Shock Wave Solutions for Some Nonlinear Flow Models Arising in ...
Transcript of Shock Wave Solutions for Some Nonlinear Flow Models Arising in ...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 602902, 6 pageshttp://dx.doi.org/10.1155/2013/602902
Research ArticleShock Wave Solutions for Some Nonlinear Flow Models Arisingin the Study of a Non-Newtonian Third Grade Fluid
Taha Aziz, R. J. Moitsheki, A. Fatima, and F. M. Mahomed
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics,University of the Witwatersrand, Wits 2050, South Africa
Correspondence should be addressed to R. J. Moitsheki; [email protected]
Received 28 February 2013; Revised 6 June 2013; Accepted 7 June 2013
Academic Editor: Waqar Khan
Copyright Β© 2013 Taha Aziz et al.This is an open access article distributed under theCreativeCommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in thestudy of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinearpartial differential equations into nonlinear ordinary differential equations.The reduced equations are then solved analytically, andthe shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally inthe process of the derivation of the solutions.
1. Introduction
A shock wave is a disturbance that propagates through amedia. Shock is yielded when a disturbance is made tomove through a fluid faster than the speed of sound (thecelerity) of the medium. This can occur when a solid objectis forced through a fluid. It represents a sharp discontinuityof the parameters that delineate the media. Unlike solutionswhere the energy is a conserved quantity and thus remainsconstant during its propagation, shockwave dissipates energyrelatively quickly with distance. One source of a shock waveis when the supersonic jets fly at a speed that is greaterthan the speed of sound. This results in the drag force onaircraft with shocks. These waves also appear in variousinteresting phenomena in real life situations. For example,solitons appear in the propagation of pulses through opticalfibers. Another example is where cnoidal waves appear inshallow water waves, although this is an extremely scarcephenomena. Some interesting communications dealing withthe shock wave solutions are found in [1β3].
During the past several decades, the study of the nonlin-ear problems dealing with the flow of non-Newtonian fluidshas attracted considerable attention. This interest is due toseveral important applications in engineering and industrysuch as reactive polymer flows in heterogeneous porousmedia, electrochemical generation of elemental bromine
in porous electrode systems, manufacture of intumescentpaints for fire safety applications, extraction of crude oilfrom the petroleum products, synthetic fibers, and paperproduction [4]. Due to the diverse physical nature of non-Newtonian fluids, there is no single constitutive expressionwhich describe the physical behavior of all non-Newtonianfluid models. Because of this important issue, several modelsof non-Newtonian fluids have been proposed in the literature.Togetherwith this factor, themathematicalmodelling of non-Newtonian incompressible fluid flows gives rise to nonlinearand complicated differential equations. As a consequenceof this nonlinearity factor, the exact (closed-form) solutionsof these sort of problems are scarce in general. Severaltechniques and methods have been developed in the recentfew years to construct the solutions of these non-Newtonianfluid flow problems. Some of these useful methods are vari-ational iteration method, Adomian decomposition method,homotopy perturbationmethod, homotopy analysis method,and semi-inverse variational method. Literature survey wit-nesses that, despite all thesemethods, the exact (closed-form)solutions of the non-Newtonian fluid flow problems are stillrare in the literature, and it is not surprising that new exact(closed-form) solutions are most welcome, provided theycorrespond to physically realistic situations. Some interestingand useful communications in this area are made in thestudies [5β14].
2 Mathematical Problems in Engineering
y-axis
x-axis
x-axis (y = 0)
y > 0
u(0, t) = V(t)
Third grade fluid filling theporous half space y > 0
Infinite porous/rigid plate
Figure 1: Geometry of the physical models and coordinate system.
