Shock Wave Solutions for Some Nonlinear Flow Models Arising in the Study of a Non-Newtonian Third Grade Fluid

Abstract

This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in the study of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinear partial differential equations into nonlinear ordinary differential equations. The reduced equations are then solved analytically, and the shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally in the process of the derivation of the solutions.

1. Introduction

A shock wave is a disturbance that propagates through a media. Shock is yielded when a disturbance is made to move through a fluid faster than the speed of sound (the celerity) of the medium. This can occur when a solid object is forced through a fluid. It represents a sharp discontinuity of the parameters that delineate the media. Unlike solutions where the energy is a conserved quantity and thus remains constant during its propagation, shock wave dissipates energy relatively quickly with distance. One source of a shock wave is when the supersonic jets fly at a speed that is greater than the speed of sound. This results in the drag force on aircraft with shocks. These waves also appear in various interesting phenomena in real life situations. For example, solitons appear in the propagation of pulses through optical fibers. Another example is where cnoidal waves appear in shallow water waves, although this is an extremely scarce phenomena. Some interesting communications dealing with the shock wave solutions are found in [1–3].

During the past several decades, the study of the nonlinear problems dealing with the flow of non-Newtonian fluids has attracted considerable attention. This interest is due to several important applications in engineering and industry such as reactive polymer flows in heterogeneous porous media, electrochemical generation of elemental bromine in porous electrode systems, manufacture of intumescent paints for fire safety applications, extraction of crude oil from the petroleum products, synthetic fibers, and paper production [4]. Due to the diverse physical nature of non-Newtonian fluids, there is no single constitutive expression which describe the physical behavior of all non-Newtonian fluid models. Because of this important issue, several models of non-Newtonian fluids have been proposed in the literature. Together with this factor, the mathematical modelling of non-Newtonian incompressible fluid flows gives rise to nonlinear and complicated differential equations. As a consequence of this nonlinearity factor, the exact (closed-form) solutions of these sort of problems are scarce in general. Several techniques and methods have been developed in the recent few years to construct the solutions of these non-Newtonian fluid flow problems. Some of these useful methods are variational iteration method, Adomian decomposition method, homotopy perturbation method, homotopy analysis method, and semi-inverse variational method. Literature survey witnesses that, despite all these methods, the exact (closed-form) solutions of the non-Newtonian fluid flow problems are still rare in the literature, and it is not surprising that new exact (closed-form) solutions are most welcome, provided they correspond to physically realistic situations. Some interesting and useful communications in this area are made in the studies [5–14].

One of the widely accepted models amongst non-Newtonian fluids is the class of Rivlin-Ericksen fluids of differential type [15]. Rivlin-Ericksen fluids of differential type have acquired the special status in order to describe the several nonstandard features such as normal stress effects, rod climbing, shear thinning, and shear thickening. In the literature much attention has been given to the flow of a second grade fluid [16–19]. A second grade fluid model is the simplest subclass of non-Newtonian fluids for which one can reasonably hope to obtain an analytical solution. In most of the flow situations, the governing equations for a second grade fluid are linear. Although a second grade fluid model is able to predict the normal stress differences, it does not take into account the shear thinning, and shear thickening phenomena that many fluids reveal. Therefore some experiments may be well described by third grade fluid [20–24]. The mathematical model of a third grade fluid represents a more realistic description of the behavior of non-Newtonian fluids. A third grade fluid model represents a further attempt towards the study of the flow properties of non-Newtonian fluids. Therefore, a third grade fluid model has been considered in this study. This model is known to capture the non-Newtonian effects such as shear thinning or shear thickening as well as normal stress.

In this particular study, we have constructed the shock wave solutions of some nonlinear PDEs arising in the study of third grade fluid flow in porous medium. We know that the flow of non-Newtonian fluids in a porous medium has wide range of engineering applications. These include enhanced oil recovery, ceramic processing, and geothermal engineering. Motivated by these facts, we have developed some shock wave solutions of three nonlinear problems dealing with the unsteady flow of third grade fluid in a porous half space.

2. Geometry of the Models

Consider a Cartesian coordinate frame of reference 𝑂𝑋𝑌𝑍 with 𝑥-axis along the direction of the flow and 𝑦-axis pointing in the vertically upward direction. The third grade fluid occupies the porous space 𝑦>0 and is in contact with an infinite plate at 𝑦=0. Since the plate is infinite in the 𝑋𝑍-plane, therefore all the physical quantities except the pressure depend on 𝑦 only. We have taken three different problems on the same flat plate geometry. The geometry of the problems is shown in Figure 1.

Figure 1 

Geometry of the physical models and coordinate system.

3. Problems to Be Investigated

3.1. Unsteady Flow of a Third Grade Fluid over a Flat Rigid Plate with Porous Medium

Following the methodology of [22, 23], the unsteady incompressible flow of a third grade fluid over the rigid plate with porous medium is governed by 𝜌+𝛼₁𝜑𝜅𝜕𝑢𝜕𝜕𝑡=𝜇²𝑢𝜕𝑦²+𝛼₁𝜕³𝑢𝜕𝑦²𝜕𝑡+6𝛽₃𝜕𝑢𝜕𝑦²𝜕²𝑢𝜕𝑦²−2𝛽₃𝜑𝜅𝜕𝑢𝜕𝑦²𝜑𝑢−𝜅𝜇𝑢, (1) where 𝑢 is the velocity component in 𝑥-direction, 𝑡 is time, 𝑦 is spatial variable, 𝜌 is the density, 𝜇 is the coefficient of viscosity, 𝛼₁ and 𝛽₃ are the material constants (for details on these material constants and the conditions that are satisfied by these constants, the reader is referred to [24]), 𝜑 is the porosity and 𝜅 is the permeability of the porous medium.

In order to solve (1), the relevant boundary and initial conditions are 𝑢(0,𝑡)=𝑢₀𝑔(𝑡)𝑢()𝑢𝑦,𝑡>0,∞,𝑡=0,𝑡>0,𝑦,0=𝑓,𝑦>0, (2)(3)(4) where 𝑢₀ is the reference velocity and 𝑔(𝑡) and 𝑓(𝑦) are yet functions to be determined. The first boundary condition (2) is the no-slip condition, and the second boundary condition (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 (4) indicates initially that the fluid is moving with some nonuniform velocity 𝑓(𝑦).

3.1.1. Reduction of the Governing Equation

We know that from the principal of Lie symmetry methods [25, 26] that if a differential equation is explicitly independent of any dependent or independent variable, then this particular differential equation remains invariant under the translation symmetry corresponding to that particular variable. We noticed that (1) admits Lie point symmetry generators, 𝜕/𝜕𝑡 (translation in 𝑡) and 𝜕/𝜕𝑦 (translation in 𝑦). Let 𝑋₁ and 𝑋₂ be time-translation and space-translation symmetry generators, respectively. Then the solution corresponding to the generator 𝑋=𝑋₁+𝑚𝑋₂=𝜕𝜕𝜕𝑡+𝑚,()𝜕𝑦𝑚>0 (5) would represent travelling wave solution with constant wave speed 𝑚. Travelling wave solutions are characterized by the fact that the profiles of these solutions at different time instants are obtained from one another by appropriate shifts (translations) along the 𝑦-axis. Consequently, a Cartesian coordinate system moving with the constant speed can be introduced in which the profile of the derived quantity is stationary. 𝑚=0 represents the stationary or steady-state solutions.

The characteristic system corresponding to (5) is 𝑑𝑦𝑚=𝑑𝑡1=𝑑𝑢0. (6) Solving (6), invariant solutions are given by 𝑢𝑦,𝑡=𝐹𝜂with𝜂=𝑦−𝑚𝑡, (7) where 𝐹(𝜂) is an arbitrary function of the characteristic variable 𝜂=𝑦−𝑚𝑡. Making use of (7) into (1) results in a third-order ordinary differential for 𝐹(𝜂) as follows: 𝜌+𝛼₁𝜑𝜅𝑚𝑑𝐹𝑑𝑑𝜂+𝜇²𝐹𝑑𝜂²−𝑚𝛼₁𝑑³𝐹𝑑𝜂³+6𝛽₃𝑑𝐹𝑑𝜂²𝑑²𝐹𝑑𝜂²−2𝛽₃𝜑𝜅𝐹𝑑𝐹𝑑𝜂²𝜑−𝜇𝜅𝐹=0. (8) Thus the original third-order nonlinear PDE (1) is reduced to a third-order ODE (5) along certain curves in the 𝑦-𝑡 plane. These curves are called characteristic curves or just the characteristic.

3.1.2. Shock Wave Solution

In this section, we show that the travelling wave solutions of (1) approach a shock wave solution. Now we construct the shock wave solution of the reduced equation (5). The starting hypothesis for shock wave solution is given by 𝐹𝜂=𝑎exp𝑏𝜂, (9) where 𝑎 and 𝑏 are the free parameters to be determined. Substituting (9) into (8), we obtain 𝑚𝜌+𝛼₁𝜑𝜅𝑏+𝜇𝑏²−𝑚𝛼₁𝑏³𝜑−𝜇𝜅+𝑒^2𝑏𝜂6𝛽₃𝑎²𝑏⁴−2𝛽₃𝜑𝜅𝑎²𝑏²=0. (10) Separating (10) in the powers of 𝑒⁰ and 𝑒^2𝑏𝜂, we find 𝑒⁰∶𝑚𝜌+𝛼₁𝜑𝜅𝑏+𝜇𝑏²−𝑚𝛼₁𝑏³𝜑−𝜇𝜅𝑒=0,_2𝐵𝜂∶𝛽₃𝑎²3𝑏⁴−𝜑𝜅𝑏²=0,with𝛽₃𝑎²≠0. (11)(12) From (12), we deduce 𝑏=𝜑.3𝜅 (13) Using the value of 𝑏 in (11), we obtain 𝑚𝜌+𝛼₁𝜑𝜅𝜑𝜑3𝜅+𝜇3𝜅−𝑚𝛼₁𝜑3𝜅𝜑𝜑3𝜅−𝜇𝜅=0. (14) Finally, the solution for 𝐹(𝜂) (provided the condition (14) holds) is written as 𝐹𝜂=𝑎exp𝜑𝜂.3𝜅 (15) So the solution 𝑢(𝑦,𝑡) which satisfies the condition (14) is written as 𝑢𝑦,𝑡=𝑎exp𝜑3𝜅𝑦−𝑚𝑡with𝑚>0. (16)

Remark 1. Note that the solution (16) is the shock wave solution to the governing PDE (1). The previous solution is valid under the particular condition on the physical parameters of the flow given in (14) (which is some kind of dispersion relation in 𝑚). This solution does show the hidden shock wave behavior of the flow problem with slope of the velocity field or the velocity gradient approaches to infinity such that 𝜕𝑢𝜕𝑦→∞as𝑦→∞. (17)

Remark 2. Note that the solution (16) also satisfies the particular initial and the boundary condition; that is, 𝑢(0,𝑡)=𝑔(𝑡)=exp−𝜑𝑢𝑦3𝜅𝑚𝑡,𝑡>0,𝑦,0=𝑓=exp𝜑𝑦3𝜅,𝑦>0, (18) with 𝑔(0)=𝑓(0)=𝑎=1. (19) The functions 𝑔(𝑡) and 𝑓(𝑦) depend on the physical parameters of the flow.

Remark 3. We also observe that the physical significance of the imposing condition (14) is that it gives the speed of travelling shock wave. From (14), we deduce 𝑚=2𝜇𝜑/3𝜅^1/2𝜌+2𝛼₁𝜑/3𝜅>0. (20)

3.2. Unsteady Magnetohydrodynamic (MHD) Flow of Third Grade Fluid in a Porous Medium

By employing the same geometry as we have explained in Section 2, in this problem we extend the previous model by considering the fluid to be electrically conducting under the influence of a uniform magnetic field applied transversely to the flow. We provide the closed-form solution of the problem by reducing the governing nonlinear PDE into an ODE with the help of Lie reduction technique.

The time-dependent magnetohydrodynamic flow of a third grade fluid in a porous half space in the absence of the modified pressure gradient takes the form 𝜌+𝛼₁𝜑𝜅𝜕𝑣𝜕𝜕𝑡=𝜇²𝑣𝜕𝑦²+𝛼₁𝜕³𝑣𝜕𝑦²𝜕𝑡+6𝛽₃𝜕𝑣𝜕𝑦²𝜕²𝑣𝜕𝑦²−2𝛽₃𝜑𝜅𝜕𝑣𝜕𝑦²𝜑𝑢−𝜅𝜇𝑣−𝜎𝐵²₀𝑣, (21) where 𝑣 is the velocity component in 𝑥-direction, 𝜎 is the electrical conductivity, and 𝐵₀ is the uniform applied magnetic field. In order to solve (21), the relevant time and space dependent velocity boundary conditions are 𝑢(0,𝑡)=𝑢₀𝑔(𝑡)𝑢()𝑢𝑦,𝑡>0,∞,𝑡=0,𝑡>0,𝑦,0=𝑓,𝑦>0. (22) As it can be seen (21) also admits Lie point symmetry generators, 𝜕/𝜕𝑡 (translation in 𝑡) and 𝜕/𝜕𝑦 (translation in 𝑦). Let 𝑋₁ and 𝑋₂ be time-translation and space-translation symmetry generators, respectively. The invariant solution corresponding to the generator 𝑋=𝑋₁+𝑚𝑋₂ is given by 𝑣𝑦,𝑡=𝐺(𝜉)with𝜉=𝑦−𝑚𝑡. (23) Using (23) into (21) yields 𝜌+𝛼₁𝜑𝜅𝑚𝑑𝐺𝑑𝑑𝜉+𝜇²𝐺𝑑𝜉²−𝑚𝛼₁𝑑³𝐺𝑑𝜉³+6𝛽₃𝑑𝐺𝑑𝜉²𝑑²𝐺𝑑𝜉²−2𝛽₃𝜑𝜅𝐺𝑑𝐺𝑑𝜉²𝜑−𝜇𝜅𝐺−𝜎𝐵²₀𝐺=0. (24) Following the same methodology adopted to solve the previous problem, the reduced ODE (24) admits an exact solution of the form 𝐺(𝜉)=𝑎exp𝜑𝜉,3𝜅 (25) provided that 𝑚𝜌+𝛼₁𝜑𝜅𝜑𝜑3𝜅+𝜇3𝜅−𝑚𝛼₁𝜑3𝜅𝜑𝜑3𝜅−𝜇𝜅−𝜎𝐵²₀=0. (26) Thus the solution of the PDE (21) which satisfies the condition (26) is written as 𝑣𝑦,𝑡=exp𝜑3𝜅𝑦−𝑚𝑡with𝑚>0. (27)

Remark 4. Note that the previous solution (27) also satisfies the boundary and initial conditions given in (22). The imposing physical condition (26) gives the speed of travelling shock wave. From (26), we find 𝑚=2𝜇𝜑/3𝜅+𝜎𝐵²₀𝜑/3𝜅^1/2𝜌+2𝛼₁𝜑/3𝜅>0. (28) If we set 𝐵₀=0 (no magnetic field), we recover the condition given in (14).

3.3. Unsteady Magnetohydrodynamic (MHD) Flow of Third Grade Fluid in a Porous Medium with Plate Suction/Injection

This particular model is an extension of previous two problems with combined effects of plate suction/injection and MHD nature of the fluid. Thus, for flow under consideration, we seek a velocity of the form 𝐕=𝑤𝑦,𝑡,−𝑊,0, (29) where 𝑤 denotes the velocity of the fluid in 𝑥-direction and 𝑊>0 indicates suction velocity and 𝑊<0 blowing or injection velocity.

The unsteady MHD flow of a third grade fluid in a porous half space with plate suction/injection is governed by 𝜌+𝛼₁𝜑𝜅𝜕𝑤=𝜕𝑡𝜌+𝛼₁𝜑𝜅𝑊𝜕𝑤𝜕𝜕𝑦+𝜇²𝑤𝜕𝑦²+𝛼₁𝜕³𝑤𝜕𝑦²𝜕𝑡+6𝛽₃𝜕𝑤𝜕𝑦²𝜕²𝑤𝜕𝑦²−𝛼₁𝑊𝜕³𝑤𝜕𝑦³−2𝛽₃𝜑𝜅𝜕𝑤𝜕𝑦²𝜑𝑢−𝜅𝜇𝑤−𝜎𝐵²₀𝑤. (30)

The PDE (30) is solved subject to the same boundary and initial conditions specified for the previous models. The invariant solution of the previous nonlinear PDE under the linear combination of time-translation and space-translation symmetry generators is given by 𝑤𝑦,𝑡=𝐻(𝜃)with𝜃=𝑦−𝑚𝑡. (31) Inserting (31) in (30), we get a third-order nonlinear ODE in 𝐻(𝜃), namely, 𝜌+𝛼₁𝜑𝜅𝑚𝑑𝐻+𝑑𝜃𝜌+𝛼₁𝜑𝜅𝑊𝑑𝐻𝑑𝑑𝜃+𝜇²𝐻𝑑𝜃²−𝑚𝛼₁𝑑³𝐻𝑑𝜃³+6𝛽₃𝑑𝐻𝑑𝜃²𝑑²𝐻𝑑𝜃²−𝛼₁𝑊𝜕³𝐻𝜕𝜃³−2𝛽₃𝜑𝜅𝐻𝑑𝐻𝑑𝜃²−𝜇𝜑𝜅+𝜎𝐵²₀𝐻(𝜃)=0. (32) Following the same procedure used to tackle the first problem, (32) admits the exact solution of the form 𝐻(𝜃)=𝑎exp𝜑𝜃,3𝜅 (33) provided 0=𝑚𝜌+𝛼₁𝜑𝜅𝜑+3𝜅𝜌+𝛼₁𝜑𝜅𝑊𝜑𝜑3𝜅+𝜇−()𝛼3𝜅𝑚+𝑊₁𝜑3𝜅𝜑−𝜇𝜑3𝜅𝜅+𝜎𝐵²₀. (34) The solution of 𝑤(𝑦,𝑡) is written as 𝑤𝑦,𝑡=exp𝜑3𝜅𝑦−𝑚𝑡with𝑚>0. (35)

Remark 5. The previous solution is valid only under the particular condition on the physical parameters given in (34). The condition (34) also gives the speed of shock wave. Thus from (34), we obtain 𝑚=2𝜇𝜑/3𝜅+𝜎𝐵²₀−𝜌+𝛼₁𝜑/3𝜅𝜑/3𝜅𝑊𝜑/3𝜅𝜌+2𝛼₁𝜑/3𝜅−𝛼₁𝜑/3𝜅𝑊>0. (36) Note that, if we set 𝑊=𝐵₀=0 (with no porosity and magnetic field), we recover the previous two conditions given in (14) and (26).

Remark 6. We note that the shock wave solutions (16), (27), and (35) are the same, but the imposing conditions on the physical parameters of the flow given in (14), (26), and (34) under which these solutions are valid are different. This means that in each case the speed of the travelling shock wave is different. Therefore, the graphical behavior of these solutions is the same which shows the shock wave behavior of the flow problems. However, the imposing conditions contain the magnetic field, suction/blowing, porosity, and second grade and the third grade parameters. Thus these closed-form shock wave solutions are valid for the particular values of these parameters.

4. Shock Wave Behavior of the Solutions

Figures 2 and 3 show the shock wave behavior of the solutions (16), (27), and (35) in 2D and 3D, respectively. From the graphs it is quite clear that the slope of the velocity profile is approaching to infinity representing the sudden sharp discontinuity in the velocity field.

Figure 2 

Profile of the shock wave solutions (16), (27), and (35) with 𝜑=4, 𝜅=0.2, 𝑚=1, and 𝑡=𝜋/2.

Figure 3 

3D profile of the shock wave solutions (16), (27), and (35) with 𝜑=4, 𝜅=0.2, 𝑚=1, and 𝑡=𝜋/2.

5. Final Comments

In this paper, we have presented closed-form shock wave solutions for some nonlinear problems which describe the phenomena of third grade fluids. In each case the governing nonlinear PDEs are reduced to nonlinear ODEs by using the Lie point symmetry (which is translation) in the 𝑡 and 𝑦 directions. The reduced ODEs are then solved analytically. We observe that the shock wave solutions (16), (27), and (35) are the same, but the imposing conditions on the physical parameters of the flow models given in (14), (26), and (34) under which these solutions are valid are different. These solutions do not directly contain the parameter which is responsible for showing the behavior of third grade fluid parameter on the flow. However, the imposing conditions under which these solutions are valid do contain the third grade parameter. To emphasize, we can say that these solutions are valid for the particular values of third grade fluid parameter. The results obtained describe the mathematical structure of the shock wave behavior of the flow problems. The models considered in this study are prototype, but the obtained solutions are going to be very helpful in carrying out further analysis of the shock wave characteristic associated with the non-Newtonian fluid flow models. The method that we have adopted is also prosperous for tackling wide range of nonlinear problems in non-Newtonian fluid mechanics.

Acknowledgments

T. Aziz and A. Fatima would like to thank the School of Computational and Applied Mathematics and the Financial Aid and Scholarship office, University of the Witwatersrand, for financial support and scholarships.