Similarity Solution for Free Convection Flow of a Micropolar Fluid under Convective Boundary Condition via Lie Scaling Group Transformations

The free convective flow of an incompressible micropolar fluid along permeable vertical plate under the convective boundary condition is investigated. The Lie scaling group of transformations is applied to get the similarity representation for the system of partial differential equations and then the resulting systems of equations are solved using spectral quasi-linearisation method. A quantitative comparison of the numerical results is made with previously published results for special cases and the results are found to be in good agreement. The results of the physical parameters on the developments of flow, temperature, concentration, skinfriction, wall couple stress, heat transfer, and mass transfer characteristics along vertical plate are given and the salient features are discussed.


Introduction
In the past few decades, most of the researchers considered convective heat transfer problems with either constant wall temperature (CWT), constant heat flux (CHF), or Newtonian heating (NH) in a Newtonian and/or non-Newtonian fluid.Recently, a novel mechanism for the heating process has drawn the involvement of many researchers, namely, convective boundary condition (CBC), where the heat is supplied to the convecting fluid through a bounding surface with a finite heat capacity.Further, this results in the heat transfer rate through the surface being proportional to the local difference in temperature with the ambient conditions (Merkin [1]).Besides, it is more general and realistic, particularly in various technologies and industrial operations such as transpiration cooling process, textile drying, and laser pulse heating.Aziz [2] reported similarity solution for thermal boundary layer flow over a flat plate in a uniform stream of fluid with the convective boundary condition and he concluded that a similarity solution is possible if the convective heat transfer related to hot fluid on the lower surface of the plate is proportional to the inverse square root of the axial length.In the presence of an internal heat generation local similarity solution for free convection heat transfer from a moving vertical plate with the convective boundary condition is discussed by Makinde [3].The laminar natural convection flow over a semi-infinite moving vertical plate under the convective boundary condition is examined by Ibrahim and Bhashar Reddy [4].RamReddy et al. [5] investigated the influence of the prominent Soret effect on mixed convection in a nanofluid under the convective boundary conditions.The nonsimilar result has been presented for the free convection boundary layer flow along a solid sphere under the convective boundary conditions by Alkasasbeh et al. [6].More recently, a note on the natural convection along convectively heated vertical plate is given by Pantokratoras [7].
One of the best established theories of fluids with microstructure is the theory of micropolar fluids and this theory can be found in the books by Lukaszewicz [8] and Eremeyev et al. [9].It has gathered a good deal of attention due to the obvious reasons that the Navier Stokes equation for Newtonian fluids cannot successfully explain the attributes of 2 Advances in High Energy Physics fluids with a substructure.Physically, the micropolar fluids may be treated as non-Newtonian fluids consisting of dumbbell molecules or rigid cylindrical element, polymer fluids, fluid suspension, animal blood, and so forth.Further, the theory of micropolar fluids includes microrotation as well as microinertia effects.This theory studies viscous fluids in which microconstituents are rigid and spherical or randomly oriented as well.The subject of free convection boundary layer flow in a micropolar fluid has been keyed out by several investigators due to its immense applications in the engineering problems such as solar energy collecting devices, air conditioning of a room, material processing, and passive cooling of nuclear reactors.The boundary layer flow over a semi-infinite flat plate is considered for analyzing theory of micropolar fluid and its application to low concentration suspension flow by Ahmadi [10].Rees and Pop [11] discussed the free convection boundary layer flow of a micropolar fluid from a vertical flat plate.The nonsimilarity transformations are used to analyze the effects of double stratification on free/mixed convective transport in a micropolar fluid by Srinivasacharya and RamReddy [12][13][14] (also see the references cited therein).The problems of a steady laminar stagnation point flow towards a stretching/shrinking sheet in an incompressible micropolar fluid under the convective surface boundary condition are discussed by Yacob and Ishak [15] and Zaimi and Ishak [16].Merely from the literature, it is noted that the majority of the researchers have found the local similarity or nonsimilarity solutions for the problems involving convective boundary conditions, since most of the researchers have taken a convective heat transfer coefficient as a function  for getting the similarity solutions in their problems.Nevertheless, the assumption of a heat transfer coefficient varying along the plate as a function of  is not realistic and very difficult to be obtained in practice.For that cause, it could be supposed that the above works have only theoretical value.
In the recent past, several researchers are focused on obtaining the similarity solutions of the convective transport phenomena problems arising in fluid dynamics, aerodynamics, plasma physics, meteorology, and some branches of engineering by using different procedures.One such procedure is Lie group analysis.The concept of Lie group analysis also called symmetry analysis is developed by Sophius Lie to determine transformations which map a given differential equation to itself and it unifies almost all known exact integration techniques (see [17][18][19]).It provides a potent, sophisticated, and systematic tool for generating the invariant solutions of the system of nonlinear partial differential equations (PDEs) with relevant initial or boundary conditions.A special form of Lie group transformations, known as the scaling group, has been suggested by various researchers to study convection flows of different flow phenomena (see Tapanidis et al. [20], Hassanien and Hamad [21], Kandasamy et al. [22], Aziz et al. [23], Mutlag et al. [24], etc.; they are worth observing).
From the literature survey, it seems that the problem of the free convective heat and mass transport along permeable vertical plate in a micropolar fluid under the convective boundary condition has not been investigated so far.Motivated by all these works, this paper attempts to present the new similarity transformations and corresponding similarity solution to investigate the free convection flow of a micropolar fluid under the convective boundary condition using the Lie group transformations.The mathematical model involving the convective boundary conditions becomes slightly more complicated leading to the complex interactions of the flow, heat, and mass transfer mechanism.Further, the analytical solution is out of scope in the present set-up and hence a numerical solution is obtained for the current problem.Also, the influence of important parameters, namely, micropolar, suction/injection, and convective heat transfer parameters, on the physical quantities of the flow, heat, and mass transfer rates is analyzed in different flow situations.

Mathematical Formulation
Consider the steady, laminar, and free convective flow of an incompressible micropolar fluid with the free stream temperature and concentration,  ∞ and  ∞ , respectively.Choose the coordinate system such that the -axis is along the vertical plate and -axis normal to the plate, as shown in Figure 1.The suction/injection velocity distribution is assumed to be V  .The plate is either heated or cooled from left by convection from a fluid of temperature   with   >  ∞ corresponding to a heated surface (assisting flow) and   <  ∞ corresponding to a cooled surface (opposing flow), respectively.On the wall concentration is taken to be constant and is given by   .
By employing Boussinesq approximation and making use of the standard boundary layer approximations, the governing equations for the micropolar fluid [10] are given by where  and V are the velocity components in  and  directions, respectively,  is the component of microrotation whose direction of rotation lies in the  -plane,  is the temperature,  is the concentration,  * is the acceleration due to gravity,  is the density,  is the dynamic coefficient of viscosity,   () is the volumetric coefficient of thermal expansion,   () is the volumetric coefficient of solutal expansions,  is the vortex viscosity,  is the microinertia density,  is the spin-gradient viscosity,  is the thermal diffusivity, and  is the solutal diffusivity of the medium.The boundary conditions are where subscripts  and ∞ indicate the conditions at the wall and at the outer edge of the boundary layer, respectively, ℎ  is the convective heat transfer coefficient,  is the thermal conductivity of the fluid, and  is a material constant.Further, we follow the work of many recent authors by assuming that  = ( + /2).This assumption is invoked to allow the field of equations to predict the correct behavior in the limiting case when the microstructure effects become negligible and the total spin  reduces to the angular velocity [10].

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Now boundary conditions (6a) and (6b) become where   = −(/]Gr 1/4 )V  is the suction/injection parameter.It is worth mentioning that   determines the transpiration rate at the surface, with   > 0 for suction and   < 0 for injection, and   = 0 corresponds to an impermeable surface.Further, Bi = ℎ  /Gr 1/4 is the Biot number.It is a ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance of the hot fluid at the bottom of the surface.

Similarity Equations via Lie Scaling Group Transformations
A one-parameter Lie scaling group of transformations, which is a simplified form of Lie group transformation, is selected as (for more, see [26][27][28][29][30][31]) Here  ̸ = 0 is the parameter of the group and   (where  = 1, 2, 3, . . ., 8) are arbitrary real numbers whose interrelationship will be determined by our analysis.Transformations in (11) may be treated as a point transformation, transforming the coordinates We now investigate the relationship among the exponents   (where  = 1, 2, 3, . . ., 8) such that This is the requirement that the differential forms Δ 1 , Δ 2 , Δ 3 , and Δ 4 are conformally invariant under transformation (11).Substituting transformations ( 11) in ( 9), we have Advances in High Energy Physics 5 Now, boundary conditions (10a) and (10b) become The system remains invariant under the group transformation Γ.We then have the following relationships for the parameters: Solving linear system (16), we have the following relationship among the exponents: The set of transformations Γ reduces to Expanding by the Taylor series in power of , keeping the term up to the first degree (neglecting higher power of ), we get The characteristic equations are Solving the above characteristic equations, we have the following similarity transformations: where   0 and   0 are constant thermal and mass coefficient of expansion.Using ( 21) into (9), we get the following similarity equations: where the primes indicate differentiation with respect to  alone and Boundary conditions (10a) and (10b) in terms of , , , and  become

Skin Friction, Wall Couple Stress, and Heat and Mass Transfer Coefficients
The wall shear stress and the wall couple stress are and the heat and mass transfers from the plate, respectively, are given by The nondimensional skin friction   = 2  / where  2 * is the characteristic velocity and Gr  =  *   0 (  −  ∞ ) 3 /] 2 is the local Grashof number.

Numerical Solution Using the Spectral
Quasi-Linearization Method (SQLM) In this section, we describe the quasi-linearization method (QLM) for solving the governing system of ( 22) along with boundary conditions (23a) and (23b).This QLM is a generalization of the Newton-Raphson method and was proposed by Bellman and Kalaba [32] for solving nonlinear boundary value problems.
Assume that the solutions   ,   ,   , and   of ( 22) at the (+1)th iteration are  +1 ,  +1 ,  +1 , and  +1 .If the solutions at the previous iteration are sufficiently close to the solutions at the present iteration, the nonlinear components of ( 22) can be linearised using one-term Taylor series of multiple variables so that (22) give the following iterative sequence of linear differential equations: where the coefficients   1 , ( 1 = 1, 2, 3),   2 , ( 2 = 1, 2, . . ., 4),   3 , ( 3 = 1, 2),   4 , ( 4 = 1, 2), and   5 , ( 5 = 1, 2, . . ., 4) are known functions (from previous iterations) and are defined as Advances in High Energy Physics 7 subject to boundary conditions System ( 26) constitutes a linear system of coupled differential equations with variable coefficients and can be solved iteratively using any numerical method for  = 1, 2, 3, . ... In this work, as will be discussed below, the Chebyshev pseudospectral method was used to solve the QLM scheme (26) (for more details, refer to the works of Motsa et al. [33,34]): and starting from these sets of initial approximations  0 ,  0 ,  0 , and  0 , the iteration schemes (26) can be solved iteratively for  +1 (),  +1 (),  +1 (), and  +1 () when  = 0, 1, 2, . ... For this, we discretise the equation using the Chebyshev spectral collocation method.The unknown functions are approximated by the Chebyshev interpolating polynomials in such way that they are collocated at the Gauss-Lobatto collocation points defined as where  is the number of collocation points.The physical region [0, ∞) is transformed into the region [−1, 1] using the domain truncation technique in which the problem is solved on the interval [0,  ∞ ] instead of [0, ∞).This leads to the mapping where  ∞ is the scaling parameter used to invoke the boundary condition at infinity.The functions , , , and  are approximated at the collocation points by where   is the th Chebyshev polynomial defined as The derivatives of the variables at the collocation points are represented as where  is the order of the derivative and D = 2D/ ∞ is the Chebyshev spectral differentiation matrix and its entries are clearly defined in Canuto et al. [35].Substituting ( 31)-( 34) into (26) leads to the matrix equation subject to the boundary conditions In (35) where 23 = 0, 32 = 0, 42 = 0, and here I is an identity matrix, the size of the matrix 0 is (+1)×1, and diag[ ] is a diagonal matrix of size (+1)×(+ 1).Subscript r denotes the iteration number.After modifying matrix system (35) to incorporate boundary condition (36), the solution is obtained as

Results and Discussions
It is noticed that the present problem reduces to free convection heat transfer along an impermeable vertical plate in a micropolar fluid without the convective boundary condition when   = 0, Bi → ∞, and B = 0. Also in the limit as  → 0, governing equations (2)-( 5) reduce to the corresponding equations for a free convection heat and mass transfer in a viscous fluid.In order to validate the code generated the results of the present problem have been compared with the results obtained by Merkin [36], Nazar et al. [37], and Molla et al. [38] as a special case by taking  = 0,  = 0, B = 0, Pr = 1, Bi → ∞, and   = 0 and it was found that they are in good agreement, as presented in Table 1.Also, the comparison of heat transfer coefficient has been made with the results obtained by Nazar et al. [37] as shown in Table 2 when  = 0.5, B = 0, Pr = 1, Bi → ∞, and   = 0.
To study the effects of coupling number , suction/injection parameter   , Biot number Bi, and material parameter , computations were carried out in the cases of B = 1.0,Pr = 0.71, and Sc = 0.22.The effects of the coupling number  on the dimensionless velocity, microrotation, temperature, and concentration are illustrated in Figures 2(a  fluid particles.In the case of  = 0 (i.e., as  tends to zero) the micropolarity is absent and fluid becomes nonpolar fluid.With a large value of  effect of microstructure becomes significant, whereas with a diminished value of  the individuality of the substructure is much less articulated.
As  increases, it is found from Figure 2(a) that the maximum velocity decreases in amplitude and the location of the maximum velocity moves farther away from the wall.
Since  → 0 corresponds to viscous fluid, the velocity in case of a micropolar fluid has been less compared to that of viscous fluid case.It can be observed from Figure 2(b) that, as  increases, initially microrotation profiles tend to become flatter and then approach their free stream values far away from the wall.This happens due to the vanishing Advances in High Energy Physics of antisymmetric part of the stress on the boundary that corresponds to a weak concentration of microelements.This is because an increment in the value of  implies a higher vortex viscosity of fluid which promotes the microrotation of micropolar fluids.It is seen from Figures 2(c) and 2(d) that thermal and concentration boundary layers of the fluid increase with increase in coupling number .Hence, temperature and concentration in case of micropolar fluids are more than those of the viscous fluid case.
The Biot number Bi is the ratio of internal thermal resistance of a solid to boundary layer thermal resistance.When Bi = 0 the plate is totally insulated, internal thermal resistance of the plate is extremely high, and no convective heat transfer to the cold fluid on the upper part of the plate takes place.Figure 3(a) depicts fluid velocity profiles for different values of the Biot number with  = 0.5,   = 0.5, and  = 0.5.Generally, fluid velocity is zero at plate surface and increases gradually away from plate to free stream value satisfying boundary conditions.It is interesting to note that an increase in the intensity of convective surface heat transfer Bi produces significant enhancement in fluid velocity within the momentum boundary layer.In Figure 3 Given that convective heating increases with Biot number, Bi → ∞ simulates the isothermal surface which is clearly seen from Figure 3(c), where (0) = 1 as Bi → ∞.In fact, a high Biot number indicates higher internal thermal resistance of the plate than boundary layer thermal resistance.In this fluid temperature is maximum at the plate surface and decreases exponentially to zero value far out from the plate satisfying boundary conditions.As a consequence, an increment in the Biot number leads to increase of fluid temperature efficiency.The effect of   on velocity profile is depicted in Figure 4(a).Here,   > 0 represents suction and   < 0 denotes injection.The lower velocity is noticed in case of suction when compared to case of injection.From Figure 4(b), we note that microrotation is showing reverse rotation near two boundaries with both suction and injection parameter.The dimensionless temperature for different values of suction/injection parameters is drawn in Figure 4(c).It is readable that the temperature of the fluid is more in case of injection, whereas it is less in case of suction in comparison with the impermeable surface case (  = 0).Figure 4(d and  = 0.5 on local heat transfer coefficient is exhibited in Figure 6.It is found from Figure 6 that local heat transfer rate increases nonlinearly with the increase in Biot number Bi.The influence of the Biot number Bi on local mass transfer coefficient is shown in Figure 6. Figure 6 reveals that the local mass transfer coefficient is enhanced by the growth in the Biot number Bi.In Figure 7, the effect of the suction/injection parameter   with fixed Bi = 0.1 and  = 0.5 on local heat and mass transfer coefficients is displayed.It is found from Figure 7 that the local heat and mass transfer coefficients are less in the case of injection   < 0 in comparison with the case of suction   > 0.  friction factor compared to viscous fluid, the results as well suggest larger values of coupling number  and lower wall couple stresses.Since the skin friction coefficient   (0) and wall couple stress coefficient as well as high temperature and mass transport rates are more depressed in the micropolar fluid comparing to the viscous fluid, which may be beneficial in flow, temperature, and concentration control of polymer processing, thus, the presence of microscopic effects arising from the local structure and of the fluid elements reduces the high temperature and mass transfer coefficients.The effect of Bi on   Gr 1/4  and   Gr 1/2  for   = 0.5,  = 0.5, and  = 0.5 is illustrated in Table 3.It can be noticed that the skin friction and wall couple stress coefficients are increasing with increase of Bi for fixed values of other parameters.This notice is consistent with physical profiles presented in Figure 3.The effects of suction/injection parameter on the skin friction and wall couple stress coefficients are also shown in Table 3.It is noted that the skin friction and wall couple stress coefficients are less with injection case, whereas they are more with suction case when compared to the case of impermeable surface.Finally, the detailed behavior of the material parameter  is given in Table 3.The skin friction decreases and wall couple stress increases with increase of material parameter .

