A novel technique for solving unsteady three-dimensional brownian motion of a thin film nanofluid flow over a rotating surface

The motion of the fluid due to the swirling of a disk/sheet has many applications in engineering and industry. Investigating these types of problems is very difficult due to the non-linearity of the governing equations, especially when the governing equations are to be solved analytically. Time is also considered a challenge in problems, and times dependent problems are rare. This study aims to investigate the problem related to a transient rotating angled plate through two analytical techniques for the three-dimensional thin film nanomaterials flow. The geometry of research is a swirling sheet with a three-dimensional unsteady nanomaterial thin-film moment. The problem's governing equations of the conservation of mass, momentum, energy, and concentration are partial differential equations (PDEs). Solving PDEs, especially their analytical solution, is considered a serious challenge, but by using similar variables, they can be converted into ordinary differential equations (ODEs). The derived ODEs are still nonlinear, but it is possible to approximate them analytically with semi-analytical methods. This study transformed the governing PDEs into a set of nonlinear ODEs using appropriate similarity variables. The dimensionless parameters such as Prandtl number, Schmidt number, Brownian motion parameter, thermophoretic parameter, Nusselt, and Sherwood numbers are presented in ODEs, and the impact of these dimensionless parameters was considered in four cases. Every case that is considered in this problem was demonstrated with graphs. This study used modified AGM (Akbari–Ganji Method) and HAN (Hybrid analytical and numerical) methods to solve the ODEs, which are the novelty of the current study. The modified AGM is novel and has made the former AGM more complete. The second semi-analytical technique is the HAN method, and because it has been solved numerically in previous articles, this method has also been used. The new results were obtained using the modified AGM and HAN solutions. The validity of these two analytical solutions was proved when compared with the Runge–Kutta fourth-order (RK4) numerical solutions.

In science, especially chemistry, condensate production from a cooling and saturated vapor is very substantial.Many researchers investigated this phenomenon under various circumstances.Sparrow and Gregg 1 analyzed film condensation on a rotating plate on pure saturated steam.The centrifugal field associated with the rotation moves the condensate outward along the disc's surface without requiring gravitational forces.In this problem, the governing equations were solved numerically, and finally, results were given for heat transfer and condensate layer thickness, torque, temperature, and velocity profiles.Beckett et al. 2 investigated the problem of laminar condensation on a swirling disk in a large volume of static vapor for low and high cooling rates on the disk surface.The governing equations were converted into a set of ODEs using similarity transformation and solved numerically, and solutions were compared via previously published results.Chary and Sarma 3 considered the problem of vapor-to-liquid transition in the presence of constant axial suction at a permeable condensing surface.The governing equations were reduced into a set of ODEs.The Runge-Kutta numerical method was used to calculate the heat transfer coefficient, and limiting solutions for very thin condensate films were obtained.They

Mathematical description
The geometry of the study is a swirling sheet with a three-dimensional transient nanomaterial thin-film moment, as illustrated in Fig. 1.The plate swirls with the angular velocity of , and the inclined plate has an angle of β with the horizon.The nanomaterial thickness of the sheet is indicated by h , and the speed of the sprayed fluid is denoted by W .The terminal effect is neglected because the thickness of the fluid film in comparison to the radius of the disc is not thick enough.The gravitation force exists, and it is denoted by g , and its direction is illustrated in the following figure.The temperature of the film surface is denoted by T 0 .The temperature of the inclined swirling surface is denoted by T w .The concentration of the film surface is denoted by C 0 .The concentration of the inclined swirling surface is denoted by C w .
The thickness of the fluid film is very thin, and the pressure at the surface of the surface is denoted by p 0 and it is just a function of z .The viscous dissipation function in the energy equation is negligible.The governing equations of the problem are as follows 2,3,5,6,8,37 : The equation of conservation of mass: The equation of conservation of momentum in x direction: The equation of conservation of momentum in y direction: The equation of conservation of momentum in z direction: The equation of conservation of energy: The equation of conservation of concentration: (1) (3) where D/Dt denotes the total derivative to the variable of time, ∇ is the gradient operator, u , v, and z are the velocities in the x , y, and z directions, respectively, ∇ 2 is the Laplacian operator, T is the temperature, C is the concentration, ρ f , is the density of the base fluid, µ is the dynamic viscosity, α is the thermal diffusivity, c p , is the specific heat capacity at a constant pressure of nanofluid, ρc p p / ρc p f , is the ratio of nanoparticles' heat capac- ity to the base fluid heat capacity, D B is the Brownian diffusion coefficient, and D T , is the thermophoretic diffusion coefficient.
The corresponding boundary conditions of Eqs.(1)-( 6) are as follows: The similarity transformations are considered as follows 8,11,37 : Here, ν is the kinematic viscosity, θ is the dimensionless temperature, and φ is the dimensionless concentra- tion.The similarity variables of Eq. ( 8) can be substituted in Eqs. ( 2)-( 6) for converting a system of nonlinear PDEs into a system of nonlinear dimensionless coupled ODEs: Substituting the similarity variables of Eq. ( 8) in Eq. ( 7) will be as follows: where Pr is the Prandtl number, Sc is the Schmidt number, Nb is the Brownian motion parameter, S is the parameter that depends on the angular velocity of the rotating surface, and Nt is the thermophoretic parameter, which is defined as 37,38 : The constant normalized thickness of δ is as follows 37 : The dimensionless Nusselt and Sherwood numbers are as follows 37 : Vol.:(0123456789) www.nature.com/scientificreports/Methodology.Description of the HAN method.Jalili et al. [26][27][28] developed the HAN method for approximating an analytical solution for a differential equation.In this part, the explanation of the HAN method is as follows: Consider an ODE of the m th order as follow: Equation ( 20) is a nonlinear differential equation, and Ŵ is the function of ζ and its derivatives to ξ .The parameter ζ is the function of the independent variable ξ .The derivatives of the function ζ (ξ) with respect to ξ at ξ = 0 and ξ = L are denoted as follows: The solution of Eq. ( 20) is considered as follows: Here, a 0 , a 1 , …, a n are n + 1 constant coefficients which n > m .By solving a system of n + 1 unknowns and n + 1 equations, constant coefficients will be determined.The boundary conditions of the problem can construct some of these equations as follows: The constructed equations from boundary conditions of the problem as they can be seen in Eqs.(23), (24)  are limited because we assume the value of n is higher than m earlier in this methodology.But more boundary equations are needed, and a numerical method (no matter which numerical method and no matter what kind of software package) can approximate these additional boundary conditions for making the remaining needed equations.So, the new approximated boundary conditions are as follows: For instance, the following equations are constructed from approximated boundary conditions of Eq. ( 25): Vol:.( 1234567890)  26)-( 29), it can be derived as many equations as possible are needed to create a system with n + 1 equations and n + 1 unknowns.The limitation of the HAN method is just in the numerical method that is used, and this means that if no numerical method could solve a problem, the HAN method could not be used because this method seriously needs a numerical solution.To summarize the mentioned method in a more compact form, the following Fig. 2, the flow chart is presented for the HAN method: Application of the HAN method.For applying the HAN method, let us assume the following functions are the semi-analytical solutions of Eqs. ( 9)-( 14): Based on Eq. ( 30), there are 43 unknown coefficients, and 43 equations are needed to obtain them.Equation (15) makes only 13 equations, and the remaining 30 must be made numerically.This study used the numerical solution of Zeeshan et al. 37 .Finally, according to Table 1, the system of ODEs of Eqs. ( 9)-( 14) for 4 cases can be solved by calculating the system of 46 equations and 46 unknowns and the solutions of Eqs. ( 9)-( 14) for all available cases in Table 1 are as follows: Solutions of case 1 where Pr = 6.6 , Nt = 0.2 , Nb = 0.2 , Sc = 2.0 , S = 0.0 , δ = 1.0 are are demostrated in Eqs. ( 31)- (36) as follows:    www.nature.com/scientificreports/Description of the modified Akbari-Ganji method.The Akbari-Ganji Method was developed for solving nonlinear differential equations analytically.This method has solved many problems [21][22][23][24][25]39,40 for which no exact analytical method exists. This aper introduces the modification of this method due to needing more accurate solutions.
To explain the main idea of modified AGM, the general form of the m th order differential equation is assumed as: With boundary conditions: To solve Eq. (55), we can consider the answer as the following polynomial of degree n with unknown constant coefficients: Here, a 0 , a 1 , …, a n are n + 1 constant coefficients which n > m .By solving a system of n + 1 unknowns and n + 1 equations, constant coefficients will be determined.The boundary conditions of the problem can construct some of these equations as follows: The constructed equations from boundary conditions of the problem as they can be seen in Eqs.(58), (59) are limited because we assume the value of n is higher than m earlier in this methodology.But more equations are needed to construct a system of n + 1 unknowns and n + 1 equations.So, the remaining equations can be made by substituting Eq. (57) in Eq. (55) as follows: So, it can be derived as many equations as possible from Eqs. (60)-(62) to construct a system of n + 1 unknowns and n + 1 equations.Finally, series constant coefficients and, thus, the solution to the problem will be determined by solving the equations.Unlike the HAN method, AGM does not depend on the numerical solution and is more independent, but the limitation of this method is that the more nonlinear the problem, the more difficult it is to solve with the AGM method.To summarize the mentioned method in a more compact form, the following Fig. 3, the flow chart is presented for the modified AGM: www.nature.com/scientificreports/Application of the modified Akbari-Ganji method.In this part, Eqs. ( 9)-( 14) are solved with the modified Akbari-Ganji method for cases (1, 2) according to Table 1.For applying the Modified Akbari-Ganji Method, the following functions are to be assumed the semi-analytical solutions of Eqs. ( 9)-( 14) for cases (1, 2): Based on Eq. ( 63), there are 31 unknown coefficients, and 43 equations are needed to obtain them.Equation (15) makes only 13 equations, and the remaining 18 equations must be made through Eq. (60), and the solutions for cases (1, 2) are as follows: Solutions of case 1 where Pr = 6.6 , Nt = 0.2 , Nb = 0.2 , Sc = 2.0 , S = 0.0 , δ = 1.0 are are demostrated in Eqs. ( 64  www.nature.com/scientificreports/But it can reach more accurate solutions by increasing n in assumed functions.The following functions are the semi-analytical solutions of Eqs. ( 9)-( 14) for cases (3, 4): Based on Eq. ( 76), 49 unknown coefficients and 49 equations are needed to obtain them.Since the number of constant coefficients has increased, the number of equations that we must create to make a system of n equa- tions and n unknowns increases, and in addition to Eqs. ( 60), (61) should also be used.Equation ( 15) makes only 13 equations, and the remaining 36 equations must be made through Eqs. ( 60), (61) and the solutions for cases (3, 4)

Results and discussion
Heat and mass transfer in an unsteady rotating inclined plane has been investigated using 3D thin film nanomaterials flow.The solutions were obtained using the modified AGM and HAN methods.These two analytical solutions' validity was proved when compared with the Zeeshan et al. 37 Runge-Kutta fourth-order (RK4) numerical solutions.Figures 4, 5, 6, 7, 8, 9, 10, 11 and 12 show the accuracy of the modified AGM and HAN results.The following table relates to the four cases considered in this study.
As the impact of four cases is shown in Figs. 4, 5, 6, 7, 8, 9 and variations of Sherwood number, heat transmission, and radial velocity profiles are illustrated in Figs. 10, 11 and 12.In this study, the average value of some arbitrary scalar function of χ(ξ) , is defined as follows: where the a and b are integer numbers.According to Eq. (89), the average values of f (ξ) , f ′ (ξ),g(ξ) , k(ξ) , h(ξ) , θ(ξ) , φ(ξ) , θ ′ (ξ) , and φ ′ (ξ) are denoted by f avg , f ′ avg ,g avg , k avg , h avg , θ avg , φ avg , θ ′ avg , and φ ′ avg , respectively.According to Figs. 4, 5, 6, 7, 8, 9, 10, 11 and 12, when cases (1-4) related to Table 1 occur, the average values of f avg , f ′ avg , g avg , k avg , h avg , θ avg , φ avg , θ ′ avg , and φ ′ avg , are given in the following tables: In Tables 2, 3 and 4, some average results increased when the conditions changed from case 1 to case 4, and some decreased.The decrease or increase of these values is calculated using the following relationship: where is the amount of percentage increase or decrease values, Z 2 is the second value and Z 1 is the first value.According to Table 2, when the constant coefficients of cases in Table 1 change from case 1 to case 4, the average values of the results from Ref. 37 change respectively.When the conditions change from case 1 to case 4, f avg , (89)  will decrease by 34.71362167%, f ′ avg will decrease by 35.78465385%, g avg , will decrease by 8.197818397%, k avg will decrease by 9.504907967%, h avg will increase by 23.38976383%, θ avg will increase by 9.441723369%, θ ′ avg will increase by 7.742384880%, and φ ′ avg will decrease by 9.974419462% but according to Table 2, φ avg will increase by 1.541136126% when it changes from case 1 to case 2 but, φ avg , will decrease by 1.839033024% when it changes from case 2 to case 4. According to Table 3, when the constant coefficients of cases in Table 1 change from case 1 to case 4, the average values of the results from Modified AGM change respectively.When the conditions change from case 1 to case 4, f avg , will decrease by 34.60665497%, f ′ avg will decrease by 35.68188954%, g avg , will decrease by 8.140379566%, k avg will decrease by 9.388178325%, h avg will increase by 23.82760391%, θ avg will increase by 9.259738964%, θ ′ avg will increase by 7.367258115%, and φ ′ avg will decrease by 9.528160070% but according to Table 2, φ avg will increase by 1.313715539% when it changes from case 1 to case 2 but, φ avg , will decrease by 1.109559162% when it changes from case 2 to case 4. According to Table 4, when the constant coefficients of cases in Table 1 change from case 1 to case 4, the average values of the results from the HAN Method change respectively.When the conditions change from case 1 to case 4, f avg , will decrease by 34.71362166%,  f ′ avg will decrease by 35.78461613%, g avg , will decrease by 8.197818386%, k avg will decrease by 9.504907931%, h avg will increase by 23.38976385%, θ avg will increase by 9.441723405%, θ ′ avg will increase by 7.592028101%, and φ ′ avg will decrease by 9.661365531% but according to Table 2, φ avg will increase by 1.541136126% when it changes from case 1 to case 2 but, φ avg , will decrease by 1.839033024% when it changes from case 2 to case 4.

Conclusion
This study investigates the problem of heat and mass transfer in an unsteady rotating inclined plane using 3D thin film nanomaterial flow.The governing equations were set PDEs, and by using suitable similarity transformation, the PDEs were reduced into a set of nonlinear ODEs.The ODEs in four cases were solved with two semi-analytical techniques of Modified AGM and HAN.The Modified AGM that is used in this study is a novel technique, and the novelty of current work is related to solving this problem analytically.Unlike the former AGM, the Modified Agm has solved the previous issues and can replace the previous method of AGM.The HAN Method is another semi-analytical method that transforms a numerical solution into an analytical one.Technically, if the numerical solution exists for some problem, then HAN Method can be applied to obtain an   37 when compared with the modified AGM, but at the same time, the modified AGM is not dependent on any numerical methods for approximating analytical solutions.So, this paper is concluded that: • A new semi-analytical is introduced by modifying the former AGM technique.
• The exact analytic solutions were obtained through HAN Method.
• The solutions of both analytical solutions were compared with previously published papers.
• The results of both analytical solutions were presented quantitatively.
• The Sherwood number of the film surface and inclined swirling surface will decrease as the Schmidt number increases and the angular velocity of the rotating surface decreases.• The Nusselt number of inclined swirling surfaces will increase as the Prandtl number increases and the angular velocity of the rotating surface decreases.• The Nusselt number of film surfaces will decrease as the Prandtl number increases and the angular velocity of the rotating surface decreases.

Figure 3 .
Figure 3.The flow chart of the modified AGM.

Figure 4 .
Figure 4.The impact of different cases on f (ξ).

Figure 5 .
Figure 5.The impact of different cases on g(ξ).

Figure 6 .
Figure 6.The impact of different cases on h(ξ).

Figure 7 .
Figure 7.The impact of different cases on k(ξ).

Table 1 .
Different cases of the study.

Table 3 .
The average values of results from Modified AGM in different cases.

Table 4 .
The average values of results from the HAN Method in different cases.

Table 2 .
37e average values of the results from Ref.37in different cases.