Closed-Form Solutions for a Nonlinear Partial Differential Equation Arising in the Study of a Fourth Grade Fluid Model

Abstract

The unsteady unidirectional flow of an incompressible fourth grade fluid bounded by a suddenly moved rigid plate is studied. The governing nonlinear higher order partial differential equation for this flow in a semiinfinite domain is modelled. Translational symmetries in variables 𝑡 and 𝑦 are employed to construct two different classes of closed-form travelling wave solutions of the model equation. A conditional symmetry solution of the model equation is also obtained. The physical behavior and the properties of various interesting flow parameters on the structure of the velocity are presented and discussed. In particular, the significance of the rheological effects are mentioned.

1. Introduction

There has been substantial progress in the study of the behavior and properties of viscoelastic fluids over the past couple of years. This progress is due to the fact that these viscoelastic materials are involved in many manufacturing processes in today’s industry. Modelling viscoelastic flows is important for understanding and predicting the behavior of processes and thus for designing optimal flow configurations and for selecting operating conditions. Most of the viscoelastic fluids in industry do not adhere to the commonly accepted assumption of a linear relationship between the stress and the rate of strain and thus are characterized as non-Newtonian fluids. As a result of their complex physical structure and diversity in nature, these materials cannot have a single constitutive relation which describes all the properties of viscoelastic fluids. The flow properties of viscoelastic (non-Newtonian) fluids are quite different from those of viscous and Newtonian fluids. Therefore, in practical applications, one cannot replace the behavior of non-Newtonian fluids with Newtonian fluids and it is very important to analyze the flow behavior of non-Newtonian fluids in order to obtain a thorough understanding and to improve the utilization in various manufactures.

The most interesting and important task that we need to address when dealing with the flow problems of non-Newtonian fluids is that the governing equations of these models are of higher order and much more complicated as compared with Navier-Stokes equations. Such fluids have been modelled by constitutive equations which vary greatly in complexity. The resulting nonlinear equations are not easy to solve analytically. However, various researchers [1–11] in the field have recently engaged themselves in finding analytical solutions for such flow problems.

To date, very little attention has been given on the flows of fourth grade fluids [12, 13]. This model is the most generalized model among the differential type fluids. The fourth grade fluid model is known to capture most of the non-Newtonian flow properties at one time. This model is known to have interesting non-Newtonian flow properties such as shear thinning and shear thickening that many other non-Newtonian models do not exhibit. This model is also capable of predicting the normal stress effects that lead to phenomena like “die-swell” and “rod-climbing” [14]. With these facts in mind, we have considered a fourth grade fluid model in this study. In general, the governing equations for the flow problems of fourth grade fluids are up to fifth-order nonlinear equations. Literature survey shows that very limited studies are reported and these investigations further narrow down when we speak about closed-form solutions of these problems. However, there are few investigations available in the literature in which researchers have utilized various approaches to construct solutions of fourth grade fluid flow problems. For example, [15] Wang and Wu have tackled the problem for the unsteady flow of fourth grade fluid due to an oscillating plate using numerical methods. Recently, Siddiqui et al. [16] have obtained an optimal homotopy type solution for the thin film flow of a fourth grade fluid down a vertical cylinder by using the homotopy perturbation method (HPM). Hayat et al. [17–20] studied the fourth grade fluid problems in different types of flow situations by using the homotopy analysis method (HAM). The steady flow of a fourth grade fluid past a porous plate was treated by Marinca et al. [21] with the help of the optimal homotopy asymptotic method (OHAM). Despite all of these works in recent years, the exact closed-form solutions for the problems dealing with the flow of fourth grade fluids are still rare in the literature.

Lie’s theory of differential equations (see [22, 23]) was inaugurated and utilized in the solution of differential equations by the Norwegian mathematician Marius Sophus Lie in the 1870s. The main motive of the Lie symmetry analysis formulated by Lie is to find one or several parameters local continuous transformation groups leaving the equations invariant. Thus, the symmetry method for differential equations provide a rigorous way to classify them according to their invariance properties. This allows us to obtain group invariant and partially invariant solutions of differential equations in a tractable manner. The Lie symmetry approach has thus become of great importance in the theory and applications of differential equations and widely applied by several authors to solve difficult nonlinear problems particularly dealing with the flows of non-Newtonian fluids [24–28]. The conditional symmetry approach or what is called the nonclassical symmetry approach, which is an extension of the Lie approach, was originated in the work of Bluman and Cole [29]. There are equations arising in applications that do not admit Lie point symmetries but have conditional symmetries. Thus, this method is also powerful in obtaining exact solutions of such problems. In recent years, interest in the conditional symmetry approach has intensified. This method has been used successfully to obtain new exact solutions for a number of interesting nonlinear PDEs [30–32].

