Thin Film Flow of Micropolar Fluid in a Permeable Medium

The thin film flow of micropolar fluid in a porous medium under the influence of thermophoresis with the heat effect past a stretching plate is analyzed. Micropolar fluid is assumed as a base fluid and the plate is considered to move with a linear velocity and subject to the variation of the reference temperature and concentration. The latitude of flow is limited to being two-dimensional and is steadily affected by sensitive fluid film size with the effect of thermal radiation. The basic equations of fluid flow are changed through the similarity variables into a set of nonlinear coupled differential equations with physical conditions. The suitable transformations for the energy equation is used and the non-dimensional form of the temperature field are different from the published work. The problem is solved by using Homotopy Analysis Method (HAM). The effects of radiation parameter R, vortex-viscosity parameter ∆, permeability parameter Mr, microrotation parameter Gr, Soret number Sr, thermophoretic parameter τ, inertia parameter Nr, Schmidt number Sc, and Prandtl number Pr are shown graphically and discussed.


Introduction
Fluids, generally, have a major role in many problems related to industrial and engineering applications like crystal growing, glass blowing, polymer extrusion processes, metallurgical processes, and so on.In the extrusion process, the heated liquid stretching into a cooling system, as well as the phenomenon in which the tiny sized particles are transferred from a hot surface to a cool surface, is called thermophoresis.In gasses, tiny particles like dust exert force parallel to the temperature gradient called thermophoretic force, and the motion gained by these particles is known as thermophoretic velocity.In thermophoresis, tiny particles are transferred towards cold surfaces, whereas hot surface particles also resist taking place and, as a result, a particle free layer is observed around the hot surface, as analyzed by Goldsmith and May [1].The most important application of this phenomenon is to remove tiny particles from the path of gas particles used in turbine blades.The same phenomenon was used by Goren [2] in the study of aerosol particles, and this idea was extended by Jayaraj et al. [3] in the natural convection.The idea of mass transfer in this phenomenon was investigated by Selim et al. [4].They analyzed the effects of physical parameters involved in the model.key parameter Pr and momentum boundary layer vanish and, therefore, the energy equation becomes meaningless.Therefore, we have tried to avoid this situation by using a transformation that is the same as in the works of [27,29,32] for the same problem as cited in the literature [17][18][19][20] with the addition of concentration.In recent research, most researchers used homotopy analysis method (HAM) to solve higher order nonlinear problems, and credit goes to Liao [33][34][35], who investigated such a wonderful technique to solve nonlinear higher order differential equations.Gul et al. [36,37] used the HAM method for the suitable range of parameters.Analytical solutions in series form are calculated using HAM.The effects of all parameters on velocity, microrotation, temperature, and concentration fields are shown graphically.

Mathematical Formulation
Consider the thin film micropolar fluid flow on a stretched plate, which is being stretched with a linear velocity w U ax = .Here, 0 a > is a constant and shows the stretching rate and x displays the direction of the flow.The thickness δ of the thin film is chosen uniform and the medium is considered porous, as displayed in Figure 1.The stretching plate is kept at temperature w T and concentration w C .The temperature  The basic flow equations of our proposed model are as follows: The basic flow equations of our proposed model are as follows: ρc p uT x + vT y = kT yy − (q r ) y (4) The modeled boundary conditions for the two-dimensional liquid film are as follows: The Rosseland approximation is defined as follows: where q r is radiative heat flux, σ * is Stefan-Boltzman constant, and k * represents the mean absorption coefficient.The flux is assumed to be small, such that T 5 1 and higher terms are ignored, as in the existing literature.After expanding by Taylor's series, T 4 is reduced to the following form: T 1 is used as the temperature at the free surface.Using Equations ( 8) and ( 9), Equation ( 4) is reduced as follows: Abo-Eldahab and Ghonaim [17], Rashidi et al. [18,19] and Heydari et al. [20] introduced the following transformations: In the recent research of Khan [27] and Qasim et al. [29], the thin film flows are modeled using reference temperature and concentration for steady and unsteady problems, respectively.
where T 0 is temperature at the stretched surface and T re f is used as a constant reference temperature, such that 0 ≤ T re f ≤ T 0 .Similarly, C 0 is the concentration at the stretched surface and C re f is used as a constant reference concentration, such that 0 ≤ C re f ≤ C 0 .Substituting Equations (11) and (12) into Equations ( 1)-( 7), the basic governing equations of velocity, velocity rotation, and temperature with boundary conditions yield the following forms: where f is a dimensionless velocity function and g is a dimensionless microrotation angular velocity function, θ is the temperature function, φ is the concentration function, β is the non-dimensional thickness of the liquid film, ∆ = k 1 υ is the vortex-viscosity parameter, Mr = Ka 2φυ is the permeability parameter, Nr = 2φC r U w a is the inertia coefficient parameter, Gr = G is the thermophoretic parameter and is same as in the works of [17][18][19][20].
Coatings 2019, 9, 98 5 of 17 The important physical quantities are skin friction coefficient C f , local Nusselt number Nu, and Sherwood number, which are defined as follows: where µ u y y=0 , −k T y y=0 , and −D m C y y=0 are shear stress, heat, and mass fluxes at the surface, respectively.Using the variables in (11), the expressions for dimensionless skin friction, Nusselt number, and Sherwood number are obtained as follows: Here, Re = U w x υ represents the Reynold number based on the stretching velocity.The calculated values for the skin friction coefficient and local Nusselt number are shown in Tables 1-3.

