MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation18596510.1155/2009/185965185965Research ArticleExtended Stokes' Problems for Relatively Moving Porous Half-PlanesLiuChi-MinPellicanoFrancescoDivision of MathematicsGeneral Education CenterChienkuo Technology University Changhua City 500Taiwanctu.edu.tw200910012010200904072009011220092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

1. Introduction

After Stokes presented the significant paper  on pendulums in which the problems of the impulsive and oscillatory motions of a plane were studied, numerous studies based on his idea were subsequently carried out. In the field of fluid mechanics, flows driven by a plate of an impulsive or oscillatory motion are usually referred to the Stokes’ problems. Using the integral transform, one can obtain the exact solution of the Stokes’ problems which can pursue the fluid motion not only at larger times but also at small times . In addition to the aforementioned papers dealing with the Newtonian fluid, lots of efforts were also made for the flows of the non-Newtonian fluids . The effects of the mass influx (or efflux) on the velocity profiles were investigated as well [9, 14, 15, 17]. Though previous papers have made a comprehensive and in-depth contribution to the understanding of Stokes’ problems, studies on the flow driven by more complicated motions of boundaries are comparatively few. Usually, the main concern of changing boundary conditions is to simulate the practical flows occurred in natural environments. For this purpose, Zeng and Weinbaum  theoretically studied the Stokes’ problems for moving half-planes. They provided the steady-state solutions which can be applied to many practical problems, for example, the flow induced by either earthquakes or fracture of ice sheets. In general, the exact solution consists of two parts, the steady-state solution and the transient solution. Though the transient part usually decays with time, the flow at the very early stage, however, cannot be precisely described by the steady-state solution only, especially for most of the earthquakes and other specific problems usually occurring in a very short period. Recently, Liu  reinvestigated the Stokes problems for the cases bounded by modified boundary conditions as well as the finite-depth cases.

Due to above descriptions, the viscous flow generated by relatively moving porous half-planes, which has not been studied yet, is analyzed in this paper. Mathematical formulation is firstly given in Section 2. The detailed derivation for the first and second problems is provided in Section 3. Results and discussions are given in Section 4. Finally, conclusions are made in Section 5.

2. Mathematical Formulation

The extended Stokes’ problems for relatively moving porous planes are depicted in Figure 1. The fluid occupies the positive-y domain bounded by a porous plate located at y=0. The fluid is initially at rest everywhere, and then is suddenly driven by the positive-z plate moving with either a constant speed (the first problem) or a harmonic motion (the second problem). The negative-z plate remains at rest for all times. The symbols η, u, and v denote the kinematic viscosity and the flow velocities in the x and y directions, respectively. A constant porous velocity on the plate is denoted as v0. For this flow system, the governing equation isut+v0uy=η(2uy2+2uz2), and the boundary and initial conditions areu(y=0,z>0,t)=f(t),u(y=0,z<0,t)=0,u(y,z,t)=0,u(y,z±,t)  is  finite,u(y,z,t=0)=0, wheref(t)=u0 is for the first problem, andf(t)=u0cos(σt+θ) indicates the second problem. To solve this problem, if one employs the integral transforms which are often applied to solve the Stokes’ problems, it will result in the solution in a quite inconvenient form because the inverse transform cannot be further simplified. Therefore, in this paper, an important mathematical technique which divides the original system equations (2.1) and (2.2) into two subsystems is introduced to overcome this situationu1t+v0u1y=η2u1y2,u1(y=0,t)=f(t)2,u1(y,t)=0,u1(y,t=0)=0, andu2t=η(2u2y2+2u2z2),u2(y=0,z>0,t)=f(t)2,u2(y=0,z<0,t)=-f(t)2,u2(y,z,t)=0,u2(y,z,t=0)=0, where the total solution is u=u1+u2. Note that for the latter subsystem, since the velocity u2 is antisymmetrical with respect to z=0, an additional condition is requiredu2(z=0)=0.

Diagram of the extended Stokes’ problems.

3. Derivation of the Extended Stokes’ Problems

The detailed derivation for solutions to the flow systems governed by (2.5), (2.6), and (2.7) will be presented in this section. For the first subsystem, that is, the traditional Stokes’ problem associated with the porous effect on the boundary, Liu et al.  provided the exact solutionu1=u04[erfc(y2ηt-v0t2η)+ev0y/η·erfc(y2ηt+v0t2η)], for the first problem. They demonstrated that the mass injection on the plate hastens the velocity profile to be uniform with the plate speed. On the contrary, the mass suction retards the energy transferring from the plate to the fluid. Accordingly, one can conclude that the mass inflow is advantageous to the energy transfer and the mass outflow hinders the development of the flow. As for the second problem, the solution is u1=u04ev0y/2η·{ei(σt+θ+Π-(σ/η)y)+Π+(σ/η)y·erfc[y2ηt+(Π++iΠ-)σt2]+ei(σt+θ-Π-(σ/η)y)-Π+(σ/η)y·erfc[y2ηt-(Π++iΠ-)σt2]}, where Π±=((v024ση)2+1±v024ση)0.5. It is also found that the effects of mass inflow and outflow on the energy transferred from the plate to the far-field fluid are similar to the phenomena occurred in the first problems.