Motivated by the above-mentioned analysis, the aim of the present work is to venture further in the regime of fourth grade fluid. We investigate the time-dependent flow of an incompressible fourth grade fluid over a flat rigid plate. The motion is caused due to the motion of the plate in its own plane (𝑥𝑧-plane) with an arbitrary velocity 𝑉(𝑡). Reductions and exact solutions of the governing nonlinear PDE for the unidirectional flow of a fourth grade fluid are established using the classical and the conditional symmetry approaches. Finally, the influence of physically applicable parameters of the flow model are studied through several graphs and appropriate conclusions are drawn.

2. Fundamental Equations

The basic equations governing the time-dependent flow of an incompressible fluid are the continuity equation and the momentum equation, namely, div𝐕=𝟎,𝜌𝑑𝐕𝑑𝑡=div𝐓,(2.1) where 𝐕 is the velocity vector, 𝜌 the density of the fluid, 𝑑/𝑑𝑡 the material time derivative, and 𝐓 the Cauchy stress tensor. For a fourth grade fluid, the Cauchy stress tensor satisfies the constitutive equations [17, 18] 𝐓=−𝑝𝐈+^𝑛_𝑗=1𝐒_𝑗with𝑛=4,(2.2) where 𝑝 is the pressure, 𝐈 the identity tensor, and 𝐒_𝑗 the extra stress tensor 𝐒₁=𝜇𝐀₁,𝐒₂=𝛼₁𝐀₂+𝛼₂𝐀²₁,𝐒₃=𝛽₁𝐀₃+𝛽₂𝐀₁𝐀₂+𝐀₂𝐀₁+𝛽₃tr𝐀²₁𝐀₁,𝐒₄=𝛾₁𝐀₄+𝛾₂𝐀₃𝐀₁+𝐀₁𝐀₃+𝛾₃𝐀²₂+𝛾₄𝐀₂𝐀²₁+𝐀²₁𝐀₂+𝛾₅tr𝐀₂𝐀₂+𝛾₆tr𝐀₂𝐀²₁+𝛾₇tr𝐀₃+𝛾₈tr𝐀₂𝐀₁𝐀₁.(2.3) Here 𝜇 is the dynamic viscosity, 𝛼_𝑖 (𝑖=1,2), 𝛽_𝑖 (𝑖=1,2,3), and 𝛾_𝑖 (𝑖=1,2,…,8) are material constants. The Kinematical tensors 𝐀₁ to 𝐀₄ are the Rivlin-Ericksen tensors defined by 𝐀₁=grad𝐕+grad𝐕^𝑇,𝐀(2.4)_𝑛=𝑑𝐀_𝑛−1𝑑𝑡+𝐀_𝑛−1grad𝐕+grad𝐕^𝑇𝐀_𝑛−1;(𝑛>1),(2.5) in which grad is the gradient operator.

