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Applications of Group Theoretical Methods to Non-Newtonian Fluid Flow Models: Survey of Results
The present review is intended to encompass the applications of symmetry based approaches for solving non-Newtonian fluid flow problems in various physical situations. Works which deal with the fundamental science of non-Newtonian fluids that are analyzed using the Lie group method and conditional symmetries are reviewed. We provide the mathematical modelling, the symmetries deduced, and the solutions obtained for all the models considered. This survey includes, as far as possible, all the articles published until . Only papers published by a process of peer review in archival journals are reviewed and are grouped together according to the specific non-Newtonian models under investigation.
The scientific and applications appeal of non-Newtonian fluid mechanics has necessitated a deeper study of its theory. There has been considerable focus in the study of the physical behavior and properties of non-Newtonian fluids over the past several decades. One particular reason for this interest is the wide range of applications of such models, both natural and industrial. These applications range from the extraction of crude oil from petroleum products to the polymer industry. Spin coating is a classic example where the coating fluids are typically non-Newtonian. A non-Newtonian fluid is one whose flow curve (shear stress versus shear rate) is nonlinear or does not pass through the origin, that is, where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry and shear rate and sometimes even on the kinematic history of the fluid element under consideration. Such fluids may be conveniently grouped into three general classes as follows:
Fluids for which the rate of shear at any point is determined only by the value of the shear stress at that point at that instant: these fluids are variously known as time independent, purely viscous, inelastic, or generalized Newtonian fluids (GNF).
More complex fluids on which the relation between shear stress and shear rate depends: in addition, based upon the duration of shearing and their kinematic history, they are known as time-dependent fluids.
Those substances exhibiting characteristics of both ideal fluids and elastic solids and showing partial elastic recovery, after deformation, are categorized as viscoelastic fluids.
Due to the complex physical structure of non-Newtonian fluids, there is not a single constitutive expression which describes the physical and mathematical properties of all nonlinear fluids. For this reason, many non-Newtonian fluid models for constitutive equations are available with most of them being empirical and semiempirical.
There are three diverse motivations for analyzing the flow behavior of non-Newtonian fluids: firstly, to extend the results of the flow models of Newtonian fluids to various classes of non-Newtonian fluids; secondly, to study the flow structure of non-Newtonian fluids as they occur in industry under conditions which arise there; thirdly, to construct solutions of complicated nonlinear equations as exact solutions: these, when reported, facilitate the verification of complicated numerical codes and are also helpful in stability analysis. Consequently, the exact (closed-form) solutions of the flow models of non-Newtonian fluids are physically very significant. The most challenging task that we need to address when dealing with flow problems of non-Newtonian fluids is that the governing equations of these models are of a high order, nonlinear, and complicated in nature. Such fluids are modelled by constitutive equations which vary greatly in complexity. Thus, the resulting nonlinear equations are not easy to solve exactly. Several methods have been developed in recent years to obtain the solutions of these fluid models. Some of the techniques are the variational iteration method, Adomian decomposition method, homotopy analysis method, homotopy perturbation method, simplest equation method, semi-inverse variational method, and the exponential function method, amongst others. There are also the Lie symmetry and conditional symmetry group methods which are the main focus of this review.
Lie symmetry methods for differential equation were originated in the 1870s and were introduced by the Norwegian mathematician Marius Sophus Lie. Lie’s theory is useful for solving differential equations that admit sufficient number of symmetries in a systematic way. Lie group methods are capable of handling a large number of equations. The application of this method neither depends on the type of the equation nor on the number of variables involved in the equations. Lie’s theory is a general procedure which can be applied to any class of differential equations. However, if one peruses the literature on Lie’s methods, we observe that this method and its extensions have rarely been applied in comparison with the wealth of differential equations in practical and theoretical problems.
