Group Invariant Solutions for Flow and Heat Transfer of Power-Law Nanofluid in a Porous Medium

*e present work covers the flow and heat transfer model for the power-law nanofluid in the presence of a porous medium over the penetrable plate. *e flow is caused by the impulsive movement of the plate embedded in Darcy’s type porous medium. *e flow and heat transfer model has been examined with the effect of linear thermal radiation and the internal heat source or sink in the flow regime. *e Rosseland approximation is utilized for the optically thick nanofluid. To form the closed-form solutions for the governing partial differential equations of conservation of mass, momentum, and energy, the Lie symmetry analysis is used to get the reductions of governing equations and to find the group invariants. *ese invariants are then utilized to obtain the exact solution for all three cases, i.e., shear thinning fluid, Newtonian fluid, and shear thickening fluid. In the end, all solutions are plotted for the cu-water nanofluid and discussed briefly for the different emerging flow and heat transfer parameters.


Introduction
Nanotechnology has gained valuable attention since the concept of dispersing nanosized particles into the fluid stream was introduced. e improvements made in the thermal properties of ordinary fluids by adding the nanoscaled particles give a new direction for the study of heat transfer processes. Several studies reported that the thermal performance of nanofluids depends on the dispersion, stability, size, shape, and type of nanoparticles present in the system. e demand for new thermal devices in the electronics, automotive industries, heat exchangers, cooling devices, solar collectors, and many more leads to extensive research carried out to explain the processes of heat transfer in nanofluids [1][2][3][4]. e nanofluid flow and heat transfer over the flat surface had some interesting applications in industrial setups. e survey articles of Eastman [5], Wang et al. [6], Keblinski et al. [7], Buongiorno [8], Das et al. [9], Mahanthesh and Mackolil [10], Vanaki et al. [11], and the references therein cover significant literature on the nanofluid's flow over the flat surface.
In the aforementioned literature, nanofluids are often considered as the Newtonian fluids. Recently, researchers believe non-Newtonian fluid models are more appropriate for the analysis of nanofluids. is assumption is in line with the overall molecular behavior/characteristics of nanofluids. e comprehensive literature is available on the studies related to non-Newtonian nanofluid flow and heat transfer models. Madhu et al. [12] studied the effects of convective radiation for the second law in Carreau fluid flow. e author applied the numerical technique for the solution of the model and found the results for the various emerging parameters for flow and heat transfer. e nonsimilar modeling for mixed convection in magnetized second-grade nanofluid flow is given by Raees et al. [13]. e BVP4C solver is used to show that the graphical results and inferences are drawn for important quantities that are substantially affected by the physical parameters.
e power-law model is one of the most comprehensible non-Newtonian models as it covers a broad range of fluid properties. It is beyond the scope of this work to discuss vast literature on flow and heat transfer of power-law nanofluid under different thermophysical conditions. Some significant studies on power-law nanofluids flow and heat transfer are mentioned here. Santra et al. [14] investigated power-law nanofluid flow along with heat transfer. ey consider water-based nanofluid including cu nanoparticles. e flow takes place within the rectangular channel, and the numerical solutions are computed for the modeled partial differential equations using the finite volume method. e results show that the addition of nanoparticles increases the heat transfer rate. Khana and Golra [15] present similar solutions of heat and mass transfer model for power-law nanofluid over a moving flat surface. An experiment is setup by Hojjat et al. [16] to study the impact of uniform distribution of nanoparticles in the ordinary fluid. ey observed that the rheological behavior of nanofluids varies with temperature variations and an increase in the concentration of nanoparticles. Deng [17] and Deng et al. [18] presented the study of power-law nanofluid in a rectangular microchannel. eir results show that the heat transfer rate of electroosmotic flow is enhanced for shear thickening nanofluids. Aziz and Jamshed [19] reported heat transfer analysis for the MHD power-law nanofluid slip flow over a moving surface with variable thermal conductivity. Readers are recommended to study the latest research [20][21][22][23][24][25][26][27][28] on power-law nanofluids for further understanding. ermal radiation and heat source/sink effects are very important in the interpretation of many physical problems. e collected work for the non-Newtonian nanofluid with thermal radiation and heat source is available. Venkatesan and Reddy [29] studied the impact of the pulsating flow of Al 2 O 3 -blood non-Newtonian nanofluid in a channel in the presence of the magnetic field and thermal radiation. e effects of Joule heating and heat source are taken into account. e perturbation is applied to observe the effects of flow variables. Ramzan et al. [30] studied the effects of Newtonian heating and Fourier and Fick's laws on MHD dusty Casson fluid with variable heat source/sink. e numerical solutions are obtained via a BVP4C solver, and graphical results are also discussed for novel physical parameters. Some new research articles available in the literature elaborate on the effects of thermal radiation and heat source/sink on fluid velocity and temperature. See for example and the references therein [31][32][33].
Mathematical models involving non-Newtonian nanofluids are mostly solved using numerical and experimental techniques. e nonlinear nature of modeled differential equations made them difficult to solve analytically. e exact (closed-form) solutions are often preferred over numerical or experimental results to study the physical behavior of the fluid flow. Moreover, they are also used to determine the reliability of both computational and iterative approaches.
ere are few studies available in which exact solutions are sought for the nonlinear models. e study of MHD flow and radiative heat transfer characteristics of non-Newtonian nanofluids is reported by Maghsoudi et al. [34]. e author uses the Galerkin method to get the exact solution. e results show that the increasing magnetic force decreases the thermal efficiency and in turn decreases the skin friction coefficient and Nusselt number. e non-Newtonian nanofluid flow between two vertical plates is studied by Biswal et al. [35]. In this study, two methods are incurred to find the exact solutions, that is, Galerkin's method and the homotopy perturbation method. Aziz in his research found the exact solution of Stokes' flow of a non-Newtonian nanofluid in a porous medium by considering Navier's slip condition [36]. e author uses the Lie symmetry and generalized group approach to find the solutions. Tahiri and Mansouri [37] presented the analytical solutions using the Laplace-Ritz variational method for the flow of non-Newtonian nanofluid inside a circular tube. Aziz and Javaid [38] and Aziz et al. [39] applied the Lie symmetry method on the flow and heat transfer of MHD third grade nanofluid in the presence of thermal radiation and with uniform internal heat source/sink, respectively. e authors extract the exact closed-form solutions for governing partial differential equations. Kezzar et al. [40] established the study of natural convection of non-Newtonian nanofluid flow between two vertical flat plates. e exact solutions are obtained by using the Adomian generalized method (GDM), and numerical results are given by Runge-Kutta fourth-order. e symmetry analysis is a systematic way of calculating the self-similar solutions of nonlinear partial differential equations. Recently, some studies found are reported in literature in this context that are available in [41][42][43][44][45][46][47].
e literature survey reveals that no research reported exact solutions for the power-law nanofluid flow along with the process of heat transfer over the porous plate within the porous medium. In the present model, the flow is induced by the motion of the penetrable plate. e effects of Darcy's resistance and linear thermal radiation are incorporated in the flow model. e Tiwari and Das model [48] is used for the thermophysical properties of nanofluid. e exact solutions are found by using the Lie symmetry approach. Afterward, the solutions are plotted for cu-water nanofluid to observe the flow and heat transfer behavior of the fluid depending on the physical parameters. is prototype model is directly applicable in industrial processes such as glass and paper manufacturing, hot rolling and wire drawing, and thermal solar collectors. e final quality of products and the processes of heat transfer depend on the type of operating fluid and the stretching of bounding surfaces. Further applications of this mathematical model are in the fields of aerodynamic heating, crystal growing, and extraction of oil and gas.