One of the widely accepted models amongst non-Newtonian fluids is the class of Rivlin-Ericksen fluids ofdifferential type [15]. Rivlin-Ericksen fluids of differentialtype have acquired the special status in order to describe theseveral nonstandard features such as normal stress effects,rod climbing, shear thinning, and shear thickening. In theliterature much attention has been given to the flow of asecond grade fluid [16β19]. A second grade fluid model isthe simplest subclass of non-Newtonian fluids for whichone can reasonably hope to obtain an analytical solution.In most of the flow situations, the governing equations fora second grade fluid are linear. Although a second gradefluid model is able to predict the normal stress differences,it does not take into account the shear thinning, and shearthickening phenomena that many fluids reveal. Thereforesome experiments may be well described by third gradefluid [20β24]. The mathematical model of a third grade fluidrepresents amore realistic description of the behavior of non-Newtonian fluids. A third grade fluid model represents afurther attempt towards the study of the flow properties ofnon-Newtonian fluids. Therefore, a third grade fluid modelhas been considered in this study. This model is known tocapture the non-Newtonian effects such as shear thinning orshear thickening as well as normal stress.
In this particular study, we have constructed the shockwave solutions of some nonlinear PDEs arising in the studyof third grade fluid flow in porousmedium.We know that theflow of non-Newtonian fluids in a porous medium has widerange of engineering applications.These include enhanced oilrecovery, ceramic processing, and geothermal engineering.Motivated by these facts, we have developed some shockwave solutions of three nonlinear problems dealing with theunsteady flow of third grade fluid in a porous half space.
2. Geometry of the Models
Consider a Cartesian coordinate frame of reference ππππ
withπ₯-axis along the direction of the flow andπ¦-axis pointingin the vertically upward direction. The third grade fluid
occupies the porous space π¦ > 0 and is in contact with aninfinite plate at π¦ = 0. Since the plate is infinite in the ππ-plane, therefore all the physical quantities except the pressuredepend on π¦ only. We have taken three different problems onthe same flat plate geometry.The geometry of the problems isshown in Figure 1.
3. Problems to Be Investigated
3.1. Unsteady Flow of a Third Grade Fluid over a Flat RigidPlate with Porous Medium. Following the methodology of[22, 23], the unsteady incompressible flow of a third gradefluid over the rigid plate with porous medium is governed by
(π + πΌ1
π
π )ππ’
ππ‘= π
π2π’
ππ¦2+ πΌ1
π3π’
ππ¦2ππ‘+ 6π½3(ππ’
ππ¦)
2π2π’
ππ¦2
β 2π½3
π
π (ππ’
ππ¦)
2
π’ βπ
π ππ’,
(1)
where π’ is the velocity component in π₯-direction, π‘ is time,π¦ is spatial variable, π is the density, π is the coefficient ofviscosity, πΌ
1and π½
3are the material constants (for details on
these material constants and the conditions that are satisfiedby these constants, the reader is referred to [24]), π is theporosity and π is the permeability of the porous medium.
In order to solve (1), the relevant boundary and initialconditions are
π’ (0, π‘) = π’0π (π‘) , π‘ > 0, (2)
π’ (β, π‘) = 0, π‘ > 0, (3)
π’ (π¦, 0) = π (π¦) , π¦ > 0, (4)
where π’0is the reference velocity and π(π‘) and π(π¦) 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 arestrictive assumption since we can always measure velocity
Mathematical Problems in Engineering 3
relative to themain stream.The initial condition (4) indicatesinitially that the fluid is moving with some nonuniformvelocity π(π¦).
3.1.1. Reduction of the Governing Equation. We know thatfrom the principal of Lie symmetry methods [25, 26] thatif a differential equation is explicitly independent of anydependent or independent variable, then this particulardifferential equation remains invariant under the transla-tion symmetry corresponding to that particular variable.We noticed that (1) admits Lie point symmetry generators,π/ππ‘ (translation in π‘) and π/ππ¦ (translation in π¦). Let π
1
and π2be time-translation and space-translation symmetry
generators, respectively. Then the solution corresponding tothe generator
π = π1+ ππ
2=
π
ππ‘+ π
π
ππ¦, (π > 0) (5)
would represent travelling wave solution with constant wavespeed π. Travelling wave solutions are characterized by thefact that the profiles of these solutions at different timeinstants are obtained from one another by appropriate shifts(translations) along the π¦-axis. Consequently, a Cartesiancoordinate system moving with the constant speed can beintroduced in which the profile of the derived quantity isstationary. π = 0 represents the stationary or steady-statesolutions.