The Lie symmetries of differential equations naturally form a group. Such groups are called Lie groups and are invertible point transformations of both the dependent and independent variables of the differential equations. Lie pointed out in his work that these groups are of great importance in understanding and constructing solutions of differential equations. Lie demonstrated that many techniques for finding solutions can be unified and extended by considering symmetry groups. Today, the Lie symmetry approach to differential equations is widely applied in various fields of mathematics, mechanics, physics, and the applied sciences and many results published in these areas demonstrate that Lie’s theory is an efficient tool for solving nonlinear problems formulated in terms of differential equations. The primary objective of the Lie symmetry analysis advocated by Lie is to find one or several parameters of local continuous transformations leaving the equations invariant and then exploit them to obtain reductions and the so-called invariant or similarity solutions, and the usefulness of this approach has been widely illustrated by several researchers in different contexts. An extension of this approach is the conditional symmetry approach which is also very useful.
Motivated by the above-mentioned facts, the purpose of the present survey is to provide a detailed review of those studies which deal with the flow models of non-Newtonian fluids and solved using the group theoretic approaches. We have presented the mathematical modelling of each of the problem under review together with the symmetries deduced and the solutions obtained for that particular problem.
2. Symmetry Methods for Differential Equations
In this section, we briefly discuss the main aspects of the Lie symmetry method for differential equations with some words on conditional or nonclassical symmetries.
2.1. Symmetry Transformations of Differential Equations
A transformation under which a differential equation remains invariant (unchanged) is called a symmetry transformation of the differential equation.
Consider a th order ( system of differential equations where , called the dependent variable, is a function of the independent variable and , up to are the collection of all first-order and second-order up to th order derivatives of .
A transformation of the variables and , namely.is called a symmetry transformation of system (1) if (1) is form-invariant in the new variables and ; that is,wheneverFor example, the first-order Abel equation of the second kind has symmetry transformations
2.2. Lie Symmetry Method for Partial Differential Equations
Here we discuss the classical Lie symmetry method to obtain all possible symmetries of a system of partial differential equations.
Let us consider a th order system of partial differential equations in independent variables and dependent variable , namely.where , , denotes the set of all th order derivative of , with respect to the independent variables defined bywithFor finding the symmetries of (7), we first construct the group of invertible transformations depending on the real parameter , which leaves (7) invariant; namely,The above transformations have the closure property, are associative, admit inverses and identity transformation, and are said to form a one-parameter group.
Since is a small parameter, transformations (10) can be expanded in terms of a series expansion asTransformations (11) are the infinitesimal transformations and the finite transformations are found by solving the Lie equations with the initial conditions where and
Transformations (10) can be denoted by the Lie symmetry generatorwhere the functions and are the coefficient functions of the operator .
Operator (14) is a symmetry generator of (7) ifwhere represents the th prolongation of the operator and is given bywith In the above equations, the additional coefficient functions satisfy the following relations:where denotes the total derivative operator and is given byThe determining equation (15) results in a polynomial in terms of the derivatives of the dependent variable . After separation of (15) with respect to the partial derivatives of and their powers, one obtains an overdetermined system of linear homogeneous partial differential equations for the coefficient functions ’s and ’s. By solving the overdetermined system, one has the following cases:(i)There is no symmetry, which means that the Lie point symmetry generators given by and are all zero.(ii)The point symmetry has arbitrary constants; in this case, we obtain generators of symmetry which forms an -dimensional Lie algebra of point symmetries.(iii)The point symmetry admits some finite number of arbitrary constants and arbitrary functions, in which case we obtain an infinite-dimensional Lie algebra.
2.3. Example on the Lie Symmetry Method
Here we illustrate the use of the Lie symmetry method on the well-known Korteweg-de Vries equation given byWe seek for an operator of the formEquation (21) is a symmetry generator of (20) ifThe third prolongation in this case isTherefore, the determining equation (22) becomesUsing the definitions of , , and into (24) lead to an overdetermined system of linear homogenous system of partial differential equations given byBy solving system (25), we find four Lie point symmetries which are generated by the following generators:
2.4. Nonclassical Symmetry Method for Partial Differential Equations
Here we present a brief version of the nonclassical symmetry method for partial differential equations. In last few years, the interest in nonclassical group method has increased. There are mathematical problems appearing in applications that do not admit Lie point symmetries but have nonclassical symmetries. Therefore, this approach is helpful in obtaining exact solutions.