Model Statement
Consider the unsteady unidirectional Stokes' flow of an incompressible fluid over an infinite penetrable plate in the presence of thermal radiation and uniform heat source/sink embedded in a porous medium. e fluid motion is caused by an arbitrary motion of the plate. e flow is assumed to be in the x-direction which is chosen along with the penetrable plate and y-axis is perpendicular to it. e velocity field for the flow is given by In above equation, v 0 is the constant suction/injection velocity. v 0 < 0 specify the suction and v 0 > 0 represents the injection velocity. Considering the preceding assumptions, the governing boundary layer equations for the flow and heat transfer model are [49] zv It can be noticed that equation (2) is trivially satisfied for the given velocity field. In equations (2)-(4), ρ nf is the nanofluid density, τ xy is the stress tensor, R x is Darcy's resistance, C Pnf is the specific heat capacity, κ nf is the thermal conductivity, q r is the thermal radiation, and Q is the internal heat source/sink. e internal heat generation is represented by Q > 0, whereas Q < 0 refers the internal heat absorption. e correlations for thermophysical properties of nanofluid are given in Table 1 [50][51][52][53].
In Table 1, φ is the volume concentration of nanoparticles.
e parameters ρ f , μ f , σ f , (ρC P ) f , and κ f are density, viscosity, electrical conductivity, specific heat capacity, and thermal conductivity of base fluid, respectively. Quantities ρ s , μ s , σ s , (ρC P ) s , and κ s are density, viscosity, electrical conductivity, specific heat capacity, and thermal conductivity of nanoparticles. e expression for stress tensor in equations (3) is given by [54] τ xy � K zv zy where K is the coefficient of consistency and m is the characteristic coefficient for the non-Newtonian behavior of the fluid. e relation between the pressure gradient and the velocity for the unidirectional flow of power-law nanofluids is calculated as where ϕ is the porosity and κ is the permeability of the porous medium. e radiation term in equation (4) under Rosseland approximations [55,56] is simplified to In above expression, σ * and κ * are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. T ∞ is the ambient temperature. By employing the above simplified expressions given in equations (6) and (7), the momentum equation (3) and the energy equation (4) can be written as e respective initial and boundary conditions are where V 0 is the reference velocity, and U(t), h(t), f(y), and g(y) are the unknown functions.
To make equations (8) After neglecting bars and the substitution of dimensionless parameters in equations (8) and (9), the nondimensional forms of momentum and energy equation are In equation (12),