3. Flow Development

Let an infinite rigid plate occupy the plane 𝑦=0 and a fourth grade fluid the half-space 𝑦>0. The 𝑥-axis and 𝑦-axis are chosen parallel and perpendicular to the plate. For 𝑡>0, the plate moves in its own plane with arbitrary velocity 𝑉(𝑡). By taking the velocity field (𝑢(𝑦,𝑡),0,0), the conservation of mass equation is identically satisfied. The governing PDE in 𝑢 is obtained by substituting (2.2)–(2.5) into (2.1) and rearranging. We deduce the following governing equation in the absence of the modified pressure gradient: 𝜌𝜕𝑢𝜕𝜕𝑡=𝜇²𝑢𝜕𝑦²+𝛼₁𝜕³𝑢𝜕𝑦²𝜕𝑡+𝛽₁𝜕⁴𝑢𝜕𝑦²𝜕𝑡²+6𝛽₂+𝛽₃𝜕𝑢𝜕𝑦²𝜕²𝑢𝜕𝑦²+𝛾₁𝜕⁵𝑢𝜕𝑦²𝜕𝑡³+6𝛾₂+2𝛾₃+2𝛾₄+2𝛾₅+6𝛾₇+2𝛾₈𝜕𝜕𝑦𝜕𝑢𝜕𝑦²𝜕²𝑢.𝜕𝑦𝜕𝑡(3.1) The above equation is subject to the following boundary and initial conditions: 𝑢(0,𝑡)=𝑈₀𝑉(𝑡)𝑢()𝑢𝑦,𝑡>0,(3.2)∞,𝑡=0,𝑡>0,(3.3)𝑦,0=𝐼,𝑦>0,(3.4)𝜕𝑢𝑦,0𝑦𝜕𝜕𝑡=𝐽,𝑦>0,(3.5)₂𝑢𝑦,0𝜕𝑡²=𝐾𝑦,𝑦>0,(3.6) where 𝑈₀ is the reference velocity and 𝑉(𝑡), 𝐼(𝑦), 𝐽(𝑦), 𝐾(𝑦) are the unspecified functions. The first boundary condition (3.2) is the no-slip condition and the second boundary condition (3.3) says that the main stream velocity is zero. This is not a restrictive assumption since we can always measure velocity relative to the main stream. The initial condition (3.4) indicates that initially the fluid is moving with some nonuniform velocity 𝐼(𝑦). The remaining two initial conditions are the extra two conditions imposed to make the problem well posed.

We define the dimensionless parameters as 𝑢𝑢=𝑈₀,𝑈𝑦=⁰𝑦𝜈,𝑈𝑡=²⁰𝑡𝜈,𝛼𝛼=¹𝑈²₀𝜌𝜈²,𝛽₁=𝛽₁𝑈⁴₀𝜌𝜈³,6𝛽𝛽=²+𝛽₃𝑈⁴₀𝜌𝜈³,𝛾₁=𝛾₁𝑈⁶₀𝜌𝜈⁴,𝛾=3𝛾₂+𝛾₃+𝛾₄+𝛾₅+3𝛾₇+𝛾₈𝑈⁶₀𝜌𝜈⁴.(3.7) Under these transformations, the governing (3.1) and the corresponding initial and the boundary conditions (3.2)–(3.6) take the form 𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑦²𝜕+𝛼³𝑢𝜕𝑦²𝜕𝑡+𝛽₁𝜕⁴𝑢𝜕𝑦²𝜕𝑡²+𝛽𝜕𝑢𝜕𝑦²𝜕²𝑢𝜕𝑦²+𝛾₁𝜕⁵𝑢𝜕𝑦²𝜕𝑡³𝜕+2𝛾𝜕𝑦𝜕𝑢𝜕𝑦²𝜕²𝑢,𝑢()(𝑡)𝑢()𝑢𝑦𝜕𝑦𝜕𝑡(3.8)0,𝑡=𝑉,𝑡>0,∞,𝑡=0,𝑡>0,𝑦,0=𝑓,𝑦>0,𝜕𝑢𝑦,0𝑦𝜕𝜕𝑡=𝑔,𝑦>0,₂𝑢𝑦,0𝜕𝑡²=ℎ𝑦,𝑦>0,(3.9) where 𝑓(𝑦)=𝐼(𝑦)/𝑈₀, 𝑔(𝑦)=𝐽(𝑦)/𝑈₀ and ℎ(𝑦)=𝐾(𝑦)/𝑈₀. The functions 𝑉(𝑡), 𝑓(𝑦), 𝑔(𝑦), and ℎ(𝑦) are as yet arbitrary. These functions are constrained in the next section when we seek closed-form solutions using the symmetry technique. For simplicity, we have neglected the bars in all the nondimensional quantities. We consider classical and conditional symmetries of (3.8). Equation (3.8) only admits translational symmetries in variables 𝑡, 𝑦, and 𝑢; thus, the travelling wave solutions are investigated in Section 4. In Section 5, we provide a conditional symmetry solution as well.