We begin by considering a th order partial differential equationin independent variables and one dependent variable , with denoting the derivatives of the with respect to up to order defined bywithSuppose that is a field of vectors which consists of dependent and independent variables:where and are the coefficient functions of the vector field .
Suppose that the vector field is the nonclassical symmetry generator of (27). Then the solution of (27) is an invariant solution of (27) under a one-parameter subgroup generated by if the conditionholds together with (27). The condition given in (32) is known as an invariant surface condition. Thus, the invariant solution of (27) is obtained by solving the invariant surface condition (32) together with (27).
For (27) and (32) to be compatible, the th prolongation of the generator must be tangent to the intersection of and the surface ; that is,If (32) is satisfied, then the operator is called a nonclassical infinitesimal symmetry of the th order partial differential equation (27).
For the case of two independent variables, and , two cases arise, namely. when and .
When , the operator is and thus is the invariant surface condition.
When , the operator isand hence is the invariant surface condition.
2.5. Example on the Nonclassical Symmetry Method
We illustrate the use of the nonclassical symmetry method on the well-known heat equationConsider the infinitesimal operatorThe invariant surface condition isOne can assume without loss of generality that , so that (40) takes the form
The nonclassical symmetries determining equations arewhere is the usual third prolongation of operator
3. Power-Law Fluid Flow Problems
In this section, all those problems dealing with the flow of a power-law fluid and solved by using the Lie symmetry approach are discussed.
The Cauchy stress tensor for a power-law fluid is written aswhere is the fluid pressure, is the identity tensor, is the dynamic viscosity of the fluid, tr is the trace, and the first Rivlin-Ericksen tensor is given byin which is the fluid velocity. It should be noted that is the power-law index. If , (46) represents a viscous fluid. Furthermore, (46) represents shear thinning behavior when and shear thickening for .
3.1. Solution of the Rayleigh Problem for a Power-Law Non-Newtonian Conducting Fluid via Group Method 
Abd-el-Malek et al.  studied the magnetic Rayleigh problem where a semi-infinite plate is given an impulsive motion and thereafter moves with constant velocity in a non-Newtonian power-law fluid of infinite extent. The governing nonlinear model was solved by means of the Lie group approach.
The governing problem describing the flow model  is given bywith the boundary and initial conditionsThe method of solution employed in  depends on the application of a one-parameter group of transformations to the partial differential equation (48). The one-parameter group, which transforms the PDE (48) and the boundary conditions (49), is of the form Under transformations (50), the two independent variables reduce by one and the partial differential equation (48) is transformed into an ordinary differential equation. The reduced ordinary differential equation was then solved numerically.
3.2. Invariant Solutions of the Unidirectional Flow of an Electrically Charged Power-Law Non-Newtonian Fluid over a Flat Plate in Presence of a Transverse Magnetic Field 
Wafo Soh  investigated a boundary value problem for a nonlinear diffusion equation arising in the study of a charged power-law non-Newtonian fluid through a time-dependent transverse magnetic field. Two families of exact invariant solutions were obtained by use of the Lie symmetry method.
The governing equation describing the flow model is given by The relevant boundary and initial conditions areThe symmetry Lie algebra of PDE (51) is five-dimensional and is spanned by the operators whereWith the use of the above symmetries, the group invariant solution for the PDE (51) found in  iswith given by
3.3. Unsteady Boundary Layer Flow of Power-Law Fluid on Stretching Sheet Surface 
Yürüsoy  treated the unsteady boundary layer equations of a power-law fluid over a stretching sheet. By the use of similarity transformations, the governing system of partial differential equations reduced to a nonlinear ordinary differential equation system. Finally, the resulting system of reduced ordinary differential equations was solved using a combination of the Runge-Kutta algorithm and shooting technique.
The governing equations describing the flow model  arewhere and are the velocity components inside the boundary layer and is the velocity outside the boundary layer.
The boundary conditions for flow over a stretching sheet areBy use of the Lie group method, the similarity transformations for the reduction of the above system of PDEs are given by Transformation (59) transforms the two-dimensional unsteady boundary layer equation problem into ordinary differential equations. The reduced ordinary differential equations have been solved numerically using a variable step size Runge-Kutta subroutine combined with a shooting technique.
