Abstract
This study may be applicable in heavy power engine and cooling of a nuclear reactor, insulation for buildings, petroleum reservoir operations, and magnetic material processing solar energy collectors. In this manuscript, the slip results are evaluated for the non-Newtonian fluid on the oblique stagnation point flow of induced magnetic field over the oscillating surface. The valuation of heat flux is examined through the Fourier law of heat transfer. The metallic nanoparticle Copper is within the base fluid, and water is utilized in the analysis. Nanofluids have benefits such as steadiness of the working fluid, decreasing blockage, clogs, decreasing prices, decreasing the friction coefficient, and decreasing the size of the heat transfer system. Similarity variables are utilized to convert the developed flow into higher nonlinear coupled ordinary differential equations (ODE) which are tackled numerically using a mathematical technique such as the bvp4c method in Maple and Matlab software. According to the present geometry, the flow behavior of the operating nanofluid has analyzed by stream lines. Disparities in velocity and temperature profile are demonstrated by graphs to describe the effects of controlling parameters. The Casson fluid parameter enhances the velocity of the fluid. The system heats up by the impact of Joule heating and dissipation.
1. Introduction
Due to the small size of the nanoparticles, these fluids behave like normal fluids first introduced by Choi and Eastman [1]. Nanofluids are essentially utilized to improve heat conduction and have many applications in several business processes such as heat exchanger equipment and electronic cooling system radiators given by Khan et al. [2]. Due to narrative properties of nanofluids, it is potentially valuable in numerous applications in heat transfer, such as car engines, microelectronics, engine cooling/vehicle thermal management, chiller, nuclear reactor, heat exchanger, domestic refrigerator, fuel cells, pharmaceutical processes, coolant, in grinding, defense, in boiler flue gas temperature decline, in space technology, machining, and ship given by Turkyilmazoglu [3]. Nanofluids have benefits such as steadiness of the working fluid, decreasing blockage, clogs, decreasing prices, decreasing the friction coefficient, decreasing the size of the heat transfer system, improving steadiness related to other colloids, decreasing the power required for fluid pumping, increasing heat transfer capability, and improving the heat transfer. Several researchers have studied the nanofluid flow [4–13].
Hiemenz [14] was the first person who studied the stagnation point flow by finding the exact solution to the Navier–Stokes equations. Stagnation point flows can be unsteady or steady, inviscid or viscous, oblique or normal, two or more dimensional, and opposite or onward. Alike flow frequently compacts with the motion of fluid close to the stagnation region of a firm surface preserved in moving fluid given by Khan et al. [15]. Stagnation point flows beside the heat transfer characteristic are fairly manifested in rotating filaments, paper manufacture, melt spinning process, crystal puffing, and continuous molding given by Abbas et al. [16]. A great number of researchers have put their attention on stagnation point flows [17–26].
Non-Newtonian fluids have a huge range of industrial applications, such as high molecular weight systems, most multiphase mixtures, foods, and solutions. When air is involved in such procedures as agitation, mixing, dispersion, and kneading, then gas bubbles are isolated and assorted in fluids given by Zhan et al. [27]. Materials displaying non-Newtonian flow characteristics include natural products, food products, magmas and lavas, biological fluids, dairy wastes, and agricultural wastes, building materials, soap solutions, polymer melts and solutions, multiphase mixtures, and personal care products including cosmetics and toiletries, given by Ijaz et al. [28]. Non-Newtonian fluids are of excessive importance in engineering sciences, medical field, and industry. The increasing applications of non-Newtonian fluids are in demonstrating of extrusion of polymers, steel substances, petroleum drilling, and blasting of glasses. Few polymers are utilized in communication appliances, medical, and agriculture given by Hussain et al. [29]. Many researchers have studied the non- Newtonian fluid flows [30–40].
In the present study, we describe the non-Newtonian fluid for oblique stagnation point flow with the induced magnetic field of nanofluid over the oscillatory and slip surface. Heat transfer is exposed to Cu nanoparticles. The coupled differential equations are solved by mathematical techniques such as bvp4c in MatLab and Maple. Graphical results are achieved for velocity and temperature profiles. Moreover, the flow behavior is also demonstrated by streamlines.
2. Formulation
Assume the two-phase flow model for oscillatory, unsteady, and incompressible two-dimensional state flow. The slip results are evaluated for the non-Newtonian fluid on the oblique stagnation point flow of induced magnetic field over the oscillating surface. The stream of nanofluid occupies obliquely on a plate vacillates in its own plan. The fluid is , the upper half of the plane. The flow equations are given aswith boundary conditionswhere are the density of nanofluid, magnetic field, thermal diffusivity of nanofluid, kinematic viscosity, dynamic viscosity, slip parameters, velocity field, and pressure, respectively.
Suppose fluid obliquely affects via the plane and the velocity components are
Now, the stagnation point develops , and the streamlines are hyperbolas whose asymptotes are
Moreover, the magnetic field must be equivalent to distributing streamline due to presence of oblique stagnation flow aswhere is the uniform magnetic field and are constants.
Consider the velocity component as
The streamlines’ flow becomes hyperbolic nature when considering oblique stagnation flow, and their asymptotes may have the equation
The boundary conditions are given by
From (12), the behavior of and asymptotically at infinity is given by
The continuity equation (1) deduces the existence of a stream function such that
Applying equation (14) in (2) to (4), eliminating the pressure using this yields, also making use of equation (10) in system (2) to (4), and using boundary conditions (11) and (12), in equations (14) to (17), we obtain
We define
In the light of equation (21), governing equations (4), (15), and (16) become
In which symbolized for nanoparticles volume fraction, the heat capacity of nanofluid, the thermal conductivity of solid fractions fluid, thermal conductivity of nanofluid, densities of fluid and solid fractions, Prandtl number, dimensionless frequency, Hartmann number, and amplitude of the wave, respectively, which are defined as
The Skin friction number can be extolled asand the shear stress is declared by
Nondimensional form of equation (39) takes the formwhere is the local Reynolds number.
The nondimensional stream function is
Displayed in Figure 1, the fluid flow forms an angle α with the plate, and the slope of the straight line can be found by putting as , where which gives slope = . Hence, the γ (shearing parameter) and α (impinging angle) relationship is
The Nusslet number is presented asand heat flux is computed by
Dimensionless form of equation (40):
3. Solution Process
Numerical solution of equations (18)–(22) is achieved using BVP solution technique built in Matlab and Maple software. It can be perceived that leading classification (18)–(22) has one-way coupling, i.e., effects , and but not vice versa. Also, for minor values of , a series solution of equations (20) and (22) has been obtained followed by
We are taking the solution of the real part only, as the form
Using equation (35) in equations (20) and (23), we obtain
For equation (22), the series solution can also be explained in a similar way:
For minor values of (considering the real part only), we obtain
We have observed that , and solving by direct integration, we obtain
Finally, the series form solution of the temperature equation (22) becomes
Thermophysical properties of base fluid and nanoparticle are composed by [21].
4. Discussion and Results
The problem is solved by a mathematical technique such as bvp4c method in Matlab and maple and expresses the results by various parameter such as Hartmann number , Casson parameter , slip parameter , and thermal slip parameter . We have illuminated graphically the behavior of flow parameters. Figures 2–7 exposed the velocity profiles commonly, and the nanofluid flow on the vacillating plate is operated through the mutual action of free stream velocity and magnetic field. At the plate surface, the nanofluid velocity is zero and rises slowly till it achieves the free stream value, satisfying the given boundary conditions. Figure 2 expresses the action of the velocity gradient , which is described for . Figure 3 shows the impact of Hartmann number on the velocity profile. When rises, then the velocity field declines and the viscosity of the boundary layer increases. On the fluid in the boundary layer, a resistive nature force produced by a magnetic field drops the motion of the fluid and the existing phenomena arise when it can describe the conductive fluid. Hence, lastly, it is expected that the magnetic field is utilized to curb boundary layer departure. Figure 4 demonstrates the effect of slip parameter on the velocity field. When rises, the velocity profile enhances and the boundary layer declines. Figure 5 explored the effect of Casson parameter on the velocity profile. When Casson parameter increases, then the velocity field increases and the thickness of the boundary layer declines. On the other way, for Casson fluid, it is observed that the velocity boundary layer thickness is greater as compared to that of Newtonian fluid. Figures 6 and 7 expose the effects of the velocity profile rises and the boundary layer drops when the values of Casson parameter and Hartmann number increases. Figures 8 and 9 show the temperature profiles when ; then, choose the values of thermal slip parameter and Hartmann number . Thermal slip and hartmann number decrease the fluid temperature. The boundary layer drops, and the temperature gradient at the surface also decreases. Physically, the momentum thickness and thermal boundary layer are functioned by Prandtl number. Low Prandtl number fluids have a greater rate of heat conduction; thus, heat can spread faster than for higher fluid. Figures 10–21 demonstrate the effect of , , and on the stream line. Table 1 is made to examine the impact of and on Skin friction; when the slip parameter rises, then the Skin friction enhances; when the Hartmann number increases, then Skin friction declines; and when Casson parameter increases, then the Skin friction also drops. Table 2 is made to explore the effect of and on Nusselt number; when thermal slip parameter and volume friction increases, then the Nusselt number enhances.
5. Conclusion
In this section, we demonstrate the MHD oblique stagnation point flow of nanofluid to an oscillatory plate which contains copper () as nanoparticle within the base fluid water. Also, consider the effect of thermal jump and velocity slip. The resulting opinions are worth citing: Because it is perceived at a variant distance from the plate, and the behavior of angular velocity increases by rising volume friction parameter and slip parameter. The heat transfer rate declines at the surface with rising Hartmann number, whereas it rises with raising values of nanoparticle volume fraction and thermal slip parameter. The stagnation point and zero skin friction depend on. By expanding the Hartmann number and thermal stratification, the thermal boundary layer is dejected.
Nomenclature
Symbols; Significance and Dimensions—Dimensionless
| : | Spatial coordinates [L]—x, y |
| : | Pressure field []— |
| : | Velocity components []—u, |
| : | Free stream parameter |
| : | Ordinary component of the stream— |
| : | Shear component of flow— |
| : | Tributary function []— |
| : | Hartmann number |
| : | Temperature field [K]— |
| : | Velocity slip parameter |
| : | Thermal slip parameter |
| : | Casson parameter |
| : | Reynold number |
| : | Thickness of solid fraction and base fluid [] |
| : | Density of nanofluids |
| : | Dynamic thickness of solid fraction and base fluid [] |
| : | Dynamic thickness of nanofluid |
| : | Kinematic thickness of nanofluid |
| : | Kinematic thickness of solid fraction and base fluid [] |
| : | Thermal diffusivity of nanofluid |
| : | Thermal diffusivity of base fluid and solid fraction |
| : | Electrical conductivity of nanofluids |
| : | Electrical conductivity of solid fraction and base fluid |
| : | Thermal diffusivity of base fluid and solid fraction |
| : | Heat capacity of nanofluids |
| : | Thermal conductivity of base fluid and nanoparticles . |
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.