Darcy–Forchheimer porous medium-resistant heating and viscous dissipation on stagnation point flow of a Casson nanofluid (- and -) towards a convectively heated slippery stretching/shrinking cylindrical surface in a porous medium. The governing nonlinear model equations are obtained, analysed, and tackled numerically via the shooting technique with the Runge–Kutta–Fehlberg integration scheme. A unique solution is obtained when the surface is stretching. For shrinking cylindrical surface, the model exhibits nonunique dual solutions for a defined range of parameter values, and a temporal stability analysis is conducted to ascertain the stable and physically achievable solution. The effects of emerging thermophysical parameters on the overall flow structure and thermal management such as velocity and temperature profiles, skin friction, and Nusselt number are quantitatively discussed through graphs and in tabular form. It is found that the thermal performance heat transfer enhancement capability of - is higher than that of -. Moreover, the nanofluid thermal performance is enhanced with nanoparticles volume fraction, Casson nanofluid parameter, and Biot number but lessened with porous medium permeability.
1. Introduction
Non-Newtonian fluids have various applications: damping and braking devices, printing technology, personal protective equipment, food products, and drag-reducing agents. Casson fluids are used to characterize non-Newtonian fluids with shear-thinning properties. Nakamura and Sawada [1] studied non-Newtonian fluid and introduced the biviscosity model as a constitutive equation (1) for blood instead of the usually used Casson model equation for blood. Animasaun [2] studied incompressible Casson fluid flow with viscous dissipation and obtained that the temperature profile reduces as the Casson parameter rises. Rasool et al. [3] discussed the heat and mass transfer characteristics of Casson-type MHD nanofluid flow. Recently, Khan et al. [4] and Alkasasbeh [5] also studied Casson-type fluid flow.
Due to the advancement of thermal devices in engineering systems, the applications of nanofluids have been playing a vital role in the enhancement of heat transfer and cooling processes of electronic devices in many manufacturing industry processes (Tadesse et al.) [6]. Nanoparticles provide a huge surface area for heat transfer because of their special features, such as lower density and excellent chemical and physical stabilities. For instance, due to its hard magnetic material with high coercivity and good mechanical stabilities at higher temperature and great chemical and physical stability, cobalt ferrite is the most suitable for several applications: audio, videotape, generator, etc. (Kazemi et al.) [7] and its high thermal conductivity made nanoparticles for use as enhancements in the heat transfer rate and has great applications in the areas of paints and coatings, cooling of radiators and electronic devices, nucleate pool boiling, heat exchangers, preparation of sunscreen, catalysts, etc. (Ali et al.) [8]. Mebarek-Oudina and Chabani [9] reviewed nanofluid applications and heat transfer enhancement techniques in different enclosures, and they concluded that porous media and nanofluid properties have a direct relation with flow enhancement and heat transfer-boosting impacts. Ganesh et al. [10] carried out a boundary layer analysis to investigate the influences of slip and viscous flow on water-based MHD nanofluid, and Jawad et al. [11] investigated variable heat transmission in MHD nanofluid flow. Furthermore, the Darcy–Forchheimer flow of Sisko nanofluid with convective thermal boundary conditions and viscous dissipation was investigated, and it was revealed that Darcy number enhanced while the Forchheimer number reduced the rate of heat transfer, and the values of Darcy number controls the skin friction as discussed by Bisht and Sharma [12]. Singh et al. [13] assessed nonuniform heat source and melting heat transfer on magnetized - nanofluid and obtained that an increment in porous media parameter values, the heat transfer rate, and surface drag force diminished near the surface of the cylinder. Moreover, it was observed that an augmentation in Reynolds number declined the surface drag force and raised Nusselt number (heat transfer rate). Poornima et al. [14] mathematically studied heat transfer in boundary-layer stagnation flow past a stretching/shrinking cylinder and found that the coefficients of drag force and the heat transfer rate at the surface enhanced with an augmentation in Reynolds number. Most recently, Najib et al. [15] investigated stagnation point nanofluid flow past an exponentially shrinking/stretching cylinder inserted, and they discovered that as the slip parameter rised, the skin friction coefficient dropped, whereas the heat transfer coefficient enhanced. Moreover, the heat transfer and skin friction coefficients increased with the larger nanoparticle volume fraction and curvature parameter, and the nanoparticle has the highest coefficient of skin friction and heat transfer rate. Our recent articles Duguma et al. [16, 17] detailed about the applications of non-Newtonian Casson nanofluids.
For their various practical applications in the areas of polymer technology, metallurgy, chemical engineering, industrial processes, etc., boundary layer fluid flow due to stretching-shrinking/stretching has received due attention for the last few decades (Tadesse et al.) [6]. According to Jawad et al. [11], the stretching surface is formed by boundary layer flows, which usually occur in various engineering applications such as the sketching of plastic films, pseudofibers, permanent casting, glass blowing, metal spinning, etc. Weidman et al. [18] considered uniform shear flow past a stretching sheet surface, and Das et al. [19] characterized the fluid flow over an inclined, exponentially stretching flat surface. Fluid flow towards a shrinking case is possible due to stagnation point flow (Wang [20]). According to him, even though no possible solution is found for the unconfined fluid flow occurring in the boundary layer of the shrinking surface, due to the addition of stagnation flow, a nonunique solution exists. Lund et al. [21] studied nanofluid flow across an exponentially contracting sheet surface and demonstrated that multiple solutions exist. Moreover, they investigated that as the shrinking rate increases, convective heat transfer drops due to an augmentation in the thickness of the boundary layer. Ganesh et al. [10] carried out a boundary layer analysis to investigate the influences of slip and viscous flow of water-based MHD nanofluid past a stretching/shrinking surface and discovered dual solutions under different conditions of stretching/shrinking and suction/injection parameters. Ferdows et al. [22] studied a biomagnetic fluid (blood taken as a base fluid and as magnetic particles) flow and heat transfer through a stretching/shrinking cylinder. Najib et al. [15] investigated the impact of stretching/shrinking surfaces on the nanofluid flow and demonstrated that as slip and curvature parameters enhance, the range of the upper branch solutions expands.