3.4. Axisymmetric Spreading of a Thin Power-Law Fluid under Gravity on a Horizontal Plane 
Nguetchue and Momoniat  studied a nonlinear PDE modelling the axisymmetric spreading under gravity of a thin power-law fluid on a horizontal surface. The model equation was reduced to a nonlinear second-order ordinary differential equation for the spatial variable. Then Lie symmetry analysis applied to the nonlinear ordinary differential equation enabled its linearization and solution.
The equation modelling the height of a thin power-law fluid film on a horizontal plane in presence of gravity is given by Here is the film height and is the power-law fluid parameter. The Lie point symmetry generator for the PDE (60) is The invariant solution of PDE (60) corresponding to the symmetry generator (61) found in  is
3.5. Symmetry Reductions of a Flow with Power-Law Fluid and Contaminant-Modified Viscosity 
Moitsheki et al.  have analyzed a system dealing with nonreactive pollutant transport along a single channel. Constitutive equations obeying a power-law fluid are utilized in the description of the mathematical problem. Invariant solutions which satisfy physical boundary conditions have been constructed using the Lie group approach.
The dimensionless governing equations that describe the flow model are Here is the Schmidt number and is the imposed constant pressure axial gradient. The Lie point symmetries of the above system corresponding to different forms of the source term are given in Table of . The invariant solutions of system (63) found in  are of the form
3.6. Scaling Group Transformation under the Effect of Thermal Radiation Heat Transfer of a Non-Newtonian Power-Law Fluid over a Vertical Stretching Sheet with Momentum Slip Boundary Condition 
An analysis has been conducted to study the problem of heat transfer of a power-law fluid over a vertical stretching sheet with slip boundary condition by Mutlag et al. . The partial differential equations governing the physical model have been converted into a set of nonlinear coupled ordinary differential equations using scaling group of transformations. These reduced equations are then solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order numerical method.
The dimensionless forms of the governing equations of the flow model  areThe boundary conditions specified to solve the above system of PDEs areThe scaling symmetry operator for the system of PDEs (65) is calculated as The corresponding similarity transformations areTransformation (67) transforms the system of PDEs (65) into a nonlinear system of ODEs. The reduced ordinary differential equations are solved numerically.
3.7. Lie Group Analysis of a Non-Newtonian Fluid Flow over a Porous Surface 
Akgül and Pakdemirli  investigated the two-dimensional unsteady squeezed flow over a porous surface for a power-law non-Newtonian fluid. Lie Group theory was applied on the model equations. Then, a partial differential system with three independent variables was converted into an ordinary differential system, via application of two successive symmetry generators. The ordinary differential equations were then solved numerically.
The problem describing the flow model  is given bywithThe symmetries for the system of PDEs (69) found in  areSymmetries (71) are used to reduce the nonlinear system of PDEs (69) to a nonlinear system of ODEs which was then solved using a numerical approach.
3.8. Flow of Power-Law Fluid over a Stretching Surface: A Lie Group Analysis 
The investigation of the boundary layer flow of power-law fluid over a permeable stretching surface was made by Jalil and Asghar . The use of Lie group analysis reveals all possible similarity transformations of the problem. The similarity transformations have been utilized to reduce the governing system of nonlinear PDEs to a nonlinear boundary value problem.
The governing equations of the flow model  areThe boundary conditions areThe form of the infinitesimals is found to be Symmetries (74) are used to compute the appropriate similarity transformations which were then used to reduce the nonlinear system of the above PDEs to a nonlinear boundary value problem. The reduced boundary value problem was solved numerically.
3.9. Group Invariant Solution for a Preexisting Fracture Driven by a Power-Law Fluid in Impermeable Rock 
The effect of power-law rheology on hydraulic fracturing has been studied by Fareo and Mason . With the aid of lubrication theory and the PKN approximation, a partial differential equation for the fracture half-width was derived. By using a linear combination of the Lie symmetry generators of the governing equation, the group invariant solution was obtained and the problem was reduced to a boundary value problem for an ordinary differential equation.