The characteristic system corresponding to (5) is
ππ¦
π=
ππ‘
1=
ππ’
0. (6)
Solving (6), invariant solutions are given by
π’ (π¦, π‘) = πΉ (π) with π = π¦ β ππ‘, (7)
where πΉ(π) is an arbitrary function of the characteristicvariable π = π¦ β ππ‘. Making use of (7) into (1) results in athird-order ordinary differential for πΉ(π) as follows:
(π + πΌ1
π
π )π
ππΉ
ππ+ π
π2πΉ
ππ2β ππΌ1
π3πΉ
ππ3
+ 6π½3(ππΉ
ππ)
2π2πΉ
ππ2β 2π½3
π
π πΉ(
ππΉ
ππ)
2
β ππ
π πΉ = 0.
(8)
Thus the original third-order nonlinear PDE (1) is reducedto a third-order ODE (5) along certain curves in the π¦-π‘plane. These curves are called characteristic curves or just thecharacteristic.
3.1.2. Shock Wave Solution. In this section, we show thatthe travelling wave solutions of (1) approach a shock wavesolution. Now we construct the shock wave solution of thereduced equation (5). The starting hypothesis for shock wavesolution is given by
πΉ (π) = π exp (ππ) , (9)
where π and π are the free parameters to be determined.Substituting (9) into (8), we obtain
[π(π + πΌ1
π
π ) π + ππ
2β ππΌ1π3β π
π
π ]
+ π2ππ
[6π½3π2π4β 2π½3
π
π π2π2] = 0.
(10)
Separating (10) in the powers of π0 and π2ππ, we find
π0: π (π + πΌ
1
π
π ) π + ππ
2β ππΌ1π3β π
π
π = 0, (11)
π2π΅π
: π½3π2[3π4βπ
π π2] = 0, with π½
3π2
= 0. (12)
From (12), we deduce
π = βπ
3π . (13)
Using the value of π in (11), we obtain
π(π + πΌ1
π
π )β
π
3π + π(
π
3π ) β ππΌ
1(π
3π )β
π
3π β π
π
π = 0.
(14)
Finally, the solution for πΉ(π) (provided the condition (14)holds) is written as
πΉ (π) = π exp [βπ
3π π] . (15)
So the solution π’(π¦, π‘) which satisfies the condition (14) iswritten as
π’ (π¦, π‘) = π exp [βπ
3π (π¦ β ππ‘)] with π > 0. (16)
Remark 1. Note that the solution (16) is the shock wavesolution to the governing PDE (1). The previous solutionis valid under the particular condition on the physicalparameters of the flow given in (14) (which is some kind ofdispersion relation inπ).This solution does show the hiddenshock wave behavior of the flow problem with slope of thevelocity field or the velocity gradient approaches to infinitysuch that
ππ’
ππ¦β β as π¦ β β. (17)
Remark 2. Note that the solution (16) also satisfies the partic-ular initial and the boundary condition; that is,
π’ (0, π‘) = π (π‘) = exp [ββπ
3π ππ‘] , π‘ > 0,
π’ (π¦, 0) = π (π¦) = exp [βπ
3π π¦] , π¦ > 0,
(18)
with
π (0) = π (0) = π = 1. (19)
The functions π(π‘) and π(π¦) depend on the physical parame-ters of the flow.
4 Mathematical Problems in Engineering
Remark 3. We also observe that the physical significance ofthe imposing condition (14) is that it gives the speed oftravelling shock wave. From (14), we deduce
π =2π(π/3π )
1/2
(π + (2πΌ1π/3π ))
> 0. (20)
3.2. Unsteady Magnetohydrodynamic (MHD) Flow of ThirdGrade Fluid in a Porous Medium. By employing the samegeometry as we have explained in Section 2, in this problemwe extend the previous model by considering the fluid tobe electrically conducting under the influence of a uniformmagnetic field applied transversely to the flow. We providethe closed-form solution of the problem by reducing thegoverning nonlinear PDE into an ODE with the help of Liereduction technique.