4. Travelling Wave Solutions

Travelling wave solutions are special kinds of group invariant solutions which are invariant under a linear combination of time-translation and space-translation symmetry generators. It can easily be seen that (3.8) admits Lie’s point symmetry generators, 𝜕/𝜕𝑡 (time-translation) and 𝜕/𝜕𝑦 (space-translation in 𝑦), so that we can construct travelling wave solutions for the model equation.

4.1. Backward Wave-Front Type Travelling Wave Solutions

Let 𝑋₁ and 𝑋₂ be time-translation and space-translation symmetry generators, respectively. Then the solution corresponding to the generator 𝑋=𝑋₁−𝑐𝑋₂,(𝑐>0)(4.1) would represent backward wave-front type travelling wave solutions. In this case, the waves are propagating towards the plate. The Lagrangian system corresponding to (4.1) is 𝑑𝑦=−𝑐𝑑𝑡1=𝑑𝑢0.(4.2) Solving (4.2), invariant solutions are given by 𝑢𝑦,𝑡=𝐹(𝜉)with𝜉=𝑦+𝑐𝑡.(4.3) Making use of (4.3) in (3.8) results in a fifth-order ordinary differential for 𝐹(𝜉)𝑐𝑑𝐹=𝑑𝑑𝜉²𝐹𝑑𝜉²𝑑+𝛼𝑐³𝐹𝑑𝜉³+𝛽₁𝑐²𝑑⁴𝐹𝑑𝜉⁴+𝛽𝑑𝐹𝑑𝜉²𝑑²𝐹𝑑𝜉²+𝛾₁𝑐³𝑑⁵𝐹𝑑𝜉⁵𝑑+2𝛾𝑐𝑑𝜉𝑑𝐹𝑑𝜉²𝑑²𝐹𝑑𝜉².(4.4) Thus, the PDE (3.8) became an ODE (4.4) along certain curves in the 𝑦−𝑡 plane. These curves are called characteristic curves or just the characteristics. In order to solve (4.4) for 𝐹(𝜉), we assume a solution of the form 𝐹(𝜉)=𝐴exp(𝐵𝜉),(4.5) where 𝐴 and 𝐵 are the constants to be determined. Inserting (4.5) in (4.4) we obtain −𝑐𝐵+𝐵²+𝑐𝛼𝐵³+𝛽₁𝑐²𝐵⁴+𝛾𝑐³𝐵⁵+𝑒^2𝐵𝜉𝛽𝐴²𝐵⁴+6𝑐𝛾𝐴²𝐵⁵=0.(4.6) Separating (4.6) in powers of 𝑒⁰ and 𝑒^2𝐵𝜉, we find 𝑒⁰∶−𝑐𝐵+𝐵²+𝑐𝛼𝐵³+𝛽₁𝑐²𝐵⁴+𝛾𝑐³𝐵⁵𝑒=0,(4.7)_2𝐵𝜉∶𝛽𝐴²𝐵⁴+6𝑐𝛾𝐴²𝐵⁵=0.(4.8) From (4.8), we deduce 𝐵=−𝛽.6𝑐𝛾(4.9) Using the value of 𝐵 in (4.7), we obtain 𝛽+𝛽6𝛾6𝑐𝛾²²−𝛽³𝛼𝑐6𝑐𝛾³+𝛽₁𝛽⁴𝑐²6𝑐𝛾⁴−𝛾𝛽⁵𝑐³6𝑐𝛾⁵=0.(4.10) Thus, the exact solution for 𝐹(𝜉) (provided the condition (4.10) holds) can be written as 𝐹(𝜉)=𝐴exp−𝛽𝜉.6𝑐𝛾(4.11) So the exact solution 𝑢(𝑦,𝑡) which satisfies the condition (4.10) is 𝑢𝑦,𝑡=exp−𝛽𝑦+𝑐𝑡6𝑐𝛾with𝑐>0.(4.12) The solution (4.12) satisfies the initial and boundary conditions (3.9) for the particular values of the unspecified functions 𝑉(𝑡), 𝑓(𝑦), 𝑔(𝑦), and ℎ(𝑦). Using (4.12) in (3.9) results in 𝑢(0,𝑡)=𝑉(𝑡)=exp−𝛽𝑐𝑡,()𝑢𝑦6𝑐𝛾4.13a𝑦,0=𝑓=exp−𝛽𝑦,()6𝑐𝛾4.13b𝜕𝑢𝑦,0𝑦=𝜕𝑡=𝑔−𝛽6𝛾exp−𝛽𝑦,()𝜕6𝑐𝛾4.13c₂𝑢𝑦,0𝜕𝑡²𝑦=𝛽=ℎ²6𝛾²exp−𝛽𝑦.()6𝑐𝛾4.13d Here 𝑉(𝑡), 𝑓(𝑦), 𝑔(𝑦), and ℎ(𝑦) depend on the physical parameters of the flow. The solution (4.12) is plotted in Figures 1–4 for different values of the emerging parameters.