The study of flow at the stagnation point (which means fluid flow over a solid surface stagnation area) of nanofluids has many applications in plastic sheet extrusion, manufacturing and industrial processing, aerodynamics, cooling and drying of paper products, etc. [6]. Gorla [23] made an analysis for the steady-state heat transfer in an axisymmetric stagnation flow on a circular cylinder. Gorla [24] investigated the boundary layer solutions for the axisymmetric stagnation mixed convection flow past a vertical cylinder. Again, Harris et al. [25] considered the steady mixed convection stagnation point boundary layer flow on an impermeable surface with slip. Moreover, Shatnawi et al. [26] made a mathematical analysis of the stagnation point flow of Casson nanofluid flow over a vertical Riga plate surface and solved it using the bvp4c technique built into Matlab packages. Basha and Sivaraj [27] numerically investigated the dual solutions and stability analysis over the extending/contracting wedge and stagnation point for the Casson nanofluid flow. Murad et al. [28] solved the heat transfer properties of Casson–Carreau fluid at the stagnation point over a continuous moving plate surface. More on stagnation point boundary layer flows, the existence of dual solutions, and stability analysis are discussed in [14, 16, 29–39].
For decades, convective heat transfer through a porous medium has attracted the interest of scientists due to its numerous applications in fields such as nuclear waste repositories, thermal insulation, solar power, geophysics, pollutant dispersion in aquifers, ground hydrology, grain storage devices, high-performance building insulation, chemical catalytic reactors, cooling of electronic systems, fossil fuel beds, petroleum reservoirs, aerodynamic heat shielding, etc. (Hussain and Sheremet [40]). The fluid flow regime through a porous space was first studied by Darcy (Darcy’s law, for small Reynolds numbers) and then later developed to consider large Reynolds numbers (Darcy–Forchheimer model), as discussed in Das et al. [19]. Hayat et al. [41] studied the Cattaneo–Christov heat flux model for flow past a porous medium and obtained that as the values of the porosity parameter enhance, the velocity profile and momentum boundary layer thickness are reduced while the temperature profile falls. Mebarek and Chabani [9] reviewed nanofluid applications and heat transfer enhancement techniques in different enclosures, and they concluded that porous media and nanofluid properties have a direct relationship with flow enhancement and heat transfer boosting impacts. Another study [4] investigated theoretically the Casson nanofluids flow past a vertical Riga plate embedded in porous medium and obtained that as permeability of the porous medium (Darcy number) increased, velocity profile increases (and drops for increment in nanoparticle volume fraction), and the drag force reduced for enhancement in porous medium parameters. Another study [42] presented MHD Newtonian fluid flow on a vertical sheet in the porous medium. They observed that as the velocity slip, curvature of the cylinder, and the porosity enhanced the velocity and temperature profiles and their boundary layer, and the drag force dropped, respectively. Lund et al. [21] analysed the impact of Darcy–Forchheimer on the flow of two-dimensional MHD nanofluid across an exponentially shrinking sheet surface and deduced that the velocity profile was reduced for enhancing the permeability parameter of the porous media. More discussion on the effects of porous media on non-Newtonian Casson nanofluids was found in our recent works (Duguma et al.) [16, 17].
In recent years, great interest has been developed in the study of convective fluid flow due to its applications in geophysics, techniques of oil recovery, heat storage systems, engineering of thermal insulation, etc. Moreover, numerous industrial and environmental systems such as geothermal energy systems, heat exchanger design, geophysics, fibrous insulation, catalytic reactors, etc. involve convection flow through porous media. Merkin [43] made an investigation into the existence of dual solutions in mixed convection in a porous medium. Makinde [44] theoretically investigated the stagnation point hydromagnetic flow of -water past a convectively heated permeable shrinking/stretching sheet and revealed that dual solutions exist for a certain range of stretching/shrinking parameters and confirmed that the upper branch solution is temporally stable and physically realizable while the lower branch solution is not. Moreover, he confirmed the existence of a critical shrinking parameter value below which no real solution occurs. Alizadeh et al. [45] investigated the forced convection of heat flow past cylinders impinging in porous media. Hong et al. [46] investigated the nonlinear mixed convection of heat in a stagnation-point flow past a solid cylinder inserted in a porous medium. More work regarding the convective flow in porous media and its impacts is found in [2, 12, 13].
From the aforementioned literature, no scientific work has been done to consider the collective effects of all embedded parameters under consideration on the complex non-Newtonian Casson nanofluid flow with heat transfer characteristics. The main goal of this paper is to investigate the existence of dual solutions, which is expected due to shrinking surfaces, and apply stability analysis to determine the stable and physically reliable solutions that buttress the theoretical relevance of the work for the hydrodynamic Casson nanofluid flow past a stretching/shrinking slippery surface in a Darcy–Forchheimer porous medium with the presence of viscous dissipation and convective heating using - and - as nanoparticles in comparison, filling the gap of the articles of Duguma et al. [16, 17] with considering cylindrical geometry of flow surface, where the novelty is tested nearer to the critical shrinking parameter ||. Moreover, this study brings significant input in the field of chemical and mechanical engineering sciences. Particularly, nanofluids are widely employed in different cooling systems in engineering and in industries for effective heat removal from electronics. For the modeled boundary layer PDEs which were transformed into similar ODEs with their corresponding boundary conditions, the numerical results of similar velocity and thermal profiles, skin friction coefficient, and rates of heat transfer and enhancement were discussed both graphically and quantitatively using the shooting technique with bvp solver embedded in Maple software packages.
2. Mathematical Description of the Problem
Consider a laminar, steady, viscous, incompressible two-dimensional stagnation point flow of - and - Casson nanofluid towards a horizontal linearly stretching/shrinking cylindrical surface in a Darcy–Forchheimer porous medium with surface velocity and free stream stagnation point velocity . The flow physical model presented in Figure 1, modified from the model used by Alizadeh et al. [45], is a convectively heated horizontal cylinder embedded in porous media. A cylindrical coordinate system is used, where the axial length of the cylinder (-axis) is considered in the direction of the stretching/shrinking surface and the dimension (-axis) is the radial change of the cylinder. The free stream temperature of the fluid flow is taken as , and the convectively heated temperature of the stretching/shrinking surface of the cylinder is .
Following Nakamura and Sawada [1] and Animasaun [2], the rheological equation of an incompressible and isotropic flow of a Casson fluid is expressed as follows:where is the component of the stress tensor, is the plastic dynamic viscosity of the non-Newtonian fluid flow, is the non-Newtonian Casson parameter, is the yield stress of the Casson fluid, is the rate of deformation component (product of strain tensor rate with itself), is the strain tensor rate, and is a critical value of , that is defined based on the non-Newtonian model. In the case of Casson fluid flow, . The dynamic viscosity is computed as follows: . On substitution, the kinematic viscosity becomes . For non-Newtonian Casson fluid flow , . Assuming the Darcy–Forchheimer flow model of flows in porous media, for this analysis, the governing equations of this problem are formulated from the balance of continuity, linear momentum, and energy towards a stretching/shrinking cylindrical surface with respect to a cylindrical coordinate system and are given bysubjected to boundary conditions given bywhere , , , , , , , , , , , , , , , , , and are the direction velocity, direction velocity, effective dynamic viscosity, non-Newtonian/Casson parameter, effective density, permeability of the porous medium, Forchheimer drag force coefficient, effective thermal conductivity, effective heat capacity, specific heat at constant pressure, dynamic viscocity, slip length coefficient, thermal conductivity, convective heat transfer coefficient, radius of the cylinder, constant of strain rate at the cylinder surface, constant of free stream strain rate of the nanoparticles, and real constant () of the Casson nanofluid flow, respectively. The parameters , , , and are defined (following Makinde [44], and Tadesse et al. [6]) as follows:where is the density of the base fluid, is density of the solid nanoparticle, is the base fluid thermal conductivity, is the nanoparticles thermal conductivity, is the nanoparticles volume fraction, and is the base fluid dynamic viscosity, and these are the thermophysical properties of the nanoparticles. Note that, where is called the “sphericity” which is defined as the ratio of the surface area of the sphere to that of the particle for the same volume. For spherical particles, , and for the cylinders, . This study considers the copper particle is spherical in shape, so that , as discussed by Hamilton and Crosser [47]. The thermophysical properties of , Casson fluid, , and are given in Table 1, following Tshivhi and Makinde [48] and Shaw et al. [49].
Equations (3)–(5) represent the nanofluid flow when and the non-Newtonian Casson fluid flow when and . The governing equations (2)–(4) are transformed into dimensionless form by introducing nondimensional variables defined as follows to obtain similar solutions:where the stream function is related to a velocity component as follows:
Since the nondimensional variables in (8) and (9) satisfy the continuity equation in (2), equations (3) and (4) are converted into the following nondimensional form:
With the boundary conditions in the dimensionless form,where and are the similarity variable, Darcy number (porous media parameter), Forchheimer (second order porous resistance) parameter, velocity ratio (stretching/shrinking) parameter (where for stretching and for shrinking of the surface), free stream Reynolds number, Prandtl number, velocity slip parameter, Eckert number, and Biot number (convective parameter), respectively. These dimensionless parameters and the variables and quantities are defined as follows:
Note that(i)According to Joseph et al. [50], the pressure gradient in the flow due to porous medium is given by where is the velocity vector, is the permeability of the porous medium (), is a dimensionless form-drag constant, is the diameter of spheres of the porous medium, and is the equivalent diameter of the bed (defined in terms of the height and width of the bed). Thus, putting () confirms the dimensionlessness of .(ii), where is the curvature of the cylinderical surface, as described by Gorla [24] and Alizadeh et al. [38].
3. Physical Quantities of Engineering Interest
The heat flux and wall skin friction are computed as follows:and thus, the physical quantities of engineering interest: the reduced local Nusselt number and coefficient of reduced skin friction are given by
On simplification,where represents Reynolds number (Gorla [24]). The heat transfer enhancement (HTE) of the - and - nanoparticles are computed using the following formula:
4. Temporal Stability Analysis of the Solution
On solving problems involving boundary layer flow, the solution could be multiple, unique, or does not exist. In the case of two or more solutions, the upper branch (first) solution is given to the solution that initially satisfies the boundary condition at the far field. The temporal stability analysis has proved that the upper branch solution is the only one that is physically realizable and stable in most problems. However, according to Weidman et al. [18], there also exists a problem that has a lower branch solution that is stable. Therefore, it is necessary to execute the stability analysis and validate the reliability of the stable solutions. If an initial growth of perturbation appears in the solution, the solution is not physically realizable. The perturbation may exponentially increase or decay with time, and that is the reason for considering an unsteady (time-dependent) problem form in the stability analysis formulation. Thus, an unsteady form, Merkin [43], of equations (3) and (4) becomeswhere is time. The unsteady equations (19) and (20) are transformed as follows:where represents the nondimensional time variable. Inserting (21) into equations (15) and (16), the resulting equations becomewith the time dependent boundary conditions
To investigate the stability of the similarity solutions and that satisfy boundary value problems (10)–(12), the perturbation equation (25) is induced following Weidman et al. [51]. Here, and are small relative to and , respectively, whereas an unknown eigenvalue parameter (a small disturbance of decay or growth) is used in the formulation, which provides an infinite set of the eigenvalues …
After employing (25) into (22)–(24), the following linearized eigenvalue problem is attained such thatsubjected to modified the boundary conditions:
The initial decay or growth of the solution (25) is identified by obtaining the steady state solution taking and hence and in equations (22)–(24), where and (Weidman et al.) [51]. The stability of the solution obtained depends on the sign of the smallest eigenvalue . The fact that the eigenvalue is positive implies that the flow is real and stable and that there is an initial decay. To the contrary, the value of is negative, which indicates that the steady flow solution is unstable and that there is an initial growth of disturbance. Now, the simplified and linearized eigenvalue problem above can be rewritten as follows:subjected to the following boundary conditions:
To find the possible range of the smallest eigenvalue , equations (25) and (26) along with the boundary conditions (31) are computed using the solver functions embedded in Maple. To do this, (29) needs a modification, and hence, the code can successfully execute the computation. Therefore, is relaxed and substituted with a condition such as , following Harris et al. [25]. The modified boundary conditions in (31) becomes
5. Numerical Method
In order to apply the solver, the equations must be rewritten as a set of equivalent ordinary differential equations of first order. This is done using the substitutions, where and as follows:
For the boundary conditions (12), we get
The same procedures are followed for stability analysis. New substitutions are introduced to rewrite equations (25) and (26) and the boundary conditions (32) into first-order ordinary differential equations by letting where , , and :
For the boundary conditions, we get
To determine the unknown initial conditions , , , and (i.e., the values of , , , and , respectively), shooting the equations is performed for an arbitrary slope so that the solution of the system of ODEs satisfies the boundary conditions at , and its accuracy is checked by comparing the calculated quantities with the provided end points. After obtaining these values, the fourth-fifth order Runge–Kutta–Fehlberg techniques applied to solve the system of first-order ODEs in (33) with boundary conditions (33) and determine from (35). To get the dual solutions, different initial approximates for the values of are considered where all profiles asymptotically satisfy the boundary conditions.
6. Results and Discussion
In this research article, the cumulative effects of the velocity ratio (stretching/shrinking) parameter , free stream Reynolds number , Casson parameter (factor) , porous media parameter (Darcy number) , porous media inertial resistance parameter (Forchheimer parameter) , velocity slip (slipperiness) parameter , Prandtl number , viscous dissipation parameter (Eckert number) , convective heating parameter (Biot number) , and nanoparticle volume fractions , on the fluid velocity and temperature profiles, drag force, and heat transfer rate are illustrated graphically in a chart, and numerically computed results are presented in tables. The range of parameters used in this article are as follows: . Moreover, the appearance of dual solutions for some intervals of varying parameter values is explained for the coefficients of skin friction (surface drag force) and Nusselt number (heat transfer rate) in plots and/or tables for various numerical quantities of the involving parameters. For the governing systems of highly nonlinear ordinary differential equations (9) and (10) that cannot be solved analytically, they are numerically tackled using the shooting technique with Maple 2018 (which is coded with a finite difference fourth order accuracy level and solves the boundary value problem, bvp), subjected to initial and far field boundary conditions (12). To validate this method, the computational results of (i) dimensionless stream function () and dimensionless temperature () at , Gorla [23] and Alizadeh et al. [38] presented in Table 2, and (ii) the coefficient of skin friction, Gorla [23] and Wang [52] presented in Table 3, are compared. It is seen from the tabulated results that the comparison reveals a better agreement for each value of the dimensionless stream function, dimensionless temperature, and coefficient of skin friction. Therefore, we, the authors, are guaranteed that the method and results obtained under this study are all valid and acceptable. The velocity, temperature, local Nusselt number, and local skin friction profiles are illustrated graphically and are presented in tables.
6.1. Existence of Dual Solutions due to Shrinking Surface
For the shrinking parameter , the influences of varying embedded parameters on the skin friction are presented in Figures 2–5 for (CF) nanoparticle case only, and this result is compared with that of (TD) nanoparticle case using Table 4. Accordingly, it is revealed that dual solutions exist: the upper (solid curve) and lower (dotted curve) solution branches exist for , and no real solution exists for . That means, is the shrinking parameter value for the upper and lower solutions, which physically demonstrates the extent to which the surface is able to shrink while in processing. Moreover, Figures 2–5 shows that the skin friction is a decreasing function of the shrinking parameter for all the parameters involved in the upper branch solutions except nearer to , whereas it is purely a decreasing function for the lower branch solutions. || (the range of for which the similarity solution appears) increases with the slipperiness parameter and higher values of the nanoparticle volume fraction parameter , whereas it decreases with increasing values of the Casson factor , the porous media parameter , porous media inertia resistance parameter , and Reynolds number for the upper branch solutions. Beyond this critical value , no similarity solutions exist due to the boundary layer separation from the surface, which leads us to the difficulty of using boundary layer approximations to solve the problem. Furthermore, from these figures, it is observed that the upper and lower branch solutions for the coefficient of skin friction are in opposite trend except for increment in values of and . Comparing the graphical results with Table 4, it is revealed that the critical value is wider for TD with respect to CF nanoparticles. Figures 2(a) and 5(a) demonstrate that for the shrinking parameter , the coefficient of skin friction gets enhanced as the nanoparticle volume fraction increases for the upper branch solution, which could be due to the high coercivity CF. Physically, the augmentation in nanoparticle volume fraction implies that the base fluid and the nanoparticles collision raises the motion of the nanofluid, resulting in a diminishing of the momentum boundary layer thickness and increasing the drag force at the surface. It is noted that in Figures 5, 6(b), and 7, the thinner and bold curves represent upper and lower branch solutions, respectively.
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The coefficient of skin friction falls as the Casson parameter rises for the upper branch solution, as shown in Figures 2(b) and 5(a). The coefficient of skin friction is observed to drop upon improving the Casson factor , which means that less applied force is required to move the Casson nanofluid past the surface for higher values of the Casson factor . In other words, implies that the fluid misses its non-Newtonian properties and behaves like a Newtonian type, and hence, its velocity augments due to the reduction in the shear stress.
Figures 3(a) and 5(b) reveal that the skin friction decreases as the porous media parameter (Darcy number ) gets higher for the upper branch solution. Again, the skin friction drops as the porous media inertial parameter rises for the upper branch solution, whereas the solution interval reduced, as illustrated in Figures 3(b) and 5(b).
As demonstrated in Figures 4(a) and 5(b), the coefficient of skin friction drops with an increment in the values of Reynolds number for the upper branch solution. The skin friction (surface drag force) diminishes with a rise in the values of the slipperiness parameter , as illustrated in Figures 4(b) and 5(b) for the upper branch solution.
Generally, the above results are observed nearer to the critical shrinking parameter values, as illustrated by the plots; the computed results from Tables 4 and 5 support what is discussed above. From Table 6, far from (say at ), it is revealed that coefficient of the skin friction increases only with rising values of and but drops with other parameters. Moreover, it is observable that the skin friction (surface drag force) coefficient is higher for CF compared to TD near the critical , as it is observed from Table 5.
6.2. Rate of Heat Transfer
Figures 6–11 and Table 5 explain the effects of all the parameters under discussion on the heat transfer rate (local Nusselt number ). It is noted from the model that the energy and momentum equations are coupled, and hence, the Nusselt number characterizes a dual solution for for the case of shrinking surfaces. Moreover, it is revealed that dual solutions exist and that the upper branch solutions (represented by the solid curve) for the Nusselt number are an increasing function of the shrinking parameter , whereas the opposite trend is observed for the lower branch solutions. The critical shrinking parameter for the solution decreases for increasing values of nanoparticle volume fraction, porous media inertia parameter, and Biot numbers, whereas for the Casson, Reynolds, slipperiness, and velocity ratio parameters, Darcy, Prandtl, and Eckert numbers, it gets widened for the upper branch solutions.
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It is observable that rising values of nanoparticle volume fraction resulted in an ascending heat transfer rate (Nusselt number ) for the upper branch solution, as seen in Figures 6(b) and 8(a) nearer to the critical shrinking parameter .
Figures 6(b) and 8(b) reveal that the augmentation in the values of the Casson factor produces an intensified heat transfer rate nearer to the critical shrinking parameter values for the upper branch solution. Physically, an increment in the Casson factor augments the fluid motion and the thermal profile, resulting in an increase in the heat transfer rate. As (the Newtonian fluid case), the heat transfer rate is highly reduced compared to the non-Newtonian Casson fluid.
Figures 7(a) and 9 demonstrate that the intensification in the values of porous media parameter and porous inertia resistance parameter resulted in the dropping of the heat transfer rate for the upper branch solutions nearer to the critical shrinking parameter due to the effect of the porous media against the flow rate. For the upper branch solution, nearer to the critical shrinking parameter , increment in Reynolds number drops the convective heat transfer rate, as demonstrated in Figures 7(a) and 10(a). Physically, the added nanoparticles increases viscosity (Figure 2(a)), producing dominated inertial forces nearer to . Moreover, the retarded flow rate at this region leads to the overcoming of conductive heat transfer to the convective one due to better thermal conductivity of the used nanoparticles than the normal base fluid, and therefore, weak temperature gradient (Nusselt number) is observed, resulting in reduced heat transfer rate of the Casson Nanofluid flow.
Stepping up in the values of the slip parameter enhances the convective heat transfer rate (Nusselt number ) in the upper branch solution as we see from Figures 6(b) and 10(b), nearer to the critical shrinking parameter values. Physically, heat transfer rate increases as a result of an improvement in the slipperiness of the cylindrical surface, reducing adhesion of the nanofluid to it.
Figure 11(a) shows that as the Pradntl number increases, the convective heat transfer rate gets higher in the upper branch solution nearer to the critical shrinking parameter values. Physically, this is due to an increment in , which signifies that the momentum diffuses more quickly and the velocity boundary layer is less thick than the thermal boundary layer of the fluid, meaning more heat transfer by convection is dominant in the Casson nonofluid flow.
Figures 7(b) and 11(b) illustrate as the viscous dissipation parameter increases, the local Nusselt number (heat transfer rate) also increases nearer to the critical shrinking values for the upper branch solution. Figures 6(a) and 7(b) demonstrate the heat transfer rate (local Nusselt number ) against the Biot number , and as it can be observed, the heat transfer rate is increasing as the convective heating parameter (Biot number ) increases for the upper branch solution nearer to the critical shrinking values. Physically, it means that the coefficient of heat transfer caused by the hot fluid beneath the sheet is directly associated with the convective heating parameter. Moreover, from Figure 7(b), it is demonstrated that the heat transfer rate () escalates as the values of the shrinking parameter increases for the upper branch solution.
Generally, it is observed from the graphs and Table 6 that the heat transfer rate (the Nusselt number) is more pronounced for TD compared to CF nanoparticles. Moreover, nearer to , heat transfer rate (Nusselt number) increases for rising values of and but decreases for higher values of and ; however, the opposite trend holds far from (say at ).
6.3. Numerical Analysis of the Stability Test
From the numerical results of this problem, the dual solution exists for some interval of . The stability analysis is made to determine stable solutions within different solutions that arise due to shrinking cylindrical surfaces. As detailed in Table 5, for varying values of involving parameters, the smallest eigenvalue is calculated for the temporary change of small disturbances/perturbations regarding the basic steady flow, with respect to the fixed values of . From the table, it is observed that corresponding to the upper branch solutions, the smallest eigenvalue obtained is positive for shrinking surfaces, implicating that the upper branch solution is hydrodynamically temporally stable and therefore physically realizable. Clearly, for the lower branch solution, the negative value of revealed that it is unstable and physically unachievable. In addition, demonstrates the rate of declination of small disturbances on the upper branch solution, whereas for the lower branch solution shows the enhancement of the disturbances.
6.4. Velocity Profile
Figures 12–13(a) demonstrate the effects of different values of the embedded parameters on the fluid flow velocity profile in the case of CF nanoparticles for the UB (which is the only stable and physically realizable one) and LB (unstable and cannot be realized) solutions nearer to the critical shrinking values. In all plots, it is demonstrated that the upper and lower branch solutions are in opposite trend. Moreover, the velocity profile is an increasing function of , whereas for all rising values of and , the velocity profile is decreasing with raising of its momentum boundary layer thicknesses nearer to in the upper branch solutions. Furthermore, , and do not have observable effect on the velocity profile and coefficient of the skin friction as Table 6, and no plot is demonstrated in this regard.
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The influence of CF nanopartile volume fraction on the flow velocity profile is displayed in Figure 12(a), and it is observed that the velocity profile drops and the flow boundary layer thickness increases with enhancing nanopartile volume fraction for the upper branch solution nearer to ||, and the lower branch solution drops. Figure 12(b) depicts the influence of the Casson factor on the fluid velocity profile, and it is observed that the velocity profile decreases with rising boundary layer thickness as the values of augment for both the upper branch solutions. That is, the velocity of Casson nanofluid flow gets diminished when increases, which might be due to the inverse proportionality of the Casson factor to plastic viscosity. The Newtonian fluid case happened as got higher (), and it is observed that the velocity of the Newtonian fluid is lower than that of the Casson nanofluid with widened boundary layer thicknesses.
The diminishing velocity profiles and increasing boundary layer thicknesses are observed for the rising porous media parameter which could be due to the influence of the application of the nanoparticles into the Casson fluid and porous matrix resistance force, which will retard flow, as demonstrated in Figure 14(a) within the flow regimes of the Casson nanofluid for both the upper branch solutions. Similarly, from Figure 14(b), it is observed that the dropping of velocity profiles for rising in porous media inertial resistance parameter for the upper branch solutions with increasing boundary layer thickness. The impacts of Reynolds number (which measures the ratio of momentum/inertia to viscous forces) on the velocity profile are illustrated in Figure 15(a), and it is seen that as increases, the velocity profile drops nearer to the critical shrinking parameter value as with rising boundary layer thickness for the upper branch solutions; however, the velocity profile rises for lower branch solution. Figure 15(b) shows that as the slipperiness parameter increases, the velocity profile decreases with increasing boundary layer thickness for both the upper and lower branch solutions. Moreover, Figures 13(a) and 15(b) show that as the values of the velocity ratio parameter increases, the velocity profile increases and its boundary layer thickness diminishes in both upper and lower branch solutions.
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To generalize, it is observable from the graphs under this subsection and Table 6 that the velocity profile is higher for CF compared to TD in the case of the upper branch solution at the surface of flow. Moreover, for the shrinking surface case, unlike nearer to where the velocity profile decreases with all parameters, except for , Table 6 reveals that far from (say at ), the velocity profile increases for rising values of , and for the upper branch solution. Furthermore, fluctuating values of , and showed unobservant effects on the velocity profile and its boundary layer thickness and the skin friction coefficient, perhaps due to dominance in the thermal diffusivity, energy, and resistance for convection at the cylindrical surface of the Casson nanofluid in the upper branch solutions.