To solve the second subsystem governed by (2.6) and (2.7), one only needs to solve the solution of the positive-z domain since the flow is antisymmetrical with respect to z=0. By applying both the Laplace transform and the Fourier Sine transform respectively defined asû(s)=0u2(t)·e-stds,ũ(ω)=0û(y)·sin(ωy)dy, to this system, it yieldsũzz-(ω2+sη)ũ=-ω·f̂(s). From the boundedness of ũ as z approaches infinity and the boundary condition at z=0, the solutions to ũ is ũ=ωη(ω2η+s)[1-exp(-ω2+s/η·z)]f̂(s). The solutions to u2 can be obtained by taking the inverse transforms twice. For the first problem, the solution isu2=uo2erfc(y2ηt)+u02πt=0t=terfc(z2ηt)exp(-y24ηt)d(yηt). For the second problem, it readsu2=u0σy4πη0σtα-1.5cos(σt-α+θ)exp(-σy24αη)dα+u0σz2π3η00σtsin(σωyη)·G1(ω,σt-α)·G2(ω,α,σηz)dαdω, whereG1(ω,β)=ωω4+1[sinθ(e-ω2β-cosβ+ω2sinβ)-cosθ(sinβ+ω2cosβ-ω2e-ω2β)],G2(ω,β,Z)=β-1.5exp(-ω2β-Z24β).

4. Results

Before analyzing the solutions derived in the previous section, dimensionless parameters are required to enhance the understanding of the flow. Hence, for the first problem, applying the following dimensionless parameters:U=uu0,Y=u0ηy,Z=u0ηz,Vw=v0u0,T=u02ηt, and combining (3.1) and (3.7) together, the total solution isU=14[erfc(Y2T-VwT2)+eVwY·erfc(Y2T+VwT2)]+12πY/Terf(Zα2Y)exp(-α24)dαZ. If one further takes the new set of parameters(Ψ,Φ,Ω)=(YT,ZT,VwT), into account, the total solution equation (4.2) becomesU=14[erfc(Ψ-Ω2)+eΨ·Ω·erfc(Ψ+Ω2)]+12πΨerf(Φ2Ψα)exp(-α24)dα. For viscous flows, spatial and time variables in the solution are often coupled in certain forms. Such a solution is also know as a similarity solution in which the solution profile with respect to the spatial coordinate is similar at all times. In the present problem, the similarity not only appears in the Y and Z variables but also in the porous velocity Vw. With the idea of the similarity; four variables (Y,Z,T,Vw) in (4.2) can be reduced to three dependent variables (Ψ,Φ,Ω), as shown in (4.4). Figure 2 shows the velocity profiles for various values of Φ under the condition Ω=1. When the flow develops (i.e., T goes large), the velocity distribution at the far end of Z0 will gradually approach that of the traditional Stokes’ first problem. This implies that the effects of the still plate on the velocity distribution will become weaker for larger values of Φ. Similarly, the velocity profile will approach zero at the other far end (Z0), where the influence from the moving plate is weak as well. The porous effects on the velocity profiles are shown in Figure 3. The effects of different porous velocities on the flow for Φ=1 are plotted. It is found that the mass influx (positive Ω) hastens the development of the flow, while the mass efflux (negative Ω) retards the energy transferred from the moving plate to the far-field fluid.

Velocity profiles at various values of Φ for the case Ω=1 (the first problem).

Velocity profiles of various values of Ω for the case Φ=1 (the first problem).

For the second problem, the total dimensionless solution is obtained by combining (3.2) and (3.8) togetherU=14e(Vw/2)Y·{ei(T+θ+Π̅-Y)+Π̅+Y·erfc[Y2T+(Π̅++iΠ̅-)T2]+ei(T+θ-Π̅-Y)-Π̅+Y·erfc[Y2T-(Π̅++iΠ̅-)T2]}Y4π0Tα-1.5cos(T-α+θ)exp(-Y24α)dα+Z2π300Tsin(ωY)·G1(ω,T-α)·G2(ω,α,Z)dαdωfor  the±Z  domain, whereU=uu0,Y=σηy,Z=σηz,T=σt,Vw=v0ση,Π̅±=(Vw416+1±Vw24)0.5, and G1 and G2 are given in (3.9). From (4.5), it is noted that there exists no similarity solution for the second problem. Figure 4 shows the velocity profiles at various Z sections at T=2π for the cosine oscillation (θ=0) with the porous velocity Vw=1. Similar to the results of the first problem, the velocity distribution at the far end in the positive-Z direction will gradually approach the solution of the traditional Stokes’ second problem when T becomes large. To investigate the porous effects, Figure 5 displays the velocity profiles affected by different porous velocities at Z=1 at T=2π for the cosine oscillation. It is also obvious that the larger influx velocity leads to the faster development of the flow.

Velocity profiles at various Z sections for the case Vw=1, T=2π, and θ=0 (the second problem).

Velocity profiles of various values of Vw for the case Z=1, T=2π, and θ=0 (the second problem).

5. Concluding Remarks

A still fluid suddenly driven by relatively moving porous half-planes is theoretically analyzed in this paper. In addition to the integral transforms, it is impossible to acquire the exact solution without using an important technique which divides the original problem into two subsystems. The solution to the first subsystem is equivalent to half of the solution of the traditional Stokes’ problem. As for the second subsystem, the velocity profiles in the whole domain can be obtained by solving the flow in the positive-z domain since the flow is antisymmetrical to z=0. Based on present solutions, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

Acknowledgment

The author appreciates the financial support from National Science Council of Taiwan with Grant no. NSC-97-2221-E-270-013-MY3.

StokesG. G.On the effect of the internal friction of fluids on the motion of pendulumsTransactions of the Cambridge Philosophical Society185198106PantonR.The transient for Stokes' oscillating plate: a solution in terms of tabulated functionsThe Journal of Fluid Mechanics196831481982510.1017/S0022112068000509ErdoganM. E.Note on an unsteady flow of a viscous fluid due to an oscillating plane wallInternational Journal of Non-Linear Mechanics20003511610.1016/S0020-7462(99)00019-02-s2.0-0033687788ZBL1006.76028LiuC.-M.LiuI.-C.A note on the transient solution of Stokes' second problem with arbitrary initial phaseJournal of Mechanics20062243493542-s2.0-16244398024LiuC.-M.HwungH.-H.KongC.-H.The unsteady viscous flow generated by an oscillating porous plateJournal of Mechanics20082421451522-s2.0-0031700228ErdoganM. E.On unsteady motions of a second-order fluid over a plane wallInternational Journal of Non-Linear Mechanics20033871045105110.1016/S0020-7462(02)00051-32-s2.0-0037412170FetecauC.FetecauC.A new exact solution for the flow of a Maxwell fluid past an infinite plateInternational Journal of Non-Linear Mechanics200338342342710.1016/S0020-7462(01)00062-22-s2.0-0037374883FetecauC.FetecauC.The first problem of Stokes for an Oldroyd-B fluidInternational Journal of Non-Linear Mechanics200338101539154410.1016/S0020-7462(02)00117-82-s2.0-0037413185JordanP. M.PuriP.Stokes' first problem for a Rivlin-Ericksen fluid of second grade in a porous half-spaceInternational Journal of Non-Linear Mechanics20033871019102510.1016/S0020-7462(02)00048-32-s2.0-0037412173JordanP. M.PuriA.BorosG.On a new exact solution to Stokes' first problem for Maxwell fluidsInternational Journal of Non-Linear Mechanics20043981371137710.1016/j.ijnonlinmec.2003.12.0032-s2.0-1442284537AiL.VafaiK.vafai@engr.ucr.eduAn investigation of stokes' second problem for non-Newtonian fluidsNumerical Heat Transfer A200547109559802-s2.0-000049172310.1080/10407780590926390FetecauC.FetecauC.Starting solutions for some unsteady unidirectional flows of a second grade fluidInternational Journal of Engineering Science2005431078178910.1016/j.ijengsci.2004.12.0092-s2.0-22944437728JordanP. M.pjordan@nrlssc.navy.milPuriA.apuri@uno.eduRevisiting Stokes' first problem for Maxwell fluidsQuarterly Journal of Mechanics and Applied Mathematics20055822132272-s2.0-444427925810.1093/qjmamj/hbi008ZBL1072.76006TanW.MasuokaT.Stokes' first problem for a second grade fluid in a porous half-space with heated boundaryInternational Journal of Non-Linear Mechanics200540451552210.1016/j.ijnonlinmec.2004.07.0162-s2.0-9544246749TanW.tanwchpku@hotmail.comMasuokaT.Stokes' first problem for an Oldroyd-B fluid in a porous half spacePhysics of Fluids2005172170231012-s2.0-003737488310.1063/1.1850409AkselN.FetecauC.ScholleM.Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluidsZeitschrift fur Angewandte Mathematik und Physik200657581583110.1007/s00033-006-0063-82-s2.0-33747870105ZBL1101.76006HayatT.ShahazadF.AyubM.Stokes' first problem for the fourth order fluid in a porous half spaceActa Mechanica Sinica/Lixue Xuebao2007231172110.1007/s10409-006-0048-z2-s2.0-34547378073FetecauC.JamilM.FetecauC.SiddiqueI.A note on the second problem of Stokes for Maxwell fluidsInternational Journal of Non-Linear Mechanics200944101085109010.1016/j.ijnonlinmec.2009.08.0032-s2.0-70349810668NazarM.FetecauC.VieruD.FetecauC.New exact solutions corresponding to the second problem of Stokes for second grade fluidsNonlinear Analysis: Real World Applications201011158459110.1016/j.nonrwa.2008.10.0552-s2.0-70350725008ZengY.WeinbaumS.Stokes problems for moving half-planesJournal of Fluid Mechanics199528759742-s2.0-0028992562ZBL0843.76015LiuC.-M.Complete solutions to extended Stokes' problemsMathematical Problems in Engineering20082008187542622-s2.0-002899256210.1155/2008/754262ZBL1149.76016