Conclusions
In this composition, the similarity solution of the free convection flow on a permeable vertical plate of a micropolar fluid under the convective boundary condition is obtained using Lie group transformations.Using the similarity variables, the governing equations are transformed into a set of nondimensional parabolic equations.These equations are solved numerically using spectral quasi-linearisation method.The numerical computation is carried out for various values of nondimensional physical parameters.The main findings are summarized as follows: (i) The numerical results indicate that velocity distribution is less near the plate but it is more far away from the plate; the wall couple stress coefficient and rate of heat and mass transfers are lower but the temperature and concentration distributions and the skin friction coefficient are higher for the micropolar fluids in comparison with those of viscous fluids.Also, the reverse rotation of microrotation near two boundaries is found with the increasing value of .
(ii) An increase in Biot number Bi decreases concentration distribution, whereas it causes an increase in temperature distribution, skin friction and wall couple stress coefficients, and heat and mass transfer rates within the boundary layer.Further, enhancing in Biot number Bi enhances velocity distribution near the plate but shows the reverse behavior far away from the plate.We observe reverse rotation of microrotation near two boundaries within the boundary layer in the presence of Biot number.
(iii) Less velocity, temperature, and concentration distributions are observed, more skin friction and wall couple stress coefficients and heat and mass transfer rates in the case of suction compared to the case of injection.Further, microrotation decreases near the wall and depicts the opposite trend far away from the wall.
(iv) It is found that microrotation, temperature, and concentration distributions and skin friction coefficients are more in the case of a micropolar fluid with strong concentration (i.e.,  = 0) when compared to the case of a micropolar fluid with weak concentration (i.e.,  = 1/2).Further, velocity is less in the case of  = 0 when compared to the case of  = 1/2 near the wall and shows the opposite trend far away from the wall.

Figure 1 :
Figure 1: Physical model and coordinate system.
Figure 3(d)  illustrates the variation of dimensionless concentration for different values of Bi.It is clear that the concentration of fluid decreases with increase of Bi.

Figure 6 :Figure 7 :
Figure 6: Effect of Bi on (a) heat transfer rate and (b) mass transfer rate.

Figure 8 Figure 8 :
Figure 8: Effect of material parameter  on (a) heat transfer rate and (b) mass transfer rate.

Table 3 :
Effects of skin friction and wall couple stress for varying values of Biot numbers Bi, micropolar parameter , material parameter , and suction/injection parameter   .