The time-dependent magnetohydrodynamic flow of athird grade fluid in a porous half space in the absence of themodified pressure gradient takes the form
(π + πΌ1
π
π )πV
ππ‘= π
π2V
ππ¦2+ πΌ1
π3V
ππ¦2ππ‘+ 6π½3(πV
ππ¦)
2π2V
ππ¦2
β 2π½3
π
π (πV
ππ¦)
2
π’ βπ
π πV β ππ΅
2
0V,
(21)
where V is the velocity component in π₯-direction, π isthe electrical conductivity, and π΅
0is the uniform applied
magnetic field. In order to solve (21), the relevant time andspace dependent velocity boundary conditions are
π’ (0, π‘) = π’0π (π‘) , π‘ > 0,
π’ (β, π‘) = 0, π‘ > 0,
π’ (π¦, 0) = π (π¦) , π¦ > 0.
(22)
As it can be seen (21) also admits Lie point symmetrygenerators, π/ππ‘ (translation in π‘) and π/ππ¦ (translation inπ¦). Let π
1and π
2be time-translation and space-translation
symmetry generators, respectively. The invariant solutioncorresponding to the generatorπ = π
1+ ππ
2is given by
V (π¦, π‘) = πΊ (π) with π = π¦ β ππ‘. (23)
Using (23) into (21) yields
(π + πΌ1
π
π )π
ππΊ
ππ+ π
π2πΊ
ππ2β ππΌ1
π3πΊ
ππ3+ 6π½3(ππΊ
ππ)
2π2πΊ
ππ2
β 2π½3
π
π πΊ(
ππΊ
ππ)
2
β ππ
π πΊ β ππ΅
2
0πΊ = 0.
(24)
Following the same methodology adopted to solve the previ-ous problem, the reduced ODE (24) admits an exact solutionof the form
πΊ (π) = π exp [βπ
3π π] , (25)
provided that
π(π + πΌ1
π
π )β
π
3π + π(
π
3π )
β ππΌ1(π
3π )β
π
3π β π
π
π β ππ΅2
0= 0.
(26)
Thus the solution of the PDE (21)which satisfies the condition(26) is written as
V (π¦, π‘) = exp [βπ
3π (π¦ β ππ‘)] with π > 0. (27)
Remark 4. Note that the previous solution (27) also satisfiesthe boundary and initial conditions given in (22). Theimposing physical condition (26) gives the speed of travellingshock wave. From (26), we find
π =2π (π/3π ) + ππ΅
2
0
(π/3π )1/2
[π + (2πΌ1π/3π )]
> 0. (28)
If we set π΅0= 0 (no magnetic field), we recover the condition
given in (14).
3.3. Unsteady Magnetohydrodynamic (MHD) Flow of ThirdGrade Fluid in a Porous Medium with Plate Suction/Injection.This particular model is an extension of previous two prob-lems with combined effects of plate suction/injection andMHD nature of the fluid.Thus, for flow under consideration,we seek a velocity of the form
V = [π€ (π¦, π‘) , βπ, 0] , (29)
where π€ denotes the velocity of the fluid in π₯-direction andπ > 0 indicates suction velocity and π < 0 blowing orinjection velocity.
The unsteadyMHDflow of a third grade fluid in a poroushalf space with plate suction/injection is governed by
(π + πΌ1
π
π )ππ€
ππ‘= (π + πΌ
1
π
π )π
ππ€
ππ¦+ π
π2π€
ππ¦2+ πΌ1
π3π€
ππ¦2ππ‘
+ 6π½3(ππ€
ππ¦)
2π2π€
ππ¦2β πΌ1π
π3π€
ππ¦3
β 2π½3
π
π (ππ€
ππ¦)
2
π’ βπ
π ππ€ β ππ΅
2
0π€.