Mathematical Problems in Engineering
where

Lie Symmetry Analysis
is section covers the procedure adopted to find the Lie point symmetries for the momentum and energy equations given in (12) and (13) respectively. Symmetry transformations are then used to form the group invariant solutions. e aim is to find the symmetry operator of the form Π is the symmetry generator of second-order PDE if and only if where Π [2] is the second prolonged operator of Π, that can be written as together with e total derivatives operators are given by

Lie Symmetry Analysis of the Flow Model.
In this subsection, Lie symmetry analysis is applied to the flow equation (12). e equation (18) along with equations (12) and (19)-(22) reduces to the following overdetermined system of partial differential equations e solution of above system gives rise to two cases with respect to the power-law index m and presents a three-dimensional Lie algebra.

Exact Solution for Pseudoplastic Fluids (m > 1)
In this section, the exact solution is derived for the flow of nanofluid in the case m � 2. is gives the physical interpretation of pseudoplastic (shear thickening) fluids. e momentum equation (12) for m � 2 can be written as e symmetry operators given in Case I are considered to solve the PDE (26).

Group Invariant Solution Corresponding to
e invariant solution corresponding to the linear combination of time and space translation generators Π 1 + cΠ 2 with (c > 0) represents a forward travelling wave solution. After solving the characteristic equation, the invariant is given by Using the relation written in equation (27), the PDE (26) reduces to the first-order ordinary differential equation along with the initial condition as e solution of equation (28) subjected to the initial condition (29) gives the exact solution for G(η) as Substituting G(η) from equation (30) into equation (27), the group invariant solution will be provided that the following compatibility condition holds e solution given in equation (31) satisfies the initial and boundary conditions (14)- (16) with It can be noticed from the above equation that U(t) and F(y) are depending on the flow and permeability parameter.

Results and Discussion.
In this subsection, the solution (31) is plotted against different governing parameters that elaborate the effect on fluid velocity. Graphs are produced for the water-based cu-water nanofluid. Figure 1(a) shows the effect of nanoparticle concentration on the velocity of shear thickening fluid. e ratio φ controls the viscosity and the thermal conductivity of the nanofluid, such that the linear velocity of the flow decreases by increasing the nanoparticle concentration. e effect of permeability parameter on fluid velocity is shown in Figure 1(b). e presence of a porous medium causes higher resistance to the fluid flow due to which the fluid decelerates. is causes thinning of the momentum boundary layer. e velocity of the nanofluid gradually decreases to free stream velocity as shown in Figure 1(c). e suction effect, that is, v 0 > 0 is depicted in Figure 1(d).
e fluid velocity decreases for increasing values of v 0 through penetrable surface, whereas the opposite behavior is observed for the case of injection v 0 < 0.

Exact Solution for Newtonian Fluids (m = 1) e momentum equation (12) for the Newtonian fluid
where the exact solution of equation (34) will be formed corresponding to symmetry generators given in Case II.

Group Invariant Solution
Corresponding to Π 1 − cΠ 2 . e backward type travelling wave solution can be found by the linear combination of time and space translation generators Π 1 − cΠ 2 . Here, c > 0 gives the speed of the waves. In this case, the waves are propagated towards the plate. e invariant solution is of the form v(y, t) � Z(ξ), where ξ � y + ct. (35) Substitution of above expression in equation (34) reduces it to the first-order ODE along with the initial condition e exact solution of equation (36) with respect to equation (37) is Equation (35) e above solution satisfies the boundary condition when

Results and Discussion.
e effect of different physical parameters on velocity profile (39) is shown in Figure 2. e number of pores causes more resistance within the fluid (Figure 2(b)). In Figure 2(c), the overall effect of suction v 0 > 0 and injection v 0 < 0 is presented. e fluid velocity for the case of suction v 0 > 0 shows the increasing behavior, whereas the opposite trend is observed in the case of injection. Moreover, the fluid velocity continues to decrease with time as depicted in Figure 2(d). Finally, the ascending wave speed slows down fluid motion within the boundary layer, as shown in Figure 2(e).

Exact Solution for Dilatant Fluid (m < 1)
In Case II, if we take m � − 2, this will represent the shear thinning fluids, and the symmetry operators in this case takes the form  Mathematical Problems in Engineering e momentum equation (12) for m � − 2 can be written as e exact solution will be form corresponding to above three-dimensional Lie algebra.

Group Invariant Solution Corresponding to
Consider the symmetry generators with the linear combination of time and space translation operators. is type of combination represents the backward-type travelling wave solution. e invariants are assumed as e above relation together with equation (12) for m � − 2 gives the reduced first-order ODE as e initial condition is Considering the generalized compatibility approach given in [57], the general solution has the form where parameters A and B are to be determined. Using equation (46) into equation (44) admits the exact solution of the form Substitution of above expression into equation (43) gives the exact solution of the momentum equation (12) for the dilatant fluids. at is, where the speed of travelling wave c � − v 0 . e above solution satisfies the boundary conditions with (48) is plotted for the various emerging parameters in Figure 3. e qualitative behavior of dilatant nanofluids is the same as in case of pseudoplastic fluids.

Lie Symmetry Analysis of Energy Equation
e Lie symmetry analysis of energy equation (13) is presented in this section. In equation (18), substitute equation (13) in the place of PDE along with equations (19)- (22). is will generate a system Mathematical Problems in Engineering where 5(y, t) satisfies the equation

Exact Solution of Energy Equation
e symmetry operators given in equation (51) are utilized to find the group invariant solution of equation (13).

Group Invariant Solution Corresponding to
e linear combination of time and space translation operators gives the travelling wave solution. is type of solution was previously discussed for the flow equation. e wave front type travelling wave solution with constant wave speed c > 0 gives the invariant as subjected to Using equation (53) into equation (13) gives the secondorder ODE along with the boundary conditions Utilizing the compatibility approach [57], the exact solution of equation (55) is of the form Equation (57) together with equation (53) forms the exact solution of the energy equation: e above solution satisfies the boundary condition with One can notice that above functions depend on the different physical heat transfer parameters which describe the flow and heat transfer behavior.

Results and Discussion.
In this section, the solution (58) is plotted for different heat transfer parameters. e effect of physical parameters on temperature variation of nanofluid is depicted in Figure 4. e effect of increasing nanoparticles concentration within nanofluid is presented in Figure 4(a). e temperature of nanofluid rises with the addition of nanoparticles in the flow stream. It is clear from the figure that the increasing volume concentration of nanoparticles causes to increase in the thermal conductivity which in return decreases the temperature of the fluid. e internal heat source Φ > 0 causes to rise in temperature and thickening of the thermal boundary layer. e internal heat sink Φ < 0 reduces the temperature of the fluid which is clear from Figure 4(b). Figure 4(c) elaborates the combined effects of suction v 0 > 0 and injection v 0 on the temperature of the fluid. e temperature is increasing for the increased values of suction and thickening of the thermal boundary layer, and the opposite trend can be observed for injection. In Figure 4(d), the effect of Prandtl number on the temperature profile is presented. Since, the enhancement in Prandtl number corresponds to weaker thermal diffusivity which in return lowers the temperature. e above fact can be observed from Figure 4(d). e thermal boundary layer becomes thin for the increased Prandtl number. It can be seen that the temperature at the wall is decreased for larger values of Prandtl number. e effects of radiation parameter on the temperature field are shown in Figure 4(e). It can be observed that the increasing radiation causes to increase in the temperature and thickening of the thermal boundary layer.

Concluding Remarks
is research aims to extract exact solutions of time-dependent flow and the heat transfer model of power-law nanofluid. To observe the physical behavior of nanofluid for shear thinning and shear thickening fluids, the results are obtained separately for each case. e Lie symmetry method is utilized to get the complete Lie symmetry algebra for the flow and heat transfer model. e symmetry algebra is used to get the reductions of the governing partial differential equation for different values of power-law index m. en, the group invariant solution is presented for the flow and heat transfer model. e effects of physical parameters involving volume fraction of nanoparticles, permeability, suction/injection, and thermal radiation on the flow and heat transfer of the fluid are discussed separately. e exact solution for such type of model in which the non-Newtonian fluids are considered with nanofluids is rare in literature. e methodology adopted in this article is more precise and used a simple systematic algorithm. e generalized group and compatibility technique aided in finding the exact solution of power-law nanofluid with the heat transfer model. e graphical results are given for different physical parameters to observe the behavior of velocity and temperature distribution for the water-based cu-water nanofluid. e approach used in this article may be extended to other nonlinear problems related to wave mechanics, solid mechanics, and general relativity.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.