6.5. Temperature Profile
The overall impacts of varying values of the involving parameters on the fluid temperature profile and related boundary layer thickness for the shrinking parameter are demonstrated in Figures 13(b)–16. It is observed from the plots that the temperature profile is a decreasing function of for all parameters under consideration nearer to . The upper and lower branch solutions are in opposite trend except for the parameters and .
The temperature profile behavior in dealing with different values of nanoparticle volume fraction is illustrated in Figure 13(b), and it is observed that the temperature profile and its boundary layer thickness diminishes for increasing in both upper and lower branch solutions. Physically, increasing the nanoparticle volume fraction raises the thermal conductivity of the fluid, and hence, intensifies the temperature gradient of the Casson nanofluid, and drops the thermal behavior of the Casson nanofluid flow.
From Figure 17(a), it is observed that the temperature profile and the corresponding boundary layer thickness diminish with increasing values of the Casson factor for the upper branch solution. Physically, when the Casson factor values become higher, the strength of the yield stress of the Casson fluid is weakened, enhancing the plastic dynamic viscosity and therefore diminishing the thickness of the thermal boundary layer of the flow temperature profile. Note that the Casson nanofluid’s thermal boundary layer thickness is higher than that of the Newtonian fluid. The values of the Casson parameter get higher () which shows weaker interactions of molecular motion within the Casson nanofluid behaving as a Newtonian Casson fluid, which ultimately reduces the Casson nanofluid’s temperature profile.
(a)
(b)
Figure 17(b) shows that the temperature profile and its boundary layer thickness rises against the increment in within the flow regime for the upper branch solution. Physically, the increment in porous matrix reduces the temperature gradient and hence enhances the thermal properties of the Casson nanofluid flow. Moreover, from Figure 18(a), it is observed that the temperature profile and its boundary layer thickness rises against the increment in the values of for the upper branch solution.
(a)
(b)
Figure 18(b) reveals that rising values of Reynolds number resulted in an intensification of the temperature profile and its thermal boundary layer thickness for the upper branch solutions. Physically, it indicates that as Reynolds number rises, the temperature gradient drops due to dominance in thermal conductivity, and hence, the temperature of the fluid starts to rise. In Figure 19(a), it is demonstrated that an increment in the slipperiness parameter resulted in the dropping of the temperature profile and its thermal boundary layer thickness for the upper branch solution. Physically, the slipperiness of the surface facilitates the temperature gradient, resulting in a reduction of the temperature profile and a lessening of the corresponding thermal boundary layer thickness of the Casson nanofluid flow.
(a)
(b)
The effect of the Prandtl number on the thermal properties of the Casson nanofluid flows is revealed, and it is observed that as upsurges, the temperature and its boundary layer thickness diminish for the upper branch solutions, as presented in Figure 19(b). Figure 20(a) illustrates the influence of the viscous dissipation parameter (Eckert number ) on the thermal profile. The rising of both the temperature profile and the boundary layer thickness are observed at the expense of enhancing the viscous dissipation parameter for the upper branch solution. It is known that the kinetic energy is absorbed by viscosity from the fluid motion and converted into internal energy that raises the heating of the fluid flow, thus increasing both the temperature profile and its boundary layer thickness.
(a)
(b)
Figure 20(b) depicts that increasing the convective heating parameter drops the temperature profile and the related boundary layer thickness of Casson nanofluid flow for the upper branch solution. Physically, for , the temperature on the surface and beneath the surface (inside the cylinder) will be approximately similar, and/or the surface is a good conductor of heat, so the temperature is uniformly distributed throughout the surface of the cylinder.
As the velocity ratio increases, the temperature profile drops and the thermal boundary layer diminishes for the upper branch solutions, as depicted in Figure 16.
Generalizing based on the plots under discussion and Table 6, the temperature profile is higher for CF compared to TD for the upper branch solutions. Moreover, nearer to , temperature profile and its thermal boundary layer decrease for increasing values of and ; however, the opposite trend holds far from (say at ).
6.6. Heat Transfer Enhancement
The chart in Figure 21 demonstrates how the rate of heat transfer is enhanced for increasing values of the nanoparticles’ cobalt ferrite () and/or titanium dioxide () volume fraction in the Casson fluid for the shrinking, fixed, and stretching surfaces. The heat transfer enhancement is more pronounced for nanoparticles compared to that of nanoparticles, which could be due to the larger thermal conductivity of TD with respect to CF. Moreover, the rate of heat transfer enhancement is higher for the working Casson nanofluids with increasing nanoparticle volume fraction for a higher shrinking parameter compared to other cases. Studies revealed that nanoparticles serve as better coolants for industrial and engineering usage when compared to base fluids; for instance, TD can be used as a better enhancer of heat transfer rate and cooling of radiators and electronic devices, etc.
7. Conclusions
Numerical investigation into the stagnation point flow of --Casson nanofluid past a slippery surface stretching/shrinking through a Darcy–Forchheimer porous medium in the presence of viscous dissipation and convective heating has been worked out. By using similarity transformations, the modeled boundary layer PDEs were converted into a system of ODEs with their corresponding boundary conditions, and the shooting technique with bvp solver embedded in Maple software packages was used for the numerical computation of the solutions. The temporal stability analysis has been done to identify stable and physically reliable solutions subjected to small disturbances. The effect of various parameters on the dimensionless velocity and temperature profiles, the coefficient of skin friction, and rates of heat transfer and enhancement are obtained numerically and presented in graphs, tables, and a chart. The following findings are summarized from the discussion:(i)There is a critical value of the shrinking parameter that determines the interval of solutions, such that the critical value || widens only for higher values of nanoparticle volume fraction and slipperiness parameters, and increment in || is higher for than .(ii)The skin friction (drag force) coefficient of flow escalates only as the nanoparticle volume fraction parameter increases and reduces with an increment in the values of other parameters for the upper branch solutions.(iii)For the shrinking surface nearer to the critical shrinking parameter , the coefficient of skin friction increasingly overshot as rises for all parameters (, and ), resulting in a reduced velocity profile with increasing momentum boundary layer thickness for all parameters in the upper branch solutions.(iv)For the shrinking surface, there are unobservable effects of , and on flow velocity profile and coefficient of skin friction nearer to the fixed surface.(v)For the shrinking surface, the increment in and declines the heat transfer rate and raises the thermal behavior the Casson nanofluid near , and the reverse is observed nearer to the fixed cylindrical surface.(vi)The coefficient of skin friction (drag force) and the temperature profile are a decreasing function of the shrinking parameter ; however, the reverse is true for the velocity profile and rate of heat transfer in the upper branch solutions.(vii)For the upper branch solutions, the rate of heat transfer drops for increasing values of the nanoparticle volume fraction parameter, Forchheimer parameter, and Biot number nearer to and upsurges for the other parameters.(viii)Increasing the amount of nanoparticle volume fraction in the Casson fluid boosts the heat transfer enhancement rate, which is higher for than .(ix)The temporal stability analysis determined the smallest eigenvalue which revealed that only the upper branch solution is stable and physically realizable, whereas the lower branch solution is unstable and not realistic for the flow problem.(x)For the upper branch solutions, the velocity profile drops with all parameters nearer to and enhances with an increment in the values of the nanoparticle volume fraction parameter, slipperiness parameter, Darcy number, and Reynolds number far from for the shrinking surface.(xi)For the upper branch solutions, the temperature profile and thermal boundary layer thickness increase only with enhancing values of nanoparticle volume fractions and porous inertia resistance parameters nearer to , and increase the nanoparticle volume fraction parameter, Casson factor, Prandtl number, Eckert number, and Biot number far from .(xii)For the Newtonian flow (as ), the critical shrinking parameter , surface drag force coefficient, velocity profile, and temperature profile with its boundary layer thickness diminish, whereas the heat transfer rate and momentum boundary layer rise for the upper branch solutions nearer to .(xiii)The critical shrinking parameter and heat transfer rate and enhancement are higher for nanoparticles compared to nanoparticles, whereas the skin friction coefficient, velocity, and temperature profiles are higher for nanoparticles relative to for the upper branch solutions.
Nomenclature
| : | Radius of the cylinder |
| : | Real constants |
| : | Biot number |
| : | Coefficient of the skin friction |
| : | Specific heat at constant pressure of the fluid |
| : | Darcy number (porous media parameter) |
| : | The rate of strain tensor |
| : | Eckert number |
| : | Dimensionless stream function |
| : | Forchheimer drag force coefficient |
| : | Forchheimer parameter |
| : | Convective heat transfer coefficient |
| : | Porous medium permeability |
| : | Thermal conductivity of the base fluid |
| : | Nanoparticles’ thermal conductivity |
| : | Nanofluids’ effective thermal conductivity |
| : | Slip length coefficient |
| : | Real constant |
| : | Local Nusselt number |
| : | Prandtl number |
| : | Yield stress of the fluid |
| : | Heat flux |
| : | Reynolds number |
| : | Temperature of the fluid |
| : | Local fluid temperature |
| : | Ambient temperature of the Casson nanofluid |
| : | Velocity components along coordinates, respectively |
| : | Free stream velocity of the Casson fluid |
| : | Coordinates along the surface and the radial coordinate, respectively |
| : | Non-Newtonian/Casson parameter/factor |
| : | Velocity slip parameter |
| : | Eigenvalue parameter |
| : | Similarity variable |
| : | Dimensionless temperature |
| : | Velocity ratio (stretching/shrinking) parameter |
| : | The plastic dynamic viscosity of the Casson nanofluid |
| : | Dynamic viscocity of the base fluid |
| : | Effective dynamic viscosity of the Casson nanofluid |
| : | Base fluids’ kinematic viscocity |
| : | The component of deformation rate |
| : | Critical value of |
| : | Density of the base fluid |
| : | Density of the solid nanoparticle |
| : | Effective density of the Casson nanofluid |
| : | Effective heat capacity of the Casson nanofluid |
| : | Nondimensional time variable |
| : | Components of stress tensor |
| : | Wall skin friction |
| : | Nanoparticles volume fraction. |
Data Availability
The data used for supporting the findings of this research are included within the article.
Additional Points
Future Perspective. This present study on nanofluid heat transfer enhancement is valid for Casson nanofluid and did not incorporate the effects of other important factors such as nanoparticle shape factors, hybrid nanoparticles, thermophoresis, and Brownian motion. The authors envisage extending this study in the future to include other non-Newtonian nanofluids (both single-phase and two-phase flow models) as well as the effects of all the omitted factors.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge Adama Science and Technology University for facilitating this study.