(30)
The PDE (30) is solved subject to the same boundaryand initial conditions specified for the previous models. Theinvariant solution of the previous nonlinear PDE under thelinear combination of time-translation and space-translationsymmetry generators is given by
π€ (π¦, π‘) = π» (π) with π = π¦ β ππ‘. (31)
Mathematical Problems in Engineering 5
Inserting (31) in (30), we get a third-order nonlinear ODE inπ»(π), namely,
(π + πΌ1
π
π )π
ππ»
ππ+ (π + πΌ
1
π
π )π
ππ»
ππ+ π
π2π»
ππ2
β ππΌ1
π3π»
ππ3+ 6π½3(ππ»
ππ)
2π2π»
ππ2β πΌ1π
π3π»
ππ3
β 2π½3
π
π π»(
ππ»
ππ)
2
β (ππ
π + ππ΅2
0)π» (π) = 0.
(32)
Following the same procedure used to tackle the first prob-lem, (32) admits the exact solution of the form
π»(π) = π exp [βπ
3π π] , (33)
provided
0 = π(π + πΌ1
π
π )β
π
3π + (π + πΌ
1
π
π )πβ
π
3π + π(
π
3π )
β (π +π)πΌ1(π
3π )β
π
3π β (π
π
π + ππ΅2
0) .
(34)
The solution of π€(π¦, π‘) is written as
π€ (π¦, π‘) = exp [βπ
3π (π¦ β ππ‘)] with π > 0. (35)
Remark 5. The previous solution is valid only under theparticular condition on the physical parameters given in (34).The condition (34) also gives the speed of shock wave. Thusfrom (34), we obtain
π =2π (π/3π ) + ππ΅
2
0β (π + (πΌ
1π/3π ))βπ/3π π
βπ/3π [π + (2πΌ1π/3π ) β (πΌ
1π/3π )π]
> 0.
(36)
Note that, if we set π = π΅0= 0 (with no porosity and mag-
netic 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 thephysical parameters of the flow given in (14), (26), and (34)under which these solutions are valid are different. Thismeans that in each case the speed of the travelling shockwave is different. Therefore, the graphical behavior of thesesolutions is the samewhich shows the shock wave behavior ofthe flowproblems.However, the imposing conditions containthe magnetic field, suction/blowing, porosity, and secondgrade and the third grade parameters.Thus these closed-formshock wave solutions are valid for the particular values ofthese 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
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
y
Velo
city
pro
file β
u,οΏ½,w
β
Γ1026
Figure 2: Profile of the shockwave solutions (16), (27), and (35) withπ = 4, π = 0.2,π = 1, and π‘ = π/2.
00.5
1
1.5
2 0
10
20
30
y0
12
34
Γ1026
Γ1026
Velocity field u, οΏ½, w
Figure 3: 3D profile of the shock wave solutions (16), (27), and (35)with π = 4, π = 0.2,π = 1, and π‘ = π/2.
graphs it is quite clear that the slope of the velocity profileis approaching to infinity representing the sudden sharpdiscontinuity in the velocity field.
5. Final Comments
In this paper, we have presented closed-form shock wavesolutions for some nonlinear problems which describe thephenomena of third grade fluids. In each case the governingnonlinear PDEs are reduced to nonlinear ODEs by using theLie point symmetry (which is translation) in the π‘ and π¦
directions.The reducedODEs are then solved analytically.Weobserve that the shock wave solutions (16), (27), and (35) arethe same, but the imposing conditions on the physical param-eters of the flow models given in (14), (26), and (34) underwhich these solutions are valid are different. These solutions
6 Mathematical Problems in Engineering
do not directly contain the parameter which is responsible forshowing the behavior of third grade fluid parameter on theflow. However, the imposing conditions under which thesesolutions are valid do contain the third grade parameter. Toemphasize, we can say that these solutions are valid for theparticular values of third grade fluid parameter. The resultsobtained describe the mathematical structure of the shockwave behavior of the flow problems. The models consideredin this study are prototype, but the obtained solutions aregoing to be very helpful in carrying out further analysis of theshockwave characteristic associatedwith the non-Newtonianfluid flow models. The method that we have adopted is alsoprosperous for tackling wide range of nonlinear problems innon-Newtonian fluid mechanics.
Acknowledgments
T. Aziz and A. Fatima would like to thank the School ofComputational and Applied Mathematics and the FinancialAid and Scholarship office, University of the Witwatersrand,for financial support and scholarships.
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