On Behavioral Response of 3D Squeezing Flow of Nanofluids in a Rotating Channel

In this article, a behavioral study of three-dimensional (3D) squeezing flow of nanofluids with magnetic effect in a rotating channel has been performed. Using Navier–Stokes equations along with suitable similarity transformations, a nonlinear coupled ordinary differential system has been derived which models the 3D squeezing flow of nanofluids with lower permeable stretching porous wall where the channel is also rotating. The base fluid in the channel is considered to be water that contains different nanoparticles including silicon, copper, silver, gold, and platinum. The homotopy perturbation method (HPM) is employed for the solution of highly nonlinear coupled system. For validation purpose, system of equations is also solved through the Runge–Kutta–Fehlberg (RK45) scheme and results are compared with homotopy solutions, and excellent agreement has been found between analytical and numerical results. Also, validation has been performed by finding average residual error of the coupled system. Furthermore, the effects of various parameters such as nanoparticle volume fraction, suction parameter, characteristic parameter of the flow, magnetic parameter, rotation parameter, and different types of nanoparticles are studied graphically.


Introduction
Fluids exhibiting good thermal conductivity properties are a major requirement of many industrial applications involving heat transfer equipment. Typical examples are those of vehicular cooling systems, refrigerants, building services, and conventional industrial processing systems, such as petro-chemical, textile, paper, and food processing plants, to name a few [1][2][3]. To reduce costs and be energy efficient, the research and industrial community has continuously worked to develop and utilize fluids bearing high thermal conductivity. e concept of nanofluids was introduced in [4] which utilized suspended metallic nanoparticles (with a typical size of 100Å) in conventional heat transfer fluids such as water or engine oil. Since metallic solids reflect heat conductivity of orders greater than conventional heat transfer fluids, this engineered form of fluids was received quite well by the industrial and research community. A comparison can be made on the basis of silver, 429 W/(m K); water, 0.613 W/(m K); and engine oil, 0.145 W/(m K) at 300 k; and liquid sodium, 89.44 W/(m K) at 371 k. Moreover, since the surface area to volume ratio is inversely proportional to the radius of a particle, suspended metals of nanometer scales would give better heat conductivity as compared to micrometer-sized particles.
Since the convective heat transfer coefficient is derived from continuity, momentum, and energy equations, its behaviour is significantly dependent on the velocity and temperature profile of the nanofluid's base fluid [1,5]. Moreover, the heat and momentum transfer mechanisms are essentially the same and hence can be comparable on the basis of analogies. For instance, Reynold's analogy is applicable towards turbulent flows, Prandtl's analogy for laminar flows, von Korman's analogy for both laminar and turbulent flows with a buffer layer in between, and Chilton-Colburn analogy for transport independent flows [6]. e velocity is also dependent on the concentration of nanoparticles in a nanofluid such that a higher concentration may result in a flattening of the velocity profile [7]. Studies have also shown that the velocity profile is not uniform. For instance, near-wall velocity profile measurements have shown increased velocity gradient, with no slip, relative to the equivalent base fluid [8]. As such, the investigation of the velocity profile under different boundary conditions and flow models is of particular interest to the research community.
With this, a considerable volume of literature started to address the behaviour of nanofluids in context of different boundary conditions (convective [9], porous medium [10,11]), characterization parameters of different fluids and suspended metals (Cu/Cu-TiO 2 [12,13], Cu-H 2 O/Cu-Kerosene [14]), and thermodynamic [15,16] and magnetohydrodynamic [17][18][19] properties using analytical and/or numerical approaches. A comprehensive review of these aspects in general is presented in [1,3,20]. e study of squeezing flows that is normal to two parallel plates is an important problem in the area of fluid dynamics, having applications in hydraulic machinery, electric motors, food industry, bioengineering, and automobile engines, amongst others. e mechanics of these studies in context of turbulent, laminar, and transitional flows, and different non-Newtonian fluid models (e.g., power law [21], grade 2/3 [22], and Casson [23,24]) have proved to be a significant challenge to the research community due to the involved nonlinearities. e squeezing flow response in context of parallel plates appears in [13,25,26], parallel moving plates are given in [27][28][29], orthogonal moving plates are discussed in [10], moving surface in [30], stretching surface in [31][32][33], and stretching cylinders are given in [34]. Likewise, the response to Casson flow is discussed in [16,19,32,34], Hiemenz flow in [35], and bioconvection flows in [19,36]. e studies have also been extended to nanofluids [1,17,25,36,37]. In [38], Naz et al. solved analytically the problem of variable thermophysical features of the threedimensional flow of a non-Newtonian yield manifesting liquid with heat and mass transport in the presence of gyrotactic microorganisms over a nonlinear stretched surface.
ey utilized the boundary-layer theory to develop the governing partial differential equations. ey concluded that the mounting values of the fluid parameter and magnetic parameter retard the fluid flow. Moreover [39], Ahmed et al. investigated analytically the problem of Jeffery-Hamel flow for second-grade fluid between two nonparallel walls having a source or a sink at the cusp. Soret and Dufour effects are incorporated in the energy and concentration equations. For solution purposes, the authors used the homotopy analysis method (HAM). Variations in temperature and concentration profiles for varying in grained physical parameters in the flow model are discussed graphically. Also, Nusselt number and the skin friction coefficient along with Sherwood number are extracted numerically and analytically. In [40], Khan et al. investigated viscous incompressible fluid between two nonparallel plane walls, known as Jeffery-Hamel flow, under the influence of thermal radiation. ey used the similarity technique to solve for the governing equations. Moreover, the same problem is solved numerically and a comparison between the two methods is conducted that yields a great result. ey presented the rates of heat and mass transfer. Furthermore, the effects of the investigated parameters on the flow and heat transfer are discussed and analyzed. Additionally [41], Khan et al. studied analytically by employing similarity technique along with Runge-Kutta and homotopy analysis algorithms the unsteady magnetohydrodynamics flow with heat generation/absorption of H 2 O saturated by tiny nanosized particles with various shapes over a thin slit. ey found out that as the magnetic field increases, the nanofluid temperature increases and the motion decreases. In addition [42], Khan et al. investigated the flow of nanofluid over a curved Riga surface; they studied the impact of the freezing temperature and the diameter of the nanoparticles on the flow field and the heat transfer. ey found out that the nanofluid velocity dropped by increasing the flow parameters c and S, and an abrupt decrement occurred at the surface of the Riga sheet. Also [43], Naz et al. investigated the entropy analysis of 3D flow of Maxwell nanofluid containing gyrotactic microorganism in the presence of homogeneous-heterogeneous reactions with improved heat conduction and mass diffusion models over a stretched surface. ey found out that entropy generation increases for higher values of radiation parameter and Brinkman number, whereas the Bejan number is reduced for the higher values of radiation and magnetic parameters.
From a numerical perspective, the various behaviours are explained on the basis of solutions over discrete points distributed throughout the problem domain. Solutions for remaining points are obtained by means of interpolation. For reasons of performance and accuracy, these points appear amidst regular geometries. In context of simpler domains, reduced number of dimensions is sufficient for such studies. However, by considering richer intricacies of involved physics, the simplified problem domain is compromised, giving failed predictions. In such situations, studies on the basis of 3D domain models, even for simpler geometry configurations, are imperative. e common approach in terms of nanofluids is the fourth-order Runge-Kutta family of integration methods [44,45]. e same is also true from an analytical perspective that takes into account simplified assumptions. e usual approach for boundary value problems is the usage of perturbation 2 Complexity techniques. However, due to assumptions of small or large parameters, this is not sufficient. In this regard, a seminal work that combined these perturbation techniques with homotopy was proposed as the homotopy perturbation method (HPM) in [46][47][48][49]. Since its introduction, the method has been applied to different nonlinear equations [50][51][52][53][54][55][56]. Specifically, in the case of nanofluids, the method has been applied in [57][58][59][60]. Other approximation techniques that have been used for the case of fluid dynamics include the homotopy analysis method (HAM) [61] and optimal homotopy asymptotic method (OHAM) [62].
In this article, we provide a comprehensive description of the three-dimensional squeezing flow of nanofluids, considering a geometry involving a rotating channel. e flow is characterised on the basis of water as base fluid and different suspended nanoparticles. e study is performed on the basis of Navier-Stokes equations using similarity transforms. Both analytical and numerical solutions are obtained using the homotopy perturbation method and Runge-Kutta-Fehlberg scheme. Furthermore, validation of results has been performed by finding the average residual error of the coupled system. After validation, we present characterization of different configurations of nanofluids using parameters such as volume fraction, suction, flow, magnetism, and rotation.
In the remaining part of the paper, Section 2 includes mathematical formulation of the problem. Section 3 presents the basic theory of numerical approach. Section 4 comprises the results and discussion. Finally, conclusion is presented in Section 5.

Mathematical Formulation
We consider a 3D rotating incompressible and electrically conducting viscous nanofluid flow between two infinite horizontal plates. e lower plate is positioned at y � 0 and is stretched with a velocity U 0 (t) � ax/(1 − ct) in x, where a is the stretching rate of the lower plate, and c is a characteristic constant. e upper plate is at a variable distance

fluid is squeezed with a velocity
v h � dh/dt in negative y-axis. e angular velocity Ω between the fluid and channel around e lower plate intakes the flow with a velocity − V0/(1 − ct). A magnetic field with density B 0 / ���� � 1 − c is applied along the y-axis. e system is rotating along the y-axis (see Figure 1). ese are then introduced to obtain similarity solutions by reducing the governing equations into a system of ordinary differential equations. e governing relation for continuity and momentum of nanofluid flow in the rotating frame of reference is given as follows: where T is the Cauchy stress tensor, J is the magnetic flux, and B is the current density. e abovementioned governing equations can also be described by the following set of Navier-Stokes equations: where ρ nf is the nanofluid density, v nf � μ nf /ρ nf is the nanofluid kinematic viscosity, σ is the electrical conductivity, B 0 is the magnetic field, and c is a characteristic parameter representing inverse time, and ct < 1. e constants for the model are given as where μ f is the fluid fraction viscosity, ϕ is the nanoparticle volume fraction, and ρ f and ρ s are the densities of the fluid and of solid fractions. e thermophysical properties of different materials are given in Table 1 for reference. e boundary conditions at y � 0 are given as Figure 1: Geometry for the problem. A nanofluid is squeezed between two infinite horizontal parallel plates. e lower plate is at position y � 0 and stretched with a velocity U 0 (t), while the upper plate is at a variable height of h(t). By movement of the upper plate in negative y-axis, the nanofluid is squeezed at a velocity of v h (t). Ω represents the angular velocity, while B 0 is the magnetic field density applied along y.
While for y � h(t), are given as where a is the stretching rate of the lower plate. e governing equations subject to similarity transformation in order to give ordinary differential equations in terms of a stream function ψ are given as Substitution of these similarity transforms to the governing Navier-Stokes equations, we have where β � c/a is the characteristic parameter of the flow, Ω � w/a is the rotation parameter, M 2 � σB 2 0 /ρ f a is the magnetic parameter, and prime denotes differentiation with respect to η. To squeeze the flow, we take β > 0 for which the upper plates move downward with a velocity of V h < 0, whereas for β < 0, the upper plate moves upwards with respect to the plane y � 0. β � 0 corresponds to a steady state. To reduce the number of independent variables as well as retain the similarity solution, the above are simplified by cross-differentiation, giving us Moreover, the transformed boundary conditions take the form: where w 0 � V 0 /ah is the suction parameter. For this problem, the physical quantity of interest is the skin friction coefficient C f along the wall at the lower and upper walls, defined as 4 Complexity Substituting stream function to the abovementioned, we obtain where Re x � ρ f U 0 h/μ f is the local Reynolds number.

Basic Theory of Homotopy Perturbation Method
e basic theory of HPM can be exhibited using the following differential equation: where Υ is the boundary of the domain Ω, and f is unknown and g(r) is a known function. L, N, B are linear, nonlinear, and boundary operators, respectively. We construct a homotopy θ(r, p): where q ∈ [0, 1] is an embedding parameter, and f 0 is the initial guess of (13) that satisfies the boundary conditions. From (14), we have us, as q varies from 0 to 1, the solution θ(r, q) approaches from f 0 (r) to f(r). To obtain an approximate solution, we expand θ(r, q) in a Taylor series about q as follows: Setting q � 1, the approximate solution of (13) would be Substituting equation (17) in equation (13) will give If R is approaching zero, U will then approach towards the exact solution.

Results and Discussion
In this article, an unsteady three-dimensional squeezing flow of electrically conducting nanofluid between two infinite horizontal planes in a rotating channel is considered. e composition of the nanofluid is made on the basis of different nanoparticles, including Silicon (Si), Copper (Cu), Silver (Ag), Gold (Au), and Platinum (Pt). ese nanofluids with silicon composition are used for observing the effect of various parameters on the velocity profile. ese parameters include the nanoparticle volume fraction ϕ, suction parameter w 0 , characteristic parameter of the flow β, rotation parameter Ω, and magnetic parameter M. e formulated boundary value system described in Section 2 for these parameters is solved using HPM, which is then compared with numerical solutions obtained using the Runge-Kutta-Fehlberg method for validation purposes. A graphical representation for this validation is given in Figure 2, showing good agreement.
A detailed analysis of this validation is performed through residual errors, for ϕ � 0.9, w 0 � 0.5, Ω � 1, β � 1, and M � 1 in Tables 2 and 3 for both the HPM and RK45 methods individually. Validation with a variation of β � 2 is given in Tables 4 and 5. In both cases, the results from HPMbased solution are consistent and in good agreement with the numerical results. A comparative analysis on the basis of β � (1, 2) using both HPM and RK45 is given in Table 6. e effect of the skin friction coefficient at both lower and upper walls is also given in Table 7.
After validation of the solutions, the behaviour response of nanofluids against various parameters is investigated. In all these investigations, 0 < η < 0.5 corresponds to the lower half, while 0.5 < η < 1 represents the upper half of the channel. e effect of nanoparticle volume fraction ϕ on normal, axial, and transverse velocity components is given in Figure 3. Here, the static parameters are w 0 � 0.5, β � 1, Ω � 1, and M � 0.5, while the variational parameter ϕε[0, 1]. e general observation is that ϕ increases with respect to normal velocity. e axial velocity profile increases with an increase in the nanoparticle volume fraction in the lower half of the channel, while it decreases in the upper half. e transverse velocity decreases in the vicinity of lower surface, (0 < η < 0.25), when the nanoparticle volume fraction increases. But other than that, an opposite trend has been observed in the rest of the channel (0.25 < η < 1). ese behaviours are justified because viscosity increases by increasing the nanomaterial volume fraction; as a result, the enhanced frictional force leads to the flow resistance. e response of suction parameter (w 0 ) on the velocity profile is given in Figure 4, where it can be observed that the observable range of w 0 is at a maximum in the lower channel. e overall trend is similar to an exponential increase in this region. However, w 0 gets confined rapidly around the midpoint as η and w 0 approach 1. In the case of the axial velocity, it stands at maximum close to the lower wall at η � 0 and at the midpoint close to the upper wall at η � 1. In other cases, it decreases significantly as w 0 Complexity 5 increases. is results in a reverse flow due to discharge of a large amount of fluid particles in the vicinity of the lower wall, while on the upper wall, it becomes much more noticeable. In the case of the traverse component of velocity profile, there is almost no change as the maximum range of the velocity profile stands at 0.002. In this case, the velocity is nonexistent at both walls. In all these cases, the static parameters are ϕ � 0.9, β � 1, Ω � 1, and M � 0.5, while the variable parameter w 0 ε[0, 1]. e effect of magnetic parameter M on the velocity profile is shown in Figure 5. Here, the static parameters are ϕ � 0.2, w 0 � 0.5, β � 1, and Ω � 1, while the variable parameters are Mε[0, 0.6]. In these cases, the normal velocity presents a tailed parabolic behaviour, as M is increased, where in the region in vicinity of lower plate shows an increased flow, but a decreased flow a little before the channel midpoint. e axial velocity component also has e reason behind these behaviours is that application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force. is force has the tendency to slow down the motion of the fluid. e effect of characteristic parameter β on the velocity profile, while the static parameters are ϕ � 0.9, w 0 � 0.5, Ω � 1, and M � 0.5 and while β varies in the interval [− 2.5, 2.5], is presented in Figures 6 and 7. Positive values of β representing movement of the upper plate towards the lower plate, all three velocity components, demonstrate an   6 Complexity increase (see Figure 6). On the contrary, the inverse movement is represented by the negative β values; all velocity components record a decrease as β is decreased (see Figure 7). Effect of rotational parameter Ω on the velocity profile is depicted in Figure 8. For Ω, the normal velocity decreases as Ω is increased. e axial velocity shows a dual behaviour. In the lower part of the channel, there is a small decrease in the velocity as Ω is increased, whereas in the upper part of the channel, there is a small increase as Ω is increased. An inverse dual behaviour to the axial component is reflected in the transverse component. Here, there is small increase in velocity as Ω is increased in the lower quarter of the channel, whereas in the remaining three quarters, there is a major reversal of velocity as Ω is increased. Here, the static parameters are ϕ � 0.2, w 0 � 0.5, β � 1, and M � 0.5, while the variational parameter is Ω � [3,12]. e transverse velocity component shows a similar behaviour in the case of increased nanoparticle volume fraction ϕ and the rotation parameter Ω. Magnitudes of these velocities are found to decease within the rotating channel with augmentation. One of the forces encountered in fluid flow with a rotating  Table 3: RK45 solution along with residual errors when ϕ � 0.9, w 0 � 0.5, Ω � 1, β � 1, and M � 1. channel is the Coriolis force which acts in a direction perpendicular to the rotational axis and the velocity of the body in the rotating frame. is is due to the fact that the rotation of the channel leads to resist the flow. e behaviour of velocity profiles against different nanofluid composition is given in Figure 9. In general, the nanofluids having greater density of nanoparticles per unit area are affecting the velocity profiles significantly, where larger densities showed a decrease in normal velocity as compared to smaller densities. For a similar volume fraction of ϕ � 0.2, Si-based nanoparticles with ρ � 2330 kg/m 3 , i.e. the minimum most in the chosen group, reflected maximum normal velocity. In contrast, Pt having ρ � 21450 kg/m 3 as the highest density in the chosen group showed the minimum normal velocity. In the case of axial component, there is a dual behaviour, with high densities having a lower velocity in lower channel, and an increased velocity in the upper channel. However, the variation in velocity with Table 6: Similarity between HPM and RK45 solutions when ϕ � 0.9, w 0 � 0.5, Ω � 1, and M � 1.       14 Complexity