The mathematical problem describing the preexisting fracture driven by a power-law fluid in impermeable rock  is given bywithThe symmetry Lie algebra of (75) is spanned by the operators The group invariant solutions of the PDE (75) found in  are of the formwhere the particular values of are given in . The Lie symmetries given in (77) were also utilized to perform various reductions of PDE (75) which was then solved numerically.
4. Sisko Fluid Flow Problems
In this section, we investigate all those models which deal with the flow of a Sisko fluid and solved with the aid of the Lie group approach.
The Cauchy stress tensor for a Sisko fluid model is given by where is the velocity vector, is the first Rivlin-Ericksen tensor, and and are the material constants. The model is a combination of viscous and power-law models. For , the model exhibits power-law behavior whereas for , the flow is Newtonian and is a characteristic of the non-Newtonian behavior of the fluid.
4.1. Rayleigh Problem for a MHD Sisko Fluid 
Molati et al.  studied the problem of unsteady unidirectional flow of an incompressible Sisko fluid bounded by a suddenly moved plate. The fluid is magnetohydrodynamic (MHD) in the presence of a time-dependent magnetic field applied in the transverse direction of the flow. The nonlinear governing flow model was solved analytically using the Lie symmetry approach.
The problem describing the flow model  is given bywithThe symmetry Lie algebra of the PDE (80) is three-dimensional and spanned by the operators whereThe similarity solution from the invariants of assumes the form Invariant (84) is used to reduce the PDE (80) into a nonlinear ODE. The reduced ODE together with suitable boundary conditions was solved by employing a numerical approach.
4.2. Reduction and Solutions for MHD Flow of a Sisko Fluid in a Porous Medium 
Mamboundou et al.  obtained the analytical solutions for magnetohydrodynamic (MHD) flow of a Sisko fluid in a semi-infinite porous medium. The governing nonlinear differential equation was solved by employing the symmetry method.
The governing equation of the flow model  isThe relevant boundary and initial conditions areThe above PDE admits the Lie point symmetry generators The travelling wave solutions of the PDE (85) were constructed corresponding to the symmetry generators (87) and is given by 
4.3. Stokes’ First Problem for Sisko Fluid over a Porous Wall 
The study of time-dependent flow of an incompressible Sisko fluid over a wall with suction or blowing was performed by Hayat et al. . The magnetohydrodynamic nature of the fluid was taken into account by applying a variable magnetic field. The resulting nonlinear problem was solved by invoking the symmetry approach.
The problem governing the flow model  in a nondimensional form is given byThe symmetry analysis of (89) revealed that extra symmetries are admitted for the casesThe reductions of PDE (89) for these cases lead to nonlinear ordinary differential equations. However, the imposed boundary conditions are not invariant under the admitted Lie point symmetries. Hence, the governing model was then solved by making use of numerical techniques.
4.4. Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid 
Sari et al.  recently derived the boundary layer equations for a Sisko fluid. Using Lie group theory, a symmetry analysis of the equations was performed. A partial differential system is transferred to an ordinary differential system using symmetries and the resulting reduced equations were numerically solved.
The dimensionless form of the boundary layer equations for a Sisko fluid is The classical boundary conditions for the problem are The infinitesimals of the above system of PDEs are The corresponding similarity transformations areTransformations (97) are used to reduce the above PDE system to an ordinary differential system. The reduced ordinary differential system was solved by using a numerical method.
4.5. Analytic Approximate Solutions for Time-Dependent Flow and Heat Transfer of a Sisko Fluid 
The purpose of this study was to find analytic approximate solutions for unsteady flow and heat transfer of a Sisko fluid. Translational symmetries were utilized in  to find a family of travelling wave solutions of the governing nonlinear problem.
In dimensionless form, the governing problem takes the form with the boundary conditionsEquation (98) admits the Lie point symmetry generators and . The generator which represents a family of travelling wave with constant wave speed has been used in  to perform reduction of the above system of PDEs into nonlinear system of ODEs. The reduced system of ODEs was solved by homotopy analysis method.
4.6. Self-Similar Unsteady Flow of a Sisko Fluid in a Cylindrical Tube Undergoing Translation 
The governing equation for unsteady flow of a Sisko fluid in a cylindrical tube due to translation of the tube wall is modelled in . The reduction of the nonlinear problem was carried out by using Lie group approach. The partial differential equation is transformed into an ordinary differential equation, which was integrated numerically.
The unsteady flow of a Sisko fluid in a cylindrical tube due to impulsive motion of tube is governed by subject to the boundary conditionsThe Lie point symmetries for the PDE (101) are spanned by the operators where satisfies the linear partial differential equation (101). The operator has been used in  to deduce the similarity transformationsThe similarity transformations (104) are employed to reduce the partial differential equation (101) into a nonlinear ordinary differential equation. The reduced ordinary differential together with suitable boundary and initial conditions was solved by shooting method.
5. Jeffrey Fluid Flow Problems
Here we discuss the problems dealing with the flow of a Jeffrey fluid that are solved using the Lie group approach.
The constitutive equations for an incompressible Jeffrey fluid model arewithwhere and are the Cauchy stress tensor and the extra stress tensor, respectively, is the pressure, is the identity tensor, is the ratio of relaxation to retardation times, is the retardation time, is the shear rate, and the dots over the quantities indicate differentiation with respect to time.
5.1. Lie Point Symmetries and Similarity Solutions for an Electrically Conducting Jeffrey Fluid 
The only model available in the literature dealing with the flow of a Jeffrey fluid and solved by employing the Lie symmetry approach was studied by Mekheimer et al. . In their work, the equations for the two-dimensional incompressible fluid flow of an electrically conducting Jeffrey fluid are studied. A Lie symmetry analysis was performed and the group invariant solutions were derived.
The governing equations of the model  arewithThe relevant boundary conditions are of the form where is the velocity of the plate, is the magnetic fluid penetrating into the plate, and is the pressure deep in the magnetic fluid. The symmetries of the system of PDEs (107) found in  arewhere and are the arbitrary constants and and are the arbitrary functions of the variable only. With the use of symmetries given in (110), the group invariant solutions for the system of PDEs (107) are 
6. Williamson Fluid Flow Problems
In this section, we investigate the problems which deal with the flow of a Williamson fluid which are solved using the Lie symmetry approach.
The Cauchy stress tensor for a Williamson fluid model is given bywhereHere and are the limiting viscosities at zero and at infinite shear rate, respectively, and is a rheological parameter.
6.1. Boundary Layer Theory and Symmetry Analysis of a Williamson Fluid 
The first study available in the literature dealing with the flow of a Williamson fluid and solved by employing the Lie group approach was performed by Aksoy et al. . In , the boundary layer equations for a Williamson fluid are derived for the first time. Using Lie group theory, a symmetry analysis of the equations was performed. The partial differential system was converted to an ordinary differential system via symmetries and the resulting equations were numerically solved.
The governing problem of the flow model  iswhere , , and The classical boundary conditions for the problem areThe infinitesimals of the above system of PDEs are The corresponding similarity transformations areThe similarity transformations (117) are used to reduce the above PDE system into a system of nonlinear ordinary differential equations. The reduced ordinary differential system was solved by using numerical techniques.
6.2. Boundary Layer Flow of Williamson Fluid with Chemically Reactive Species Using Scaling Transformation and Homotopy Analysis Method 
The study of Williamson fluid flow with a chemically reactive species was made recently by Khan et al. . The governing equations of Williamson model in two-dimensional flows were constructed by using scaling group transformation. The series solution of the system of reduced nonlinear ordinary differential equations (ODEs) was obtained by using homotopy analysis method.
The equations governing the model  arewhere is the Weissenberg number and is a Reynolds number. The boundary conditions for the problem areThe Lie point symmetries of the system of PDEs (118) are The corresponding similarity transformations areThe similarity transformations (121) are utilized in  to reduce the above PDE system into a system of nonlinear ordinary differential equations. The reduced ordinary differential system was solved analytically by homotopy analysis method.
7. Second-Grade Fluid Flow Problems
In this section, we present the studies related to flow of a second-grade fluid model that are solved by the Lie symmetry reduction method.
The constitutive equation for an incompressible homogeneous Rivlin-Ericksen fluid of second grade is given by the following relation: