Abstract
This article deals with three-dimensional non-Newtonian Jeffrey fluid in rotating frame in the presence of magnetic field. The flow is studied in the application of Hall current, where the flow is assumed in steady states. The upper plate is considered fixed, and the lower is kept stretched. The fundamental equations are transformed into a set of ordinary differential equations (ODEs). A homotopy technique is practiced for a solution. The variation in the skin friction and its effects on the velocity fields have been examined numerically. The effects of physical parameters are discussed in various plots.
1. Introduction
The rotation of fluid exists in nature due to the fact that the fluid particles rotate internally and rises with fluid movement. Due to engineering and industrial applications, the scientist considers the rotational fluid coupled with various features. Rotational fluids have many applications in engineering. Taylor and Geoffrey introduced the motion of viscous fluid in the rotating system [1]. The detailed study of fluid in rotating system is done by Greenspan [2] and Goodman [3]. The effects of MHD in a rotating system and stretched and porous mediums have been studied by Attia and Kotb [4], Borkakoti and Bharali [5], and Vajravelu and Kumar [6]. This work has been magnified along with the temperature effects by Mehmood and Ali [7], Das et al. [8], and Tauseef et al. [9].
The non-Newtonian fluid is used in many industry and technology appliances. Hayat et al. studied the non-Newtonian fluid in a rotating frame, considering the effects of MHD for micropolar nanofluids [11, 12]. Jeffrey’s model was presented by Jeffrey as a subclass of non-Newtonian fluid and studied with convection term [13, 14].
Most of the physical problems are nonlinear and have rare exact solutions. The numerical methods (NMs) and analytical methods (AMs) are used to get the results. The NMs required discretization techniques which can affect the results. Among the AMs, HAM proposed by Liao is the most powerful and fast convergent [15–19]. Hall introduced Hall current and proves that, in case of strong magnetic field, the Hall current effects cannot be ignored [20]. Similar other interesting studies are provided in [21–32] for different fluid models. This article aims to elaborate the non-Newtonian nanofluid in the rotating frame with Hall effect. Hall effect is produced due to the potential difference across an electrical conductor when a magnetic field is acting in a direction vertical to that of the flow of current. So, for this aim, Jeffrey fluid flow is considered. For the proposed model, HAM is used.
2. Problem Formulation
Assume the Jeffrey fluid between two parallel plates having d separation. The plate and fluid rotate about y axis with . The lower plate is stretched by two opposite and equal forces. A uniform magnetic field is applied perpendicularly with a steady-state condition (Figure 1).
The fundamental identities are
The BCs are
The similarity transformation used is
Using equation (6) in (1)–(4), we get
Substituting equation (8) in (7), we getwhere
is given as
The dimensionless form of is
3. Solution Procedure
HAM was introduced by Liao. Let are two continuous functions defined on topological spaces , , thensuch that :
The initial guesses are
The linear terms are with differential operatorwhere represents arbitrary constants, where 6.
3.1. Zeroth-Order Problem
Express as an embedding parameter with and , where . Then,
The BCs are where
3.2. lth-Order Deformation Problem
where
4. Convergence of HAM
With the help of assisting constraints and , the convergence region is achieved. The possible region of convergence for the proposed model is given in Figure 2 and Table 1.
5. Results and Discussion
The effect of on and is given in Figures 3 and 4. An increase in decreases and . The large amounts of viscous energy reduction produce large inertial forces, which decreases and . The effect of on the and is shown in Figures 5 and 6. It is evident that an increase in increases fluid flow due to increase in Cariolis force. This fluid rotation increases kinetic energy which also increases the flow rate. The influence of and on and is given in Figures 7–10, respectively. Both reduce velocity profile. The effect is given in Figures 11 and 12, showing that the velocity profile increases by increasing . The relaxation time gets smaller by enhancing . The effects of on and are presented in Figures 13 and 14, respectively. and oppose the flow due to large relaxation time and magnetic effects. The magnetic field opposes the flow in the y direction and enhance in the z direction.
The numerical values of , and on are presented in Table 2. We see that has inverse relations with and decreases while on direct relation with .
6. Conclusion
The following conclusion is observed:(i)A rise in R causes to decline .(ii)The mass flux decreases at a lower plate and increases at upper plate.(iii) resist the velocity profile.(iv) assist the velocity profile.(v)M resists the flow along the y direction and assists the flow along the z direction.
Nomenclature
| Gravitational acceleration: | () |
| Density: | () |
| Distance between two plates: | (m) |
| Angular velocity: | () |
| Magnetic field: | |
| Ratio of time relaxation to time retardation: | |
| Shear stress: | () |
| Electrical conductivity: | (Siemens per meter (S/m)) |
| Time: | () |
| Velocity: | () |
| x-component: | () |
| y-component: | () |
| z-component: | () |
| Dynamic viscosity: | () |
| Kinematic viscosity: | () |
| Volume: | () |
| Pressure: | P (). |
Data Availability
The data used to support the findings of this study are available in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.