Figure 1 

Backward wave-front type travelling wave solution (4.12) varying 𝑡 when 𝛾=0.5, 𝛽=2, and 𝑐=1 are fixed.

Figure 2 

Backward wave-front type travelling wave solution (4.12) varying 𝑐 when 𝛾=0.6, 𝛽=4, and 𝑡=𝜋/2 are fixed.

Figure 3 

Backward wave-front type travelling wave solution (4.12) varying 𝛽 when 𝛾=0.2, 𝑐=1, and 𝑡=𝜋/2 are fixed.

Figure 4 

Backward wave-front type travelling wave solution (4.12) varying 𝛾 when 𝛽=3, 𝑐=1, and 𝑡=𝜋/2 are fixed.

4.2. Forward Wave-Front Type Travelling Wave Solutions

We look for invariant solutions under the operator 𝑋₁+𝑐𝑋₂ (with 𝑐>0) which represent forward wave-front type travelling wave solutions with constant wave speed 𝑐. In this case, the waves are propagating away from the plate. These are solutions of the form𝑢𝑦,𝑡=𝐺𝜂with𝜂=𝑦−𝑐𝑡.(4.14) Using (4.14) in (3.8) results in a fifth-order ordinary differential equation for 𝐺(𝜂), −𝑐𝑑𝐺=𝑑𝑑𝜂²𝐺𝑑𝜂²𝑑−𝛼𝑐³𝐺𝑑𝜂³+𝛽₁𝑐²𝑑⁴𝐺𝑑𝜂⁴+𝛽𝑑𝐺𝑑𝜂²𝑑²𝐺𝑑𝜂²−𝛾₁𝑐³𝑑⁵𝐺𝑑𝜂⁵𝑑−2𝛾𝑐𝑑𝜂𝑑𝐺𝑑𝜂²𝑑²𝐺𝑑𝜂².(4.15) Following the same methodology adopted for the backward wave-front type travelling wave solutions, the above equation admits exact solutions of the form 𝐺𝜂=𝐴exp𝛽𝜂,6𝑐𝛾(4.16) provided 𝛽+𝛽6𝛾6𝑐𝛾²²−𝛽³𝛼𝑐6𝑐𝛾³+𝛽₁𝛽⁴𝑐²6𝑐𝛾⁴−𝛾𝛽⁵𝑐³6𝑐𝛾⁵=0.(4.17) Thus, (3.8) subject to (4.17) admits the exact solution 𝑢𝛽𝑦,𝑡=exp𝑦−𝑐𝑡6𝑐𝛾with𝑐>0.(4.18) Note that the solution (4.18) does not satisfy the second boundary condition at infinity but satisfies the rest of the boundary conditions for the particular values of the functions 𝑉(𝑡), 𝑓(𝑦), 𝑔(𝑦), and ℎ(𝑦). Using (4.18) in (3.9) gives 𝑢(0,𝑡)=𝑉(𝑡)=exp−𝛽𝑐𝑡,()𝑢𝑦6𝑐𝛾4.19a𝑦,0=𝑓=exp𝛽𝑦,()6𝑐𝛾4.19b𝜕𝑢𝑦,0𝑦=𝜕𝑡=𝑔−𝛽6𝛾exp𝛽𝑦,()𝜕6𝑐𝛾4.19c₂𝑢𝑦,0𝜕𝑡²𝑦=𝛽=ℎ²6𝛾²exp𝛽𝑦.()6𝑐𝛾4.19d

5. Conditional Symmetry Solution

We consider the PDE (3.8) with the invariant surface condition 𝑢_𝑡=𝜆𝑢,(5.1) where 𝜆 is a constant to be found. This corresponds to the operator 𝜕𝑋=𝜕𝜕𝑡+𝜆𝑢.𝜕𝑢(5.2) The invariant solution corresponding to (5.2) is 𝑢𝑦,𝑡=exp(𝜆𝑡)𝐹𝑦,(5.3) where 𝐹(𝑦) is an undetermined function of 𝑦.

Substituting (5.3) in (3.8) leads to a linear second-order ordinary differential equation in 𝐹(𝑦), given by 𝑑²𝐹𝑑𝑦²−𝜆1+𝛼𝜆+𝛽₁𝜆²+𝛾₁𝜆³𝐹=0.(5.4) and the relation 𝛽+6𝛾𝜆=0.(5.5) From (5.5), we deduce 𝛽𝜆=−.6𝛾(5.6) Using the value of 𝜆 in (5.4), we get 𝑑²𝐹𝑑𝑦²−−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝐹=0,(5.7) The reduced ODE (5.7) is solved subject to the boundary conditions 𝐹(0)𝐹𝑦=1,→0as𝑦→∞.(5.8) Now we solve (5.7) subject to the boundary conditions (5.8) for three different cases.

Case 1 ([(−𝛽/6𝛾)/(1−𝛼(𝛽/6𝛾)+𝛽₁(𝛽/6𝛾)²−𝛾₁(𝛽/6𝛾)³)]=0). We have 𝑑²𝐹𝑑𝑦²=0.(5.9) Solution of (5.9) subject to the boundary conditions (5.8) is 𝐹𝑦=constantsay𝐶.(5.10) In this case, the solution for 𝑢(𝑦,𝑡) is 𝑢(𝑡)−𝛽=𝐶exp𝑡.6𝛾(5.11) Thus, a time-dependent solution is obtained in this case.

Case 2 ([(−𝛽/6𝛾)/(1−𝛼(𝛽/6𝛾)+𝛽₁(𝛽/6𝛾)²−𝛾₁(𝛽/6𝛾)³)]<0). In this case, the solution for 𝐹(𝑦) is written as 𝐹𝑦=𝑎cos−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦+𝑏sin−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦.(5.12) Since 𝐹(∞)=0, an unbounded solution is obtained for 𝐹(𝑦). Thus, the solution for 𝑢(𝑦,𝑡) does not exist in this case.

Case 3 ([(−𝛽/6𝛾)/(1−𝛼(𝛽/6𝛾)+𝛽₁(𝛽/6𝛾)²−𝛾₁(𝛽/6𝛾)³)]>0). In this case, the solution for 𝐹(𝑦) is given by 𝐹𝑦=𝑎exp−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦+𝑏exp−−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦,(5.13) By applying the boundary conditions (5.8), we have 𝑎=0 (for a bounded solution) and 𝑏=1 (by using first the boundary condition). Thus, the solution for 𝐹(𝑦) is 𝐹𝑦=exp−−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦.(5.14) Substituting 𝐹(𝑦) into (5.3), we obtain the solution for 𝑢(𝑦,𝑡) in the form 𝑢−𝛽𝑦,𝑡=exp6𝛾𝑡+−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦.(5.15) Note that the conditional symmetry solution (5.15) satisfies the initial and the boundary conditions (3.9)–(21) with 𝑉(𝑡)−𝛽=exp𝑡,𝑓𝑦−6𝛾=exp−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦,𝑔𝑦𝛽=−−6𝛾exp−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦,ℎ𝑦=𝛽6𝛾²exp−−𝛽/6𝛾1−𝛼𝛽/6𝛾+𝛽₁𝛽/6𝛾²−𝛾₁𝛽/6𝛾³𝑦,(5.16) where 𝑉(𝑡), 𝑓(𝑦), 𝑔(𝑦), and ℎ(𝑦) depend on physical parameters of the flow. The graphical behavior of solution (5.15) is shown in Figure 5.

Figure 5 

Conditional symmetry solution (5.15) varying 𝑡 when 𝛾=0.2, 𝛾1=0.1, 𝛽=6, 𝛽1=0.5, and 𝛼=1 are fixed.

Remark 5.1. The backward wave-front type travelling wave closed-form solution (4.12) and the conditional symmetry solution (5.15) best represent the physics of the problem considered in the sense that these solutions satisfy all the initial and the boundary conditions and also show the effects of different emerging parameters of the flow problem given in Figures 1–5. The forward wave-front type travelling wave solution (4.18) does not satisfy the second boundary condition at infinity. As a consequence, it does not show the behavior of the physical model. But this solution does show the shockwave behavior of the flow. To emphasize, we say that the forward wave-front type travelling wave solution is actually a shockwave solution with slope approaching infinity along the characteristic 𝜂=𝑦−𝑐𝑡, as shown in Figure 6.

Figure 6 

Plot of the forward wave-front type travelling wave solution (4.18) when 𝛾=0.5, 𝛽=3, 𝑐=1, and 𝑡=𝜋/2 are fixed.

6. Physical Interpretation of the Results

In order to explain and understand the physical structure of the flow problem, various graphs are plotted in Figures 1–6.

Figure 1 shows the influence of time 𝑡 on the backward wave-front type travelling wave solution (4.12). This figure depicts that velocity decreases as time increases. Clearly, the variation of velocity is observed for 0⩽𝑡⩽5. For 𝑡>5, the velocity profile remains the same. In other words, we can say that the steady-state behavior of the velocity is achieved for 𝑡>5.

Figure 2 shows the effects of the wave speed 𝑐 on the velocity profile. It is clearly observed that with the increase of wave speed 𝑐, the velocity profile is increasing. So in this way, we can remark that both time 𝑡 and the wave speed 𝑐 have opposite effects on the backward wave-front type travelling wave solution (4.12) of the governing model.

Figures 3 and 4 have been plotted to see the influence of the third grade parameter 𝛽 and fourth grade parameter 𝛾 on the backward wave-front type travelling wave solution (4.12) of the flow problem. These figures reveal that both 𝛽 and 𝛾 have an opposite behavior on the structure of the velocity, that is, with an increase in the third grade parameter 𝛽, the velocity profile decreases showing the shear thickening behavior of the fluid, whereas the velocity field increases for increasing values of the fourth grade parameter 𝛾, which shows the shear thinning behavior of the model. This is in accordance with the fact that a fourth grade fluid model predicts both shear thickening and the shear thinning properties of the flow.

In Figure 5, the conditional symmetry solution (5.15) is plotted against the increasing values of time 𝑡. The behavior of time on the conditional symmetry solution is same as observed previously for the backward wave-front type travelling wave solution. That is, with the increase in time 𝑡, the velocity decreases. However, in this case, the steady-state behavior of the velocity is achieved quicker as compared to the backward wave-front type travelling wave solution. The variation of velocity is observed for 0⩽𝑡⩽0.7. For 𝑡>0.7, and the steady-state behavior of the velocity is observed.

In Figure 6, the forward wave-front type travelling wave solution (4.18) is plotted. This figure describes the shock wave behavior of the flow with slope approaching infinity along the characteristic. This solution does not show the physics of the model but does predict the hidden shock wave phenomena in the flow. Some examples of shock waves are moving shock, detonation wave, detached shock, attached shock, recompression shock, shock in a pipe flow, shock waves in rapid granular flows, shock waves in astrophysics, and so on.

7. Concluding Remarks

The present work has been undertaken in order to investigate further the regime of a fourth grade non-Newtonian fluid model. Some reductions and exact (closed-form) solutions for the time-dependent flow of a fourth grade fluid have been established using the symmetry approach. Travelling wave and conditional symmetry solutions are obtained for the governing nonlinear PDE. Both forward and the backward wave-front type travelling wave solutions have been constructed. The better solutions from the physical point of view of the model considered are the conditional symmetry solution and the backward wave-front type travelling wave solution. This issue has also been addressed in detail in Remark 5.1. Moreover, the work contained herein is theoretical in nature and is a prototype model. The methods used will be helpful for a wide range of nonlinear problems in fluids given the paucity of known exact solutions especially in non-Newtonian fluids.

Acknowledgments

T. Aziz thanks the University of the Witwatersrand for the scholarship. The author also thanks the organizers of the conference SDEA 2012 for their warm hospitality.