Homotopy Analysis Method
The solutions of Equations ( 13)-( 16) with the related boundary conditions ( 17) and ( 18) are achieved using HAM.Consider that initial guesses on f (η), g(η), θ(η), and φ(η) satisfying the boundary conditions at η = 0 are as follows: The linear operators for the given functions are the following: satisfying the following properties: where a i (i = 1 − 10) are constants related to the general solution.
(a) Zeroth-Order Deformation Problem The main idea of HAM is explained in Equations ( 19)- (22).We formulate the zeroth-order problem from Equations ( 13)-( 16) as follows: Expanding the functions f , g, θ and φ by Taylor's series when q = 0, we have the following: Coatings 2019, 9, 98 7 of 17 where The supporting constraints h f , h g , h θ , and h φ are taken such that series (33) converges at q = 1.Substituting q = 1 in (33) we get the following: The following equations are satisfied by the problem of the w th order. where

Numerical Solution
The numerical (ND solve) solution of Equations ( 13)-( 16) with boundary conditions ( 17) and ( 18) for different values of embedded parameters are calculated and compared with HAM in Tables 4-7.

Graphical Results and Discussion
The thin film motion of a micropolar fluid through porous media with the impact of energy radiation and thermophoresis through a stretching plate is investigated.The non-linear coupled differential Equations ( 13)-( 16) with physical conditions (17) and (18) were determined through HAM.The effects of all the embedded constants on the dimensionless velocity field, dimensionless microrotation, dimensionless temperature field, and concentration fields-f (η), g(η), θ(η), and φ(η), respectively-are observed.The physical geometry of the modeled problem is demonstrated by Figure 1.Liao [33][34][35] presented h curves to measure the convergence of the series solution for accurate Coatings 2019, 9, 98 9 of 17 results of the system, so suitable h-curves are drawn for the velocity profile f (η), microrotation profile g(η), temperature profile θ(η), and concentration profile φ(η) in range of −2.0 ≤ h f ≤ 0.1, −2 ≤ h g ≤ 0, −2.1 ≤ h θ ≤ 0.1, and −2 ≤ h φ ≤ 0, respectively, in Figures 2-5.The influence of permeability parameter Mr on the velocity field is described in Figure 6.The permeability parameter should be increased at a very small level because of the small thickness of the liquid film because higher values of Mr, that is , Mr → ∞ correspond to the case in which there is no porous medium.The increasing values of Mr respond to the large opening of the porous space, which reduces retardation of the flow; so for increasing values of Mr, the velocity increases in this region.The larger values of the inertia coefficient parameter Nr increase the velocity of fluid as a result of its direct relation with fluid motion, deliberated in Figure 7.The influence of ∆ versus motion of liquid film is represented in Figure 8.As ∆ has an inverse relation with viscosity, the viscosity falls for larger values of ∆, while the velocity of the liquid film is raised.Figures 9 and 10 indicate the relationship between β with the fluid velocity profile f (η) and microrotation profile g(η).The fluid motion reduces with the increase in the liquid film thickness.The reason is clear, because larger values of β dominate the viscous forces and, as a result, the fluid velocity decreases.In other words, the thickness of the liquid film shows resistance to liquid flow, and fluid velocity causes retardation towards the free surface-this effect is very clear in the rotation velocity field g(η).The microrotation profile g(η) of the liquid film rises with the increasing microrotation Gr, as displayed in Figure 11, because the microrotation parameter has an inverse relation with the viscosity parameter.As a result, the viscosity reduces with the rising values of Gr; therefore, larger values of Gr offer low resistance to the flow and the velocity of fluid increases.Figure 12 demonstrates the variation of the inertia parameter Nr on the non-dimensional microrotation profile g(η).It is observed that the rise in the inertia parameter Nr material parameter reduces the microrotation profile.The inclusion of thermal radiation in the equation of energy is always used as a special case and, in most of the problems in the existing literature, the energy equation is used without radiation.If the thermal radiation parameter R becomes zero, the temperature field θ(η) in Abo-Eldahab and Ghonaim [17], Rashidi et al. [18,19], and Heydari et al. [20] becomes meaningless, so it is not clear when the thermal radiation parameter R becomes zero in these papers.Therefore, our case of thermal radiation is reciprocal to the above published work, and is the same as Khan [27], Qasim et al. [29], and Mahmood and Khan [32].Therefore, the temperature rises with the larger values of thermal radiation parameter, as shown in Figure 13, because the thickness of the boundary layer (thin film) is directly related to thermal radiation.Physically, the rate of energy transport increases and, as a result, the temperature of the fluid rises.The dimensionless fluid thickness β has a vital role in temperature distribution.θ(η) decreases with increasing values of β, which is obvious from Figure 14.The size of thin film absorbing heat, and thus the temperature of the fluid, decreases and, as a result, a cooling effect is produced.In other words, the thickness of the fluid decreases with the increasing temperature.Figure 15 represents the comparison of temperature and Prandtl number Pr.The temperature falls with growing values of Pr.In fact, the larger values of Pr enhance the viscous diffusion more than the thermal diffusion and, as a result, the temperature profile declines.Schmidt number verses concentration is deliberated in Figure 16.The rising values of Schmidt number Sc decrease the concentration field, because molecular diffusivity is inversely related to Sc.The contribution of the Soret number Sr is represented in Figure 17, showing that φ(η) rises when the Soret number Sr increases.In fact, the larger Soret number increases the viscosity and, therefore, φ(η) accelerates.Figure 18 shows the relationship between thermophoretic parameter τ and φ(η).They are inversely related to each other.Rising values of τ reduce the size of the boundary layer.The concentration field rises as thickness β increases, as shown in Figure 19, because of cohesive forces between molecules dominated by the increasing value of the parameter β, which result a rise in friction force and cause the fluid flow.

Conclusions
The study of the thin film flow in a permeable medium past a stretched plate was examined.The micropolar fluid was used as a base fluid with the influence of thermal radiation and thermophoresis.Modeled non-linear coupled differential equations were tackled through HAM.The HAM solution was compared with the numerical method and close agreement was observed for the validation of the problem.The effects of the physical parameters on the velocity, temperature, and concentration profiles were displayed and discussed.
The outcomes of the problem are pointed out as follows: • The increasing values of the thin film thickness parameter β improve the resistance force to decline the velocity and microrotation profiles, and enhance the concentration field.• It was observed that the rise in the Soret number Sr enhances the concentration field ( ) η φ .

Conclusions
The study of the thin film flow in a permeable medium past a stretched plate was examined.The micropolar fluid was used as a base fluid with the influence of thermal radiation and thermophoresis.Modeled non-linear coupled differential equations were tackled through HAM.The HAM solution was compared with the numerical method and close agreement was observed for the validation of the problem.The effects of the physical parameters on the velocity, temperature, and concentration profiles were displayed and discussed.
The outcomes of the problem are pointed out as follows: • The increasing values of the thin film thickness parameter β improve the resistance force to decline the velocity and microrotation profiles, and enhance the concentration field.• It was observed that the rise in the Soret number Sr enhances the concentration field ( ) η φ .

Conclusions
The study of the thin film flow in a permeable medium past a stretched plate was examined.The micropolar fluid was used as a base fluid with the influence of thermal radiation and thermophoresis.Modeled non-linear coupled differential equations were tackled through HAM.The HAM solution was compared with the numerical method and close agreement was observed for the validation of the problem.The effects of the physical parameters on the velocity, temperature, and concentration profiles were displayed and discussed.
The outcomes of the problem are pointed out as follows: • The increasing values of the thin film thickness parameter β improve the resistance force to decline the velocity and microrotation profiles, and enhance the concentration field.

•
It was observed that the rise in the Soret number Sr enhances the concentration field φ(η).

•
The temperature field rises with the increasing value of the thermal radiation parameter R because of the rate of energy and transport growth, and consequently enhances the temperature profile.

•
The increase in the thickness of the thin film β reduces the temperature profile.Physically, heat transfer is larger in the thin film as compared with the thick film, while the concentration field increases as the thin film parameter β increases.

•
The larger vortex-viscosity parameter ∆ causes the velocity of the liquid film to rise.

•
The HAM solution was validated with the numerical solution (ND-solve) and very close agreement was observed.
the surface are assumed to vary with distance x from the plate.0 T and 0 C are the temperature and concentration at the plate, while ref T and ref C are the constant reference temperature and concentration.Further, it is assumed that the liquid film is gripping and releasing radiation.The radiate heat flux is considered along the x-axis, while neglecting along the y-axis.

Figure 1 .
Figure 1.Physical geometry of the problem.

Figure 1 .
Figure 1.Physical geometry of the problem.

1 k
1 a υ represents the microrotation parameter, Pr = ρυc p k represents the Prandtl number, R = 4σ * T 3 * k represents the radiation parameter, Sc = υ D m represents the Schmidt number, Sr = D m k T (T w −T 0 ) υT m (C w −C 0 ) represents the Soret number, and τ = kU 2 w 2υa 18 shows the relationship between thermophoretic parameter τ and ( ) η φ .They are inversely related to each other.Rising values of τ reduce the size of the boundary layer.The concentration field rises as thickness β increases, as shown in Figure 19, because of cohesive forces between molecules dominated by the increasing value of the parameter β , which result a rise in friction force and cause the fluid flow.

Figure 2 .
Figure 2. f h curves for the velocity field.

Figure 3 .
Figure 3. g h curves for the velocity field in rotation.

Figure 4 .
Figure 4. θ h curves for the temperature field.

Figure 2 .
Figure 2. h f curves for the velocity field.

.
They are inversely related to each other.Rising values of τ reduce the size of the boundary layer.The concentration field rises as thickness β increases, as shown in Figure 19, because of cohesive forces between molecules dominated by the increasing value of the parameter β , which result a rise in friction force and cause the fluid flow.

Figure 2 .
Figure 2. f h curves for the velocity field.

Figure 3 .
Figure 3. g h curves for the velocity field in rotation.

Figure 4 .
Figure 4. θ h curves for the temperature field.

Figure 3 .
Figure 3. h g curves for the velocity field in rotation.

.
They are inversely related to each other.Rising values of τ reduce the size of the boundary layer.The concentration field rises as thickness β increases, as shown in Figure 19, because of cohesive forces between molecules dominated by the increasing value of the parameter β , which result a rise in friction force and cause the fluid flow.

Figure 2 .
Figure 2. f h curves for the velocity field.

Figure 3 .
Figure 3. g h curves for the velocity field in rotation.

Figure 4 .
Figure 4. θ h curves for the temperature field.

Figure 5 .
Figure 5. h φ curves for the concentration field.

Figure 5 .
Figure 5. h φ curves for the concentration field.

Figure 6 .
Figure 6.Effect of permeability parameter Mr on the velocity.

Figure 7 .
Figure 7.The comparison of dimensionless velocity with inertia coefficient parameter Nr .

Figure 6 .
Figure 6.Effect of permeability parameter Mr on the velocity.

Figure 5 .
Figure 5. h φ curves for the concentration field.

Figure 6 .
Figure 6.Effect of permeability parameter Mr on the velocity.

Figure 7 .
Figure 7.The comparison of dimensionless velocity with inertia coefficient parameter Nr .

Figure 7 .
Figure 7.The comparison of dimensionless velocity with inertia coefficient parameter Nr.

Figure 5 .
Figure 5. h φ curves for the concentration field.

Figure 6 .
Figure 6.Effect of permeability parameter Mr on the velocity.

Figure 7 .
Figure 7.The comparison of dimensionless velocity with inertia coefficient parameter Nr .

Figure 11 .
Figure 11.Microrotation profile under the effect of microrotation parameter Gr .

Figure 12 .
Figure 12.Variation of dimensionless microrotation profile with inertial parameter Nr.

Figure 11 .
Figure 11.Microrotation profile under the effect of microrotation parameter Gr .

Figure 12 .
Figure 12.Variation of dimensionless microrotation profile with inertial parameter Nr.

Figure 11 .
Figure 11.Microrotation profile under the effect of microrotation parameter Gr.

Figure 11 .
Figure 11.Microrotation profile under the effect of microrotation parameter Gr .

Figure 12 .
Figure 12.Variation of dimensionless microrotation profile with inertial parameter Nr.Figure 12. Variation of dimensionless microrotation profile with inertial parameter Nr.

Figure 16 .
Figure 16.Variation of dimensionless concentration with Schmidt number Sc .

Figure 16 .
Figure 16.Variation of dimensionless concentration with Schmidt number Sc .

Figure 16 .
Figure 16.Variation of dimensionless concentration with Schmidt number Sc .

Figure 17 .
Figure 17.Variation of dimensionless concentration with Soret number Sr.

Figure 17 .
Figure 17.Variation of dimensionless concentration with Soret number Sr .

Figure 17 .
Figure 17.Variation of dimensionless concentration with Soret number Sr .

Table 1 .
Values for the skin friction coefficient, when h

Table 2 .
Values of rate of heat transfer or the local Nusselt number, when h

Table 3 .
Values of the Sherwood number, when h
The authors would like to thank Deanship of Scientific Research, Majmaah University for supporting this work under the No. 1440-25.Conflicts of Interest:The authors declare no conflict of interest. Funding: