# A Residual Theorem Approach Applied to Stokes’ Problems with Generally Periodic Boundary Conditions including a Pressure Gradient Term

Authors: Danilo Durante; Riccardo Broglia
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The differential problem given by a parabolic equation describing the purely viscous flow generated by a constant or an oscillating motion of a boundary is the well-known Stokes’ problem. The one-dimensional equation is generally solved for unbounded or bounded domains; for the latter, either free slip (i.e., zero normal gradient) or no-slip (i.e., zero velocity) conditions are enforced on one boundary. Generally, the analytical strategy to solve these problems is based on finding the solutions of the Laplace-transformed (in time) equation and on inverting these solutions. In the present paper this problem is solved by making use of the residuals theorem; as it will be shown, this strategy allows achieving the solutions of First and Second Stokes’ problems in both infinite and finite depth. The extension to generally periodic boundaries with the presence of a periodic pressure gradient is also presented. This approach allows getting closed form solutions in the time domain in a rather fast and simple way. An ad hoc numerical algorithm, based on a finite difference approximation of the differential equation, has been developed to check the correctness of the analytical solutions.


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## Body

## 1. Introduction

The analysis of purely viscous unsteady flows, or Stokes’ flow, consists in finding analytical solutions of parabolic equations (like the heat equation) for prescribed boundary conditions. In Batchelor [1], the fluid is supposed to occupy the half plane (x,y) for y>0, with the boundary at y=0 initially at the rest and then moving in its own plane with a velocity which suddenly assumes a constant (first problem) or sinusoidal (second problem) time law.Even if considered as an old-fashioned topic, the importance in solving this class of problems for different boundary conditions lies in the wide application fields in which they are encountered. For example, both the sudden or the periodic movements of oceanic geophysical faults during an earthquake are a clear example, as well as “the oscillatory motion which arises in the design of an oscillating half-plate flow chamber for examining the effects of fluid shear stress on cultured cell monolayers” (see Zeng and Weinbaum [2]). Moreover, some actual and interesting problems, including the development of new materials, make the analysis of Stokes problems a fundamental topic; indeed, the flow over superhydrophobic surfaces is excellently modelled by a nonconvective equation with mixed (homogeneous Dirichlet-homogeneous Neumann) boundary conditions (see, for example, Lauga and Stone [3], Rothstein [4]).Ocean wave dynamics are another example: among others, Xu and Bowen [5] exploit simplified solutions of Stokes problems in order to estimate the wave stress over the ocean bottom. The great interest for these solutions lies in the examination of the net mass transport phenomena associated with an ocean wave (Stokes’ drift) generated by the wind in a rotating frame (see Janssen [6], Hanley and Belcher [7], and Polton et al. [8]). For large-scale problems, transitional effects are normally neglected, focusing the interest on long time dynamics. However the solutions from which final conclusions are drawn contain some approximations and, obviously, anything can be said on the time scale of the problem or on fast dynamics events.The solutions of these problems in one-dimensional case and for unbounded domains are well established in literature, in particular for heat diffusion problems (see, for example, Watson [9], Batchelor [1], Erdogan [10], and Liu and Liu [11]). The strategy for the solution, as in the present paper, is to consider the Laplace transformation in time; however, the inverse transformation is not simple to achieve (see, for example, Devakar and Iyengar [12] where the inverse transform is obtained through a numerical procedure); therefore a clear and a more general strategy, which would allow the achievement of the analytical solution for different boundary conditions, has to be found.An example of analytical solutions in unbounded domain can be found in Watson [9], where Stokes’ equation is considered for the study of the boundary layer past an infinite porous wall, assuming the pressure gradient as an arbitrary function of time. The two-side Laplace transform in time is used in order to obtain analytical solutions for homogeneous Dirichlet boundary condition. However the approach to the problem is carried out in a way which is difficult to extend for finite domains and time-dependent boundary conditions.The analytical solutions of parabolic equations for heat conduction problems are strictly related to Stokes problems and in the book of Carslaw [13] a wide class of interesting cases are discussed and often solved with different techniques such as Fourier series or Laplace transform. In the latter case, the inversion theorem is adopted only for simple cases and no mention is given for multidimensional problems, which are considered instead in Section 15.11 but only for parallelepipeds with a prescribed condition on one face. In this way these examples are extensions of one-dimensional problems and not two- or three-dimensional real ones.When a bounded domain is considered and two spatial dimensions are involved, the deduction of analytical solutions is a no-trivial task. In Zeng and Weinbaum [2] the two-dimensional problem is approached with anad hoc definition of similar variables for unbounded domains and extended to bounded domains through an image superimposition technique. In a technical note by Liu and Liu [11], the one-dimensionalsecond Stokes’ problem is solved in an unbounded domain and for arbitrary initial phase; a complete analysis ofFirst andSecond Stokes’ problems in one- and two-dimensional bounded and unbounded domains can be found in Liu [14]. In this paper, the author rearranges the inversion integrals in such a way that the integrands are always straight invertible, with the aid of Laplace transform tables. This has to point out that the strategy adopted by these authors makes an extension to more general boundary conditions very hard, as well as the possibility of taking into account a pressure field. In a following paper, Liu [15] used the same strategy to find the solution of Stokes’ problems with porous wall in a two-dimensional unbounded domain. A similar strategy was also adopted in Khaled and Vafai [16].In the present paper, the use of the residuals theorem is exploited; once the path along which the complex integrals of the inverse transformation should be evaluated is defined, this theorem allows a straight deduction of the time- and space-dependent solution. As it will be demonstrated, in this way the analytical treatment of more general boundary conditions becomes possible. In particular, the analytical solutions of Stokes’ flows between moving walls with constant and sinusoidal velocities with different frequency and initial phase will be given.It will be also shown that, by using the actual strategy, the introduction of a constant or sinusoidal pressure gradient offers only a slight complication which can be easily overtaken; in fact, this term is normally a function of time only, Batchelor [1], so it plays the role of a simple nonhomogeneity in the transformed equation. However, even if the pressure is a product of two functions (one of which of time only), the achievement of a complete solution is possible in some cases (see Section 6.2).Moreover the extension of the present solving technique to two-dimensional problems avoids the onset of additional complications and will be the matter of a future work.The extension to generally periodic boundary conditions and pressure gradient is also possible; the development in Fourier series of both the boundary conditions and the pressure gradient permits superimposing the solutions for constant and sinusoidal cases, the problem being linear. In this light, the definitions of the first and the second Stokes’ problem are only particular cases of ageneralized Stokes problem, where the boundary conditions are referred to generally periodic functions.As it can be seen in the following, the expression of the analytical solutions could be rather complicated. In some cases, its computation involves the evaluation of integrals, which are solved by means of series expansions whose convergence could be rather slow. In order to check the correctness of the computed analytical solutions, an algorithm which resolves numerically the differential problem has been also implemented. The algorithm is based on a finite difference approximation of the Laplace operator for a generally nonuniform orthogonal distribution of the discretization points. The time integration is achieved by means of a Crank-Nicholson scheme; the resulting tridiagonal system of algebraic equations is solved by means of the classical Thomas algorithm. It is easy to demonstrate that the algorithm is unconditionally stable and globally second-order accurate (see, for example, Morton [17]). Both the algorithms for the computation of the analytical and the numerical solutions are available on request.The paper is organized as follows: in Section2, the methodology is explained and the classical one-dimensional solution for infinite depth is recalled. In Sections 3 and 4, the solutions of First and Second Stokes’ problem for the finite depth case are obtained. In Section 3, a free slip wall is considered on the top; this will highlight how the simple solutions obtained in the previous section can be easily used in a more complex cases. In Section 4, solutions are deduced for both walls with either constant or oscillating velocities. It has to be highlighted that the linearity of the problem allows obtaining these solutions as a superposition of solutions with only one wall with a nonzero velocity. In Section 5, the presence of a constant and sinusoidal pressure gradient is considered. Some examples are given in Section 6; in 6.1, a comparison with the numerical solution is presented, whereas in 6.2 an extension of the primary wave motion solution for a gravity wave in a viscous fluid is addressed. The aim is to show how the solutions presented are very easy to handle and to be used for different problems, achieving at the same time the best possible accuracy when problems modelled with a purely viscous equation are considered. In Section 7, a discussion on the possibility of handling more general boundary conditions and pressure gradient is carried out. In Section 8, conclusions and perspectives will wind up the paper. The extension to the solutions of two-dimensional problems will be the matter of future works.


## 2. One-Dimensional Infinite Depth Flows

A Newtonian fluid with kinematical viscosityν is contained in an unbounded half space with y>0. The boundary starts to move at time t=0 with a given velocity law f(t), moving in its own plane (y=0). The problem is clearly one-dimensional and the motion of the fluid is described by the solution of the Stokes’ problem:(1)∂tu=ν∂yy2uu0,t=ft,u+∞,t≡0uy,0≡0,where u is the velocity component parallel to the plane y=0. The First and Second Stokes’ problem are defined by the velocity law of the boundary y=0 for t>0: (2)FIRST  PROBLEMSECOND  PROBLEMft=u0ft=u0cos⁡σt+θbeing f(t)=0 at initial time. The problem is sketched in Figure 1.Figure 1
Sketch of the Stokes problem with an infinite depth.The solution of the problem could be obtained through a Laplace transformation in time [1], according to which(3)u^y,s=exp⁡-βyf^s.where β=s/ν and the transformed quantities are indicated with a hat.The solutions of the classical Stokes’ problem are briefly addressed, in order to establish the methodology. In the following, the nondimensional quantities are indicated with a tilde, where needed.


### 2.1. First Stokes’ Problem

In this casef^(s)=u0/s and the time derivative of the nondimensional velocity u~=u/u0 is(4)u~t=12πilimM→∞⁡∫μ-iMμ+iMdsexp⁡ts-βyμ being an arbitrary positive real number. The function u~t(y,t) is evaluated through an application of the Cauchy theorem on the path in Figure 2(a):(5)u~t=12πi∫-∞0dxexp⁡xt+iyν-x-exp⁡xt-iyν-x=1πη~e-η~2t∫0+∞dξe-ξ-iη~2+∫0+∞dξe-ξ+iη~2,where ξ2=xt and η~=y/(2νt). The two integrals in the above relation are evaluated along the lower and upper paths of Figure 2(b) and their sum gives π. As a consequence, u~t assumes the following form:(6)u~tη~,t=η~πtexp⁡-η~2.An integration in time leads to the classical solution:(7)u~η~=2π∫η~+∞dξe-ξ2=erfcη~.As it is evident, the solution depends on a similar variable η~. The wall stress τw, made nondimensional as τ~w=τw/(ρu02), ρ being the density of the fluid, follows as(8)τ~w=νu0u~yy=0=-1πt~defining t~=tu02/ν.Figure 2
Integration paths in the plane ofs: for the evaluation of the function u~t (a) and of the integrals I1,2.
(a) (b)


### 2.2. Second Stokes’ Problem

In second Stokes’ problem the Laplace transform of the wall velocity is(9)f^s=u02e-iθs+iσ+e+iθs-iσand solution (3) becomes(10)u~y,s=12e-iσt+θ12πi∫μ-i∞μ+i∞dsexp⁡s+iσt-βys+iσ︸G1+e+iσt+θ12πi∫μ-i∞μ+i∞dsexps-iσt-βys-iσ︸G2in which the time derivatives of the functions G1,2y,t are easily evaluated in terms of (6), so that the following is obtained: (11)G1,2y,t=G1,2y,0+2π∫η~∞dξexp⁡±iTη~2ξ2-ξ2.where T=σt. An integration along the path of Figure 2(a) proves that G1,2(y,0)=0 (see Appendix A), so the initial condition uy,0≡0 is satisfied. Solution (10) is as follows:(12)u~y~,T=cos⁡T+θ2π∫η~+∞dξe-ξ2cos⁡Tη~2ξ2+sin⁡T+θ2π∫η~+∞dξe-ξ2sin⁡Tη~2ξ2=cos⁡T+θIcη~,T+sin⁡T+θIsη~,T.The above solution is the real form of the solution in Erdogan [10] for θ=0 and θ=π/2 and of the solution in Liu and Liu [11]. Integrals Ic and Is can be calculated by recursive formulas for little values of T, as discussed in Appendix B.The asymptotic expression ofIc and Is for large T is as follows:(13)Icη~,T→e-y~cos⁡y~Isη~,T→e-y~sin⁡y~where y~=yσ/(2ν). In Figure 3 integrals Ic and Is are compared with their asymptotic expression for different times. For long times, the convergence is evident.Figure 3
Behaviour ofIc and Is integrals for short (red line), medium (green line), and long (blue line) times. Circular symbols represent the asymptotic solution (13).The wall stress is given by(14)τ~wT=-cos⁡θπt~+α′sin⁡T+θC1T-cos⁡T+θS1T,where C1 and S1 are the Fresnel integrals (see Abramowitz and Stegun [18] page 300 equations 7.3.3 and 7.3.4, respectively), while α2=σν/u02 and α′=2α. Note that the first term is the wall stress of the First Stokes problem, due to the initial nonvanishing value (cos⁡θ) of the wall velocity.


## 2.1. First Stokes’ Problem

In this casef^(s)=u0/s and the time derivative of the nondimensional velocity u~=u/u0 is(4)u~t=12πilimM→∞⁡∫μ-iMμ+iMdsexp⁡ts-βyμ being an arbitrary positive real number. The function u~t(y,t) is evaluated through an application of the Cauchy theorem on the path in Figure 2(a):(5)u~t=12πi∫-∞0dxexp⁡xt+iyν-x-exp⁡xt-iyν-x=1πη~e-η~2t∫0+∞dξe-ξ-iη~2+∫0+∞dξe-ξ+iη~2,where ξ2=xt and η~=y/(2νt). The two integrals in the above relation are evaluated along the lower and upper paths of Figure 2(b) and their sum gives π. As a consequence, u~t assumes the following form:(6)u~tη~,t=η~πtexp⁡-η~2.An integration in time leads to the classical solution:(7)u~η~=2π∫η~+∞dξe-ξ2=erfcη~.As it is evident, the solution depends on a similar variable η~. The wall stress τw, made nondimensional as τ~w=τw/(ρu02), ρ being the density of the fluid, follows as(8)τ~w=νu0u~yy=0=-1πt~defining t~=tu02/ν.Figure 2
Integration paths in the plane ofs: for the evaluation of the function u~t (a) and of the integrals I1,2.
(a) (b)


## 2.2. Second Stokes’ Problem

In second Stokes’ problem the Laplace transform of the wall velocity is(9)f^s=u02e-iθs+iσ+e+iθs-iσand solution (3) becomes(10)u~y,s=12e-iσt+θ12πi∫μ-i∞μ+i∞dsexp⁡s+iσt-βys+iσ︸G1+e+iσt+θ12πi∫μ-i∞μ+i∞dsexps-iσt-βys-iσ︸G2in which the time derivatives of the functions G1,2y,t are easily evaluated in terms of (6), so that the following is obtained: (11)G1,2y,t=G1,2y,0+2π∫η~∞dξexp⁡±iTη~2ξ2-ξ2.where T=σt. An integration along the path of Figure 2(a) proves that G1,2(y,0)=0 (see Appendix A), so the initial condition uy,0≡0 is satisfied. Solution (10) is as follows:(12)u~y~,T=cos⁡T+θ2π∫η~+∞dξe-ξ2cos⁡Tη~2ξ2+sin⁡T+θ2π∫η~+∞dξe-ξ2sin⁡Tη~2ξ2=cos⁡T+θIcη~,T+sin⁡T+θIsη~,T.The above solution is the real form of the solution in Erdogan [10] for θ=0 and θ=π/2 and of the solution in Liu and Liu [11]. Integrals Ic and Is can be calculated by recursive formulas for little values of T, as discussed in Appendix B.The asymptotic expression ofIc and Is for large T is as follows:(13)Icη~,T→e-y~cos⁡y~Isη~,T→e-y~sin⁡y~where y~=yσ/(2ν). In Figure 3 integrals Ic and Is are compared with their asymptotic expression for different times. For long times, the convergence is evident.Figure 3
Behaviour ofIc and Is integrals for short (red line), medium (green line), and long (blue line) times. Circular symbols represent the asymptotic solution (13).The wall stress is given by(14)τ~wT=-cos⁡θπt~+α′sin⁡T+θC1T-cos⁡T+θS1T,where C1 and S1 are the Fresnel integrals (see Abramowitz and Stegun [18] page 300 equations 7.3.3 and 7.3.4, respectively), while α2=σν/u02 and α′=2α. Note that the first term is the wall stress of the First Stokes problem, due to the initial nonvanishing value (cos⁡θ) of the wall velocity.


## 3. One-Dimensional Finite Depth Flows

Stokes’ problem is posed in the following way:(15)∂tu=ν∂yy2uu0,t=ft,∂yuh,t≡0uy,0≡0,h being the height of the fluid (hereafter, the corresponding nondimensional height y~=y/h will be used). Note that the second boundary condition enforces vanishing viscous stresses at the interface.The Laplace transform (in time) of the solution is(16)u^y,s=cosh⁡h-yβcosh⁡hβf^s.Note that the kernel cosh⁡h-yβ/cosh⁡hβ possesses in the s-plane a branch cut along the negative real axis and also a countable set of real and negative poles of the first order: by accounting for that, for any integer k, 2k+1 will be indicated with k′ and k′π/2 with K′, they are placed on the points sk=-K′2ν/h2. Kernel (16) being an even function of β results in being continuous across the above branch cut, with the exception of the poles.


### 3.1. First Stokes’ Problem

The time derivative of the nondimensional velocity is(17)u~t=12πi∫μ-i∞μ+i∞dsestcosh⁡h-yβcosh⁡hβ≔Fwhere the integral is evaluated by applying the residue theorem on the path in Figure 2(a):(18)F=2νh2∑k=0∞K′e-K′2νt/h2sin⁡K′yh.By accounting for the identity (see Abramowitz and Stegun [18], pag. 1005),(19)∑k=0∞sin⁡K′y/hK′≡12,the solution of the problem becomes(20)u~y~,t~=1-2∑k=0∞sin⁡K′y~K′exp⁡-K′2t~,where t~=tν/h2. The wall stress is given by(21)τ~w=-2Re0∑k=0∞exp⁡-K′2t~.where Re0=u0h/ν is the Reynolds number referring to the constant velocity u0 and to the distance h between walls.


### 3.2. Second Stokes’ Problem

By inserting the Laplace transform of the wall velocity (9) inside solution (16), its nondimensional value is as follows:(22)u~=12e-iσt+θ12πi∫μ-i∞μ+i∞dses+iσts+iσcosh⁡h-yβcosh⁡hβ︸L1+e+iσt+θ12πi∫μ-i∞μ+i∞dses-iσts-iσcosh⁡h-yβcosh⁡hβ︸L2.Two functions L1,2y,t are evaluated by observing that their time derivatives are related to the function F defined in (18): (23)∂tL1,2=e±iσtF,which are integrated by starting from the initial values:(24)L1,2y~,0=-2∑k=0∞K′sin⁡K′y~K′2∓iα2+cosh⁡α1∓i/21-y~cosh⁡α1∓i/2,where α=σ/νh. It follows the nondimensional solution:(25)u~y~,t~=cy~cos⁡T+θ+dy~sin⁡T+θ+-2∑k=0∞K′sin⁡K′y~K′4+α4e-K′2T/α2K′2cos⁡θ+α2sin⁡θ,where the two functions c(y~),d(y~) are given in Appendix C.In (25), the stationary part of the solution is separated from the transient one and can be conveniently written as Fourier series: (26)cy~cos⁡T+θ+dy~sin⁡T+θ=2∑k=0∞K′K′4+α4K′2cos⁡T+θ+α2sin⁡T+θsin⁡K′y~,allowing the straightforward evaluation of the wall stress:(27)τ~w=2Re0∑k=0∞K′2K′4+α4K′2cos⁡T+θ+α2sin⁡T+θ-K′2cos⁡θ+α2sin⁡θe-K′2T/α2


## 3.1. First Stokes’ Problem

The time derivative of the nondimensional velocity is(17)u~t=12πi∫μ-i∞μ+i∞dsestcosh⁡h-yβcosh⁡hβ≔Fwhere the integral is evaluated by applying the residue theorem on the path in Figure 2(a):(18)F=2νh2∑k=0∞K′e-K′2νt/h2sin⁡K′yh.By accounting for the identity (see Abramowitz and Stegun [18], pag. 1005),(19)∑k=0∞sin⁡K′y/hK′≡12,the solution of the problem becomes(20)u~y~,t~=1-2∑k=0∞sin⁡K′y~K′exp⁡-K′2t~,where t~=tν/h2. The wall stress is given by(21)τ~w=-2Re0∑k=0∞exp⁡-K′2t~.where Re0=u0h/ν is the Reynolds number referring to the constant velocity u0 and to the distance h between walls.


## 3.2. Second Stokes’ Problem

By inserting the Laplace transform of the wall velocity (9) inside solution (16), its nondimensional value is as follows:(22)u~=12e-iσt+θ12πi∫μ-i∞μ+i∞dses+iσts+iσcosh⁡h-yβcosh⁡hβ︸L1+e+iσt+θ12πi∫μ-i∞μ+i∞dses-iσts-iσcosh⁡h-yβcosh⁡hβ︸L2.Two functions L1,2y,t are evaluated by observing that their time derivatives are related to the function F defined in (18): (23)∂tL1,2=e±iσtF,which are integrated by starting from the initial values:(24)L1,2y~,0=-2∑k=0∞K′sin⁡K′y~K′2∓iα2+cosh⁡α1∓i/21-y~cosh⁡α1∓i/2,where α=σ/νh. It follows the nondimensional solution:(25)u~y~,t~=cy~cos⁡T+θ+dy~sin⁡T+θ+-2∑k=0∞K′sin⁡K′y~K′4+α4e-K′2T/α2K′2cos⁡θ+α2sin⁡θ,where the two functions c(y~),d(y~) are given in Appendix C.In (25), the stationary part of the solution is separated from the transient one and can be conveniently written as Fourier series: (26)cy~cos⁡T+θ+dy~sin⁡T+θ=2∑k=0∞K′K′4+α4K′2cos⁡T+θ+α2sin⁡T+θsin⁡K′y~,allowing the straightforward evaluation of the wall stress:(27)τ~w=2Re0∑k=0∞K′2K′4+α4K′2cos⁡T+θ+α2sin⁡T+θ-K′2cos⁡θ+α2sin⁡θe-K′2T/α2


## 4. One-Dimensional Flow between Moving Walls

The following Stokes’ problem is considered:(28)∂tu=ν∂yy2uu0,t=ft,uh,t=gtuy,0≡0,where the second boundary condition represents a moving wall with a given velocity law g(t).The Laplace transform in time of the solution is(29)u^y,s=f^ssinh⁡βh-ysinh⁡βh+g^ssinh⁡βysinh⁡βh.As in the previous case, the ratios possess in the s-plane a branch cut along the negative real axis and a countable set of real and negative poles of the first order. By indicating with K′′=kπ, they are placed on the points sk=-K′′2ν/h2 for k=1,2,…. The ratios sinh⁡βh-y/sinh⁡βh and sinh⁡βy/sinh⁡βh are even functions of β, so they are continuous across the branch cut with exception of the poles.The case with two walls is more complicated, because it could lead to a new class of problems. If the walls have the same velocity law (either constant or periodical), one of the classic Stokes problems arises, but if the laws are different the resulting problem is a new one, which could be denoted as themixed problem. Anyhow it is easy to show that the problem with one or both walls with constant velocity can be seen as a special case of the more general periodical velocity laws with zero frequency and initial phase.


### 4.1. First Stokes Problem

By assuming constant velocity laws for the wallsf^=u0/s and g^=v0/s and indicating with γ0=v0/u0, the time derivative of the velocity is(30)u~t=12πi∫μ-i∞μ+i∞dsestsinh⁡βh-ysinh⁡βh︸F1+γ02πi∫μ-i∞μ+i∞dsestsinh⁡βysinh⁡βh︸F2.The integrals are evaluated by applying the residue theorem on the path of Figure 2(a):(31)u~t=2νh2∑k=1∞K′′e-K′′2νt/h2sin⁡K′′yh1-γ0-1k.By taking into account the identity (see Abramowitz and Stegun [18], pag.1005), (32)∑k=1∞sin⁡K′′y/hK′′-1k≡-y2hthe solution is found with a simple integration in time:(33)u~y~,t~=1+γ0-1y~-2∑k=1∞sin⁡K′′y~K′′e-K′′2t~1-γ0-1kwhere t~=tν/h2 and y~=y/h. As expected, the stationary part of the solution is a linear function of y~ and it depends only on the ratio of the velocities on the walls.The shear stresses on the walls can be easily computed taken the spatial derivative of the previous relation; for the bottom wall, it reads(34)τ~wbt~=1h~γ0-1-2∑k=1∞e-K′′2t~1-γ0-1kwhile the stress on the upper wall returns:(35)τ~wut~=1h~γ0-1-2∑k=1∞e-K′′2t~-1k-γ0.


### 4.2. Second Stokes’ Problem

By assuming periodical velocity laws for the walls,(36)f^s=u02e-iθs+iσ+eiθs-iσg^s=v02e-iϕs+iω+eiϕs-iω,the inverse Laplace transform of the velocity is(37)u~y,t=12e-iσt+θ12πi∫μ-i∞μ+i∞dses+iσts+iσsinh⁡h-yβsinh⁡hβ︸S1+e+iσt+θ12πi∫μ-i∞μ+i∞dses-iσts-iσsinh⁡h-yβsinh⁡hβ︸S2+e-iωt+ϕγ02πi∫μ-i∞μ+i∞dses+iωts+iωsinh⁡yβsinh⁡hβ︸S3+e+iωt+ϕγ02πi∫μ-i∞μ+i∞dses-iωts-iωsinh⁡yβsinh⁡hβ︸S4.Two functions S1,2(y,t) are evaluated by observing that their time derivatives are related to the function F1 defined in (30): (38)∂tS1,2=e±iσtF1,which are integrated by starting from the initial values:(39)S1,2y~,0=-2∑k=1∞K′′sin⁡K′′y~K′′2∓iα2+sinh⁡α1∓i/21-y~sinh⁡α1∓i/2.where α=σ/νh. In the same way, the time derivatives of the functions S3,4y,t are related to function F2 defined in (30),(40)∂tS3,4=e±iωtF2,and are integrated by starting from the initial values:(41)S3,4y~,0=2∑k=1∞-1kK′′sin⁡K′′y~K′′2∓iα22+γ0sinh⁡α21∓i/2y~sinh⁡α21∓i/2,where α2=ω/νh. The solution may be rewritten in terms of S1,2,3,4 as(42)u~=12S1+S2cos⁡T+θ+iS2-S1sin⁡T+θ︷AA+S3+S4cos⁡T2+ϕ+iS4-S3sin⁡T2+ϕ︸BBwhere T2=ωt is the nondimensional time with the frequency of the top wall. In order to simplify the expression of the solution, avoiding the presence of complex terms, the following quantities are evaluated:(43)AA=-4∑k=1∞K′′sin⁡K′′y~K′′4+α4e-T/α2K′′2K′′2cos⁡θ+α2sin⁡θ+2c1y~cos⁡T+θ+2d1y~sin⁡T+θBB=+4γ0∑k=1∞K′′sin⁡K′′y~K′′4+α24e-T2/α22K′′2K′′2cos⁡ϕ+α22sin⁡ϕ-1k+2γ0c2y~cos⁡Tψ+ϕ+2γ0d2y~sin⁡Tψ+ϕwhere the complete expressions of the stationary terms c1, d1, c2, d2 are given in Appendix C. It follows the nondimensional solution:(44)u~y~,T,T2=u~sty~,T,T2+u~try~,T,T2where u~st is the steady-state part of the solution:(45)u~sty~,T,T2=c1y~cos⁡T+θ+d1y~sin⁡T+θ+γ0c2y~cos⁡T2+ϕ+d2y~sin⁡T2+ϕwhile u~tr is the transient part of the solution:(46)u~try~,T,T2=-2∑k=1∞K′′sin⁡K′′y~e-T/α2K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4+2∑k=1∞K′′sin⁡K′′y~e-T2/α22K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24γ0-1k.The shear stresses on the walls can be computed by deriving expression (44) with respect to the normal direction; their value for y~=0 gives the stress on the bottom wall: (47)τ~wbT,T2=c1′0cos⁡T+θ+d1′0sin⁡T+θ++γ0c2′0cos⁡T2+ϕ+d2′0sin⁡T2+ϕ+-2∑k=1∞K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4e-T/α2K′′2++2∑k=1∞K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24e-T2/α22K′′2γ0-1kwhile the value of the derivative at y~=1 gives the shear stress on the upper wall:(48)τ~wuT,T2=c1′1cos⁡T+θ+d1′1sin⁡T+θ++γ0c2′1cos⁡T2+ϕ+d2′1sin⁡T2+ϕ+-2∑k=1∞K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4e-T/α2K′′2-1k++2∑k=1∞K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24e-T2/α22K′′2γ0The expression of the derivatives c1′, d1′, c2′, d2′ can be found in Appendix C.As it has been anticipated, the solution for theFirst Stokes’ problem can be recovered from the solution of the problem with two moving boundaries (44). As a matter of the fact, by considering that c1(y~)→1-y~ for α→0 and c2(y~)→y~ for α2→0, the steady-state solution (45) goes to the stationary part of the (33), whereas, for the transient part it can be easily demonstrated that, by taking θ, ϕ, α, and α2 in (46) equal to zero, the transient part of (33) is recovered.


## 4.1. First Stokes Problem

By assuming constant velocity laws for the wallsf^=u0/s and g^=v0/s and indicating with γ0=v0/u0, the time derivative of the velocity is(30)u~t=12πi∫μ-i∞μ+i∞dsestsinh⁡βh-ysinh⁡βh︸F1+γ02πi∫μ-i∞μ+i∞dsestsinh⁡βysinh⁡βh︸F2.The integrals are evaluated by applying the residue theorem on the path of Figure 2(a):(31)u~t=2νh2∑k=1∞K′′e-K′′2νt/h2sin⁡K′′yh1-γ0-1k.By taking into account the identity (see Abramowitz and Stegun [18], pag.1005), (32)∑k=1∞sin⁡K′′y/hK′′-1k≡-y2hthe solution is found with a simple integration in time:(33)u~y~,t~=1+γ0-1y~-2∑k=1∞sin⁡K′′y~K′′e-K′′2t~1-γ0-1kwhere t~=tν/h2 and y~=y/h. As expected, the stationary part of the solution is a linear function of y~ and it depends only on the ratio of the velocities on the walls.The shear stresses on the walls can be easily computed taken the spatial derivative of the previous relation; for the bottom wall, it reads(34)τ~wbt~=1h~γ0-1-2∑k=1∞e-K′′2t~1-γ0-1kwhile the stress on the upper wall returns:(35)τ~wut~=1h~γ0-1-2∑k=1∞e-K′′2t~-1k-γ0.


## 4.2. Second Stokes’ Problem

By assuming periodical velocity laws for the walls,(36)f^s=u02e-iθs+iσ+eiθs-iσg^s=v02e-iϕs+iω+eiϕs-iω,the inverse Laplace transform of the velocity is(37)u~y,t=12e-iσt+θ12πi∫μ-i∞μ+i∞dses+iσts+iσsinh⁡h-yβsinh⁡hβ︸S1+e+iσt+θ12πi∫μ-i∞μ+i∞dses-iσts-iσsinh⁡h-yβsinh⁡hβ︸S2+e-iωt+ϕγ02πi∫μ-i∞μ+i∞dses+iωts+iωsinh⁡yβsinh⁡hβ︸S3+e+iωt+ϕγ02πi∫μ-i∞μ+i∞dses-iωts-iωsinh⁡yβsinh⁡hβ︸S4.Two functions S1,2(y,t) are evaluated by observing that their time derivatives are related to the function F1 defined in (30): (38)∂tS1,2=e±iσtF1,which are integrated by starting from the initial values:(39)S1,2y~,0=-2∑k=1∞K′′sin⁡K′′y~K′′2∓iα2+sinh⁡α1∓i/21-y~sinh⁡α1∓i/2.where α=σ/νh. In the same way, the time derivatives of the functions S3,4y,t are related to function F2 defined in (30),(40)∂tS3,4=e±iωtF2,and are integrated by starting from the initial values:(41)S3,4y~,0=2∑k=1∞-1kK′′sin⁡K′′y~K′′2∓iα22+γ0sinh⁡α21∓i/2y~sinh⁡α21∓i/2,where α2=ω/νh. The solution may be rewritten in terms of S1,2,3,4 as(42)u~=12S1+S2cos⁡T+θ+iS2-S1sin⁡T+θ︷AA+S3+S4cos⁡T2+ϕ+iS4-S3sin⁡T2+ϕ︸BBwhere T2=ωt is the nondimensional time with the frequency of the top wall. In order to simplify the expression of the solution, avoiding the presence of complex terms, the following quantities are evaluated:(43)AA=-4∑k=1∞K′′sin⁡K′′y~K′′4+α4e-T/α2K′′2K′′2cos⁡θ+α2sin⁡θ+2c1y~cos⁡T+θ+2d1y~sin⁡T+θBB=+4γ0∑k=1∞K′′sin⁡K′′y~K′′4+α24e-T2/α22K′′2K′′2cos⁡ϕ+α22sin⁡ϕ-1k+2γ0c2y~cos⁡Tψ+ϕ+2γ0d2y~sin⁡Tψ+ϕwhere the complete expressions of the stationary terms c1, d1, c2, d2 are given in Appendix C. It follows the nondimensional solution:(44)u~y~,T,T2=u~sty~,T,T2+u~try~,T,T2where u~st is the steady-state part of the solution:(45)u~sty~,T,T2=c1y~cos⁡T+θ+d1y~sin⁡T+θ+γ0c2y~cos⁡T2+ϕ+d2y~sin⁡T2+ϕwhile u~tr is the transient part of the solution:(46)u~try~,T,T2=-2∑k=1∞K′′sin⁡K′′y~e-T/α2K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4+2∑k=1∞K′′sin⁡K′′y~e-T2/α22K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24γ0-1k.The shear stresses on the walls can be computed by deriving expression (44) with respect to the normal direction; their value for y~=0 gives the stress on the bottom wall: (47)τ~wbT,T2=c1′0cos⁡T+θ+d1′0sin⁡T+θ++γ0c2′0cos⁡T2+ϕ+d2′0sin⁡T2+ϕ+-2∑k=1∞K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4e-T/α2K′′2++2∑k=1∞K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24e-T2/α22K′′2γ0-1kwhile the value of the derivative at y~=1 gives the shear stress on the upper wall:(48)τ~wuT,T2=c1′1cos⁡T+θ+d1′1sin⁡T+θ++γ0c2′1cos⁡T2+ϕ+d2′1sin⁡T2+ϕ+-2∑k=1∞K′′2K′′2cos⁡θ+α2sin⁡θK′′4+α4e-T/α2K′′2-1k++2∑k=1∞K′′2K′′2cos⁡ϕ+α22sin⁡ϕK′′4+α24e-T2/α22K′′2γ0The expression of the derivatives c1′, d1′, c2′, d2′ can be found in Appendix C.As it has been anticipated, the solution for theFirst Stokes’ problem can be recovered from the solution of the problem with two moving boundaries (44). As a matter of the fact, by considering that c1(y~)→1-y~ for α→0 and c2(y~)→y~ for α2→0, the steady-state solution (45) goes to the stationary part of the (33), whereas, for the transient part it can be easily demonstrated that, by taking θ, ϕ, α, and α2 in (46) equal to zero, the transient part of (33) is recovered.


## 5. One-Dimensional Flow between Moving Walls with Pressure Gradient

A useful extension of the Stokes’ problems can be achieved by considering the presence of a pressure gradient in the motion equation (see Xu and Bowen [5]; Lauga and Stone [3]); this term must be a function of time only [1], so that the Laplace transform of the solution is very similar to the previous case, allowing a direct inversion with the same procedure of Section 4. Stokes’ problem considered is as follows:(49)∂tu=-∂xP+ν∂yy2uu0,t=ft,uh,t=gtuy,0≡0,where P=p/ρ and the boundary conditions refer to oscillatory moving walls, since the case with constant velocity can be seen as a particular case (see Section 4). A constant -∂xP=λ0 and sinusoidal time law -∂xP=λ0cos⁡χt+Θ are considered for the new term; in this way, as it will be shown in Section 7, the solution for the case with a generally periodical function of time for the pressure gradient can be easily obtained by exploiting the linearity of the problem and taking the Fourier series of the pressure term. The Laplace transform in time of the solution (49) is(50)u^y,s=f^ssinh⁡βh-ysinh⁡βh+g^ssinh⁡βysinh⁡βh-λ^sssinh⁡βh-ysinh⁡βh+sinh⁡βysinh⁡βh-1.The first two quantities in the right-hand side are the solution of the associated homogeneous equation (the same just discussed in Section 4); similarly the pressure term is composed by two terms with the same poles sk=-K′′2ν/h2 plus a pole on the origin and a last term with a single pole on the origin.The inverse Laplace transform of the velocity is a linear combination of the solutions deduced in Section4 (the first or second Stokes’ problem according to the expressions of f(t) and g(t)), so only the inversion of pressure term, indicated with u^p, will be here discussed: (51)upy,t=12πi∫μ-i∞μ+i∞dsestλ^ssinh⁡h-yβsinh⁡hβ+sinh⁡yβsinh⁡hβ-∫0tdtλt


### 5.1. Constant Pressure Gradient

By assuming a constant pressure gradient, the expression ofutp is as follows:(52)utp=λ02πi∫μ-i∞μ+i∞dsestssinh⁡h-yβsinh⁡hβ+sinh⁡yβsinh⁡hβ-λ0.A simple integration in time, starting by the initial value,(53)u0p=λ0h22νy~2-y~+4λ0h2ν∑k=1∞sin⁡Koy~Ko3,which can be derived from the general expression of the inverse transform of u^p for t=0, leads to the particular solution:(54)u~py~,t~=Hg2Re0y~2-y~+4HgRe0∑k=1∞sin⁡Koy~Ko3e-Ko2t~where the stationary part of the solution is the well-known Poiseuille solution between planar solid walls. In the previous relations Ko=2k-1π and Hg=λ0h3/ν2 is the Hagen number. It is worth noticing that, from the identity(55)∑k=1∞sin⁡Koy~Ko3≡18y~-y~2(see Abramowitz and Stegun [18] pag. 1005) the initial condition u~p(y~,0)=0 is satisfied.


### 5.2. Sinusoidal Pressure Gradient

By assuming a sinusoidal law for the pressure gradient, the expression ofup is as follows:(56)upy,t=λ02Z1+Z2+Z3+Z4+λ0χsin⁡Θ-sin⁡χt+Θwhere(57)Z1,2=e∓iχt+Θ12πi∫μ-i∞μ+i∞dses±iχtss±iχsinh⁡h-yβsinh⁡hβ(58)Z3,4=e∓iχt+Θ12πi∫μ-i∞μ+i∞dses±iχtss±iχsinh⁡yβsinh⁡hβ.By observing that the time derivatives of Z1,2 are similar to S1,2 and the derivatives of Z3,4 are similar to S3,4 (the sums Z1+Z2 and Z3+Z4 are given in Appendix D), a simple integration in time gives back the solution:(59)u~py~,T3=Hgα3Re0u~stp+u~trpwhere the definition of α3 is given in Appendix D. The stationary part of the solution is(60)u~stpy~,T3=-sin⁡T3+Θ+c1y~+c2y~sin⁡T3+Θ-d1y~+d2y~cos⁡T3+Θwhile the transient part is(61)u~trpy~,T3=4α32∑k=1∞sin⁡Koy~KoKo4+α34e-Ko2T3/α32Ko2cos⁡Θ+α32sin⁡Θwhereas the definition of T3 is given in Appendix D.It is useful to notice that(62)limχ→0c1y~+c2y~=1limχ→0d1y~+d2y~Hgα3Re0=Hg2Re0y~-y~2so that, for Θ=0, the limit for χ→0 of u~stp goes to the stationary part of the solution (54), while for the transient part u~trp the limit is obvious.


## 5.1. Constant Pressure Gradient

By assuming a constant pressure gradient, the expression ofutp is as follows:(52)utp=λ02πi∫μ-i∞μ+i∞dsestssinh⁡h-yβsinh⁡hβ+sinh⁡yβsinh⁡hβ-λ0.A simple integration in time, starting by the initial value,(53)u0p=λ0h22νy~2-y~+4λ0h2ν∑k=1∞sin⁡Koy~Ko3,which can be derived from the general expression of the inverse transform of u^p for t=0, leads to the particular solution:(54)u~py~,t~=Hg2Re0y~2-y~+4HgRe0∑k=1∞sin⁡Koy~Ko3e-Ko2t~where the stationary part of the solution is the well-known Poiseuille solution between planar solid walls. In the previous relations Ko=2k-1π and Hg=λ0h3/ν2 is the Hagen number. It is worth noticing that, from the identity(55)∑k=1∞sin⁡Koy~Ko3≡18y~-y~2(see Abramowitz and Stegun [18] pag. 1005) the initial condition u~p(y~,0)=0 is satisfied.


## 5.2. Sinusoidal Pressure Gradient

By assuming a sinusoidal law for the pressure gradient, the expression ofup is as follows:(56)upy,t=λ02Z1+Z2+Z3+Z4+λ0χsin⁡Θ-sin⁡χt+Θwhere(57)Z1,2=e∓iχt+Θ12πi∫μ-i∞μ+i∞dses±iχtss±iχsinh⁡h-yβsinh⁡hβ(58)Z3,4=e∓iχt+Θ12πi∫μ-i∞μ+i∞dses±iχtss±iχsinh⁡yβsinh⁡hβ.By observing that the time derivatives of Z1,2 are similar to S1,2 and the derivatives of Z3,4 are similar to S3,4 (the sums Z1+Z2 and Z3+Z4 are given in Appendix D), a simple integration in time gives back the solution:(59)u~py~,T3=Hgα3Re0u~stp+u~trpwhere the definition of α3 is given in Appendix D. The stationary part of the solution is(60)u~stpy~,T3=-sin⁡T3+Θ+c1y~+c2y~sin⁡T3+Θ-d1y~+d2y~cos⁡T3+Θwhile the transient part is(61)u~trpy~,T3=4α32∑k=1∞sin⁡Koy~KoKo4+α34e-Ko2T3/α32Ko2cos⁡Θ+α32sin⁡Θwhereas the definition of T3 is given in Appendix D.It is useful to notice that(62)limχ→0c1y~+c2y~=1limχ→0d1y~+d2y~Hgα3Re0=Hg2Re0y~-y~2so that, for Θ=0, the limit for χ→0 of u~stp goes to the stationary part of the solution (54), while for the transient part u~trp the limit is obvious.


## 6. Examples

In this section, two examples of solutions for the extended Stokes’s problem are given; namely, a general case of Stokes’ flow within two moving walls without and with a pressure gradient (see also Durante and Broglia [19]) and a more physical application in which the complete primary wave motion for gravity waves will be given as an extension of the solutions presented in Xu and Bowen [5]. The problems are outlined in Figure 4 for the sake of clearness.Figure 4
Sketches of the example problems presented in Section6.1 (left) and in Section 6.2 (right).


### 6.1. Flow within Two Moving Walls

In this section, the cases of the Stokes’ flow within two moving walls without (Section4) or with (Section 5) a pressure gradient are presented. For both cases the two walls have an oscillatory movement, in particular, the problem is defined by the following periodical boundary conditions:(63)u0,t=ft=u0cos⁡σt+θu1,t=gt=v0cos⁡ωt+ϕwith v0=0.5u0, σ=10π, ω=20π, θ=π/4 and ϕ=0. For the problem with nonzero pressure gradient, the following law has been set:(64)-∂xP=λ0cos⁡χt+Θwith λ0=10, χ=3/10π and Θ=π/2. In Figure 5, the solutions at six time instants are given; for the sake of completeness, the transient and the steady-state components of the solutions are individually plotted as well. The time instants shown are chosen between an early stage (t=0.01) (where the transient part is still of the same order of the stationary part), passing through an intermediate stage (t=0.07) (the energy associated with the transient part is only a small fraction of that associated with the steady-state part) and finally to a late stage (t=1.84) (the transient part is negligible).Figure 5
One-dimensional flow between two walls with periodical motion. Solid line: global solution; dashed line: transient part; dashed-dotted line, stationary part.In order to give a comparison with the case where a pressure gradient is enforced, Figure6 is sketched. In an initial stage the contributions coming from stationary and transient parts are higher respect to the former case. The transient part becomes practically zero at the late stage, as before. The correctness of the analytical solutions and of their evaluation has been checked by a comparison with a numerical solution of the related Stokes’ problem. The solutions and the comparison errors are given in Figure 7; the comparison error, which is substantially the truncation error of the numerical scheme, is always less than 2.510-5. It has been tested that this error goes to zero accordingly with the accuracy of the numerical scheme.Figure 6
One-dimensional flow between two walls with periodical motion and pressure gradient. Solid line: global solution; dashed line: transient part; dashed-dotted line, stationary part.Figure 7
One-dimensional flow between two walls with periodical motion. The analytical solution is drawn with a solid line, whereas the error between numerical and analytical solutions is in dashed line (the reference axis is plotted on top). Top row, zero pressure gradient; bottom row, periodical pressure gradient.


### 6.2. The Complete Primary Wave Motion for Gravity Waves

The motion field of a gravity wave has been largely investigated since Stokes in1847 found that a net material transport (called Stokes’ drift) is associated with a periodic wave field, being in this way responsible for an advection process in the dispersion of passive scalars over a fluid surface. The equations of motion of a gravity wave (described as fs=acos⁡kx-σt) in a viscous and incompressible fluid are normally written in terms of Stokes problems:(65)∂u∂t=-1ρ∂p∂x+ν∂2u∂z2∂w∂t=-1ρ∂p∂z+ν∂2w∂z2∂u∂x+∂w∂z=0where p=pd+ρgz is the total pressure, i.e., the sum of the hydrostatic and the dynamical contributions. For a finite depth flow, the inviscid solution is achieved by supposing the velocity field to be irrotational and the amplitude a of the surface wave small enough that the kinematic condition p=0, to be satisfied on the surface z=fs, is instead considered on z=0. In this way the pressure field is found to be a periodical function of time:(66)p=ρgacosh⁡kz′cosh⁡khcos⁡kx-σtwhere z′=z+h, h is the distance of mean free surface (z=0) from the bottom and k is the wave-number (see Xu and Bowen [5]). The u and w components of the velocity field are found through direct integration in time of x and z pressure derivatives, respectively, giving (67)unv=aσcosh⁡kz′sinh⁡khcos⁡kx-σtwnv=aσsinh⁡kz′sinh⁡khsin⁡kx-σtwhere σ2=gktanh⁡kh and subscript nv stands fornonviscous solutions. Normally in earth sciences the transient phenomena are not interesting on large-scale problems, often adopting simplified approaches to achieve analytical solutions. This is evident in the paper Xu and Bowen [5], where the perturbed potential approach has been exploited for resolving the problem (65). A more detailed solution can be derived without any further simplification, by a simple application of the procedure discussed in the previous sections.By considering a wave travelling alongx direction in deep water (where the effect of viscosity is not taken into account) which encounters an increasing bottom, it is clear that the solution passes from a nonviscous to a viscous one. In the classical approach of earth sciences the unsteady effect which comes from this event is normally neglected. Nonetheless, it will be pointed out that once the complete sets of the Stokes problems solutions have been deduced, a solution, which considers transient effects, is simple enough to be achieved.The boundary conditions(68)u0,x,z′=unv0,x,z′ut,x,0=0u-unv→0forlargez′for u hold, the homogeneity at z′=0 being enforced for w.The solutionu will be extensively discussed, while w will be deduced by a direct integration of the continuity equation.The viscous solution is associated with a different and unknown pressure field, which, in general, is a function oft,x,z. By considering the nonviscous solution, the pressure field must be the product of a function of z only and a function of x and t, as it will be clear later.The Laplace transform of the solution is(69)u^v=Ae-βz′+Be+βz′+f^where f^ is the particular integral. By considering that for large z′ the viscous solution must match the nonviscous one, the following condition holds:(70)Beβz′+f^=u^nvwhere a term of kind Beβz′ is not present in u^nv. Obviously, B must be zero in order to allow this identity and f^ is equal to the Laplace transform of the nonviscous solution. The other constant A is found by the no-slip boundary condition, then the antitransform is as follows:(71)u=unv-aσsinh⁡kh12πi∫μ-i∞μ+i∞dsexp⁡st-βz′2e-ikxs-iσ+e+ikxs+iσwhere the integral can be directly evaluated, having the expressions of G1,2 (see Section 2.2). The final expression of the solution can be finally found as(72)u~=u~nv-1sinh⁡khcos⁡kx-TIcη~,T-sin⁡kx-TIsη~,Twhere u~=u/(aσ), T=σt and Ic, Is are the integral expressions described in Section 2.2 and numerically investigated in Appendix (Appendix B), with η~=z~/2T (z~=z′σ/(2ν)). Solution (72) must match the nonviscous solution for z~→∞ and this is guaranteed by the behaviour of the (12) for y~ diverging.As it can be clearly noticed, the solution is composed by a termunv which resolves the nonviscous equation with a prescribed pressure gradient and by a second term uv which resolves a viscous equation without a pressure gradient but with a prescribed oscillating boundary condition on the bottom and a homogeneous boundary condition at infinite.For largeT, solution (72) recovers the simplified one:(73)u~=u~nv-e-z~sinh⁡khcos⁡kx-T+z~presented in Xu and Bowen [5], as it can be noticed from the behaviour of integrals Ic and Is sketched in Figure 3.By exploiting the continuity, thez-component of velocity w is found to be straightforward:(74)w~=w~nv-1sinh⁡khkχsin⁡kx-T∫0z~dzIc+cos⁡kx-T∫0z~dzIswhere χ2=σ/(2ν). The expression of Xu and Bowen [5] can be recovered, by considering (13).In order to establish the behaviour of the complete solution with respect to the approximate one, the velocity field may be considered as the superposition of a nonviscous and a viscous field:(75)u=unv+uvw=wnv+wvand the viscous kinetic energy(76)Ecρa3σ2=E~c=12∫0z~mdzuv2+wv2(z~m is the nondimensional distance from the bottom) can be evaluated. In Figure 8 the kinetic energy associated with the viscous part of the velocity fields (72) and (74) is shown in comparison with the same fields of approximate solutions. The difference is initially of about 10% and it reduces to 3% for T=40 and to less than 1% for T=80 (not shown). This means that, for a gravity wave with a wavelength of 62.8 meters (σ≈1s-1), which passes from a deep to a 50-meter bottom, the kinetic energy associated with the viscous part is described with a satisfactory agreement (error less than 1%) with the approximate solutions after 80 seconds.Figure 8
On the top, the viscous kinetic energy calculated with the approximate solutions of Xu and Bowen [5] is plotted with symbols and compared with the analytical solution, represented with solid line. On the top right, the magnifying of the comparison for short times. On the bottom, the difference between the two solutions. In abscissa, the time is nondimensional with frequency σ. The values of energy are multiplied for 105.To get the pressure field, it is worth noticing thatu^nv satisfies the nonhomogeneous transformed equation, so it can be inserted into together with the transformed pressure gradient, arriving to the complete expression of the pressure field:(77)p=ρgacosh⁡kz′cosh⁡khcos⁡kx-σt+νk2σsin⁡kx-σtwhich returns (66) for ν→0. The pressure field possess two components: one is in phase with the velocity field unv along the propagation direction, while the other component (the viscous) is in phase with the vertical component of velocity wnv. As the decomposition of the velocity field (75) is valid, it similarly holds for the pressure. In particular, the gradient of the viscous component equals the Laplacian of the nonviscous velocity.


## 6.1. Flow within Two Moving Walls

In this section, the cases of the Stokes’ flow within two moving walls without (Section4) or with (Section 5) a pressure gradient are presented. For both cases the two walls have an oscillatory movement, in particular, the problem is defined by the following periodical boundary conditions:(63)u0,t=ft=u0cos⁡σt+θu1,t=gt=v0cos⁡ωt+ϕwith v0=0.5u0, σ=10π, ω=20π, θ=π/4 and ϕ=0. For the problem with nonzero pressure gradient, the following law has been set:(64)-∂xP=λ0cos⁡χt+Θwith λ0=10, χ=3/10π and Θ=π/2. In Figure 5, the solutions at six time instants are given; for the sake of completeness, the transient and the steady-state components of the solutions are individually plotted as well. The time instants shown are chosen between an early stage (t=0.01) (where the transient part is still of the same order of the stationary part), passing through an intermediate stage (t=0.07) (the energy associated with the transient part is only a small fraction of that associated with the steady-state part) and finally to a late stage (t=1.84) (the transient part is negligible).Figure 5
One-dimensional flow between two walls with periodical motion. Solid line: global solution; dashed line: transient part; dashed-dotted line, stationary part.In order to give a comparison with the case where a pressure gradient is enforced, Figure6 is sketched. In an initial stage the contributions coming from stationary and transient parts are higher respect to the former case. The transient part becomes practically zero at the late stage, as before. The correctness of the analytical solutions and of their evaluation has been checked by a comparison with a numerical solution of the related Stokes’ problem. The solutions and the comparison errors are given in Figure 7; the comparison error, which is substantially the truncation error of the numerical scheme, is always less than 2.510-5. It has been tested that this error goes to zero accordingly with the accuracy of the numerical scheme.Figure 6
One-dimensional flow between two walls with periodical motion and pressure gradient. Solid line: global solution; dashed line: transient part; dashed-dotted line, stationary part.Figure 7
One-dimensional flow between two walls with periodical motion. The analytical solution is drawn with a solid line, whereas the error between numerical and analytical solutions is in dashed line (the reference axis is plotted on top). Top row, zero pressure gradient; bottom row, periodical pressure gradient.


## 6.2. The Complete Primary Wave Motion for Gravity Waves

The motion field of a gravity wave has been largely investigated since Stokes in1847 found that a net material transport (called Stokes’ drift) is associated with a periodic wave field, being in this way responsible for an advection process in the dispersion of passive scalars over a fluid surface. The equations of motion of a gravity wave (described as fs=acos⁡kx-σt) in a viscous and incompressible fluid are normally written in terms of Stokes problems:(65)∂u∂t=-1ρ∂p∂x+ν∂2u∂z2∂w∂t=-1ρ∂p∂z+ν∂2w∂z2∂u∂x+∂w∂z=0where p=pd+ρgz is the total pressure, i.e., the sum of the hydrostatic and the dynamical contributions. For a finite depth flow, the inviscid solution is achieved by supposing the velocity field to be irrotational and the amplitude a of the surface wave small enough that the kinematic condition p=0, to be satisfied on the surface z=fs, is instead considered on z=0. In this way the pressure field is found to be a periodical function of time:(66)p=ρgacosh⁡kz′cosh⁡khcos⁡kx-σtwhere z′=z+h, h is the distance of mean free surface (z=0) from the bottom and k is the wave-number (see Xu and Bowen [5]). The u and w components of the velocity field are found through direct integration in time of x and z pressure derivatives, respectively, giving (67)unv=aσcosh⁡kz′sinh⁡khcos⁡kx-σtwnv=aσsinh⁡kz′sinh⁡khsin⁡kx-σtwhere σ2=gktanh⁡kh and subscript nv stands fornonviscous solutions. Normally in earth sciences the transient phenomena are not interesting on large-scale problems, often adopting simplified approaches to achieve analytical solutions. This is evident in the paper Xu and Bowen [5], where the perturbed potential approach has been exploited for resolving the problem (65). A more detailed solution can be derived without any further simplification, by a simple application of the procedure discussed in the previous sections.By considering a wave travelling alongx direction in deep water (where the effect of viscosity is not taken into account) which encounters an increasing bottom, it is clear that the solution passes from a nonviscous to a viscous one. In the classical approach of earth sciences the unsteady effect which comes from this event is normally neglected. Nonetheless, it will be pointed out that once the complete sets of the Stokes problems solutions have been deduced, a solution, which considers transient effects, is simple enough to be achieved.The boundary conditions(68)u0,x,z′=unv0,x,z′ut,x,0=0u-unv→0forlargez′for u hold, the homogeneity at z′=0 being enforced for w.The solutionu will be extensively discussed, while w will be deduced by a direct integration of the continuity equation.The viscous solution is associated with a different and unknown pressure field, which, in general, is a function oft,x,z. By considering the nonviscous solution, the pressure field must be the product of a function of z only and a function of x and t, as it will be clear later.The Laplace transform of the solution is(69)u^v=Ae-βz′+Be+βz′+f^where f^ is the particular integral. By considering that for large z′ the viscous solution must match the nonviscous one, the following condition holds:(70)Beβz′+f^=u^nvwhere a term of kind Beβz′ is not present in u^nv. Obviously, B must be zero in order to allow this identity and f^ is equal to the Laplace transform of the nonviscous solution. The other constant A is found by the no-slip boundary condition, then the antitransform is as follows:(71)u=unv-aσsinh⁡kh12πi∫μ-i∞μ+i∞dsexp⁡st-βz′2e-ikxs-iσ+e+ikxs+iσwhere the integral can be directly evaluated, having the expressions of G1,2 (see Section 2.2). The final expression of the solution can be finally found as(72)u~=u~nv-1sinh⁡khcos⁡kx-TIcη~,T-sin⁡kx-TIsη~,Twhere u~=u/(aσ), T=σt and Ic, Is are the integral expressions described in Section 2.2 and numerically investigated in Appendix (Appendix B), with η~=z~/2T (z~=z′σ/(2ν)). Solution (72) must match the nonviscous solution for z~→∞ and this is guaranteed by the behaviour of the (12) for y~ diverging.As it can be clearly noticed, the solution is composed by a termunv which resolves the nonviscous equation with a prescribed pressure gradient and by a second term uv which resolves a viscous equation without a pressure gradient but with a prescribed oscillating boundary condition on the bottom and a homogeneous boundary condition at infinite.For largeT, solution (72) recovers the simplified one:(73)u~=u~nv-e-z~sinh⁡khcos⁡kx-T+z~presented in Xu and Bowen [5], as it can be noticed from the behaviour of integrals Ic and Is sketched in Figure 3.By exploiting the continuity, thez-component of velocity w is found to be straightforward:(74)w~=w~nv-1sinh⁡khkχsin⁡kx-T∫0z~dzIc+cos⁡kx-T∫0z~dzIswhere χ2=σ/(2ν). The expression of Xu and Bowen [5] can be recovered, by considering (13).In order to establish the behaviour of the complete solution with respect to the approximate one, the velocity field may be considered as the superposition of a nonviscous and a viscous field:(75)u=unv+uvw=wnv+wvand the viscous kinetic energy(76)Ecρa3σ2=E~c=12∫0z~mdzuv2+wv2(z~m is the nondimensional distance from the bottom) can be evaluated. In Figure 8 the kinetic energy associated with the viscous part of the velocity fields (72) and (74) is shown in comparison with the same fields of approximate solutions. The difference is initially of about 10% and it reduces to 3% for T=40 and to less than 1% for T=80 (not shown). This means that, for a gravity wave with a wavelength of 62.8 meters (σ≈1s-1), which passes from a deep to a 50-meter bottom, the kinetic energy associated with the viscous part is described with a satisfactory agreement (error less than 1%) with the approximate solutions after 80 seconds.Figure 8
On the top, the viscous kinetic energy calculated with the approximate solutions of Xu and Bowen [5] is plotted with symbols and compared with the analytical solution, represented with solid line. On the top right, the magnifying of the comparison for short times. On the bottom, the difference between the two solutions. In abscissa, the time is nondimensional with frequency σ. The values of energy are multiplied for 105.To get the pressure field, it is worth noticing thatu^nv satisfies the nonhomogeneous transformed equation, so it can be inserted into together with the transformed pressure gradient, arriving to the complete expression of the pressure field:(77)p=ρgacosh⁡kz′cosh⁡khcos⁡kx-σt+νk2σsin⁡kx-σtwhich returns (66) for ν→0. The pressure field possess two components: one is in phase with the velocity field unv along the propagation direction, while the other component (the viscous) is in phase with the vertical component of velocity wnv. As the decomposition of the velocity field (75) is valid, it similarly holds for the pressure. In particular, the gradient of the viscous component equals the Laplacian of the nonviscous velocity.


## 7. Discussion

The classical approach to Stokes’ problems consists in solving a parabolic linear differential problem with two different kind of boundary conditions: constant or sinusoidal. These problems may be solved for bounded or unbounded domains and for Dirichlet or Neumann boundary conditions.When a wall (see Section2) or two walls (Section 4) move with a generally periodical time law, the problem may be approached developing in Fourier series both the functions describing the movement of the boundaries and the periodical pressure gradient.In order to underline this point, consider the differential problem (49) and assume λ(t) to be a periodical function with a period τ; the pressure term in (50) may be developed in Fourier series and recast as(78)u^ps+λ^s=λ0s2sinh⁡βh-ysinh⁡βh+sinh⁡βysinh⁡βh+∑l=1∞λlc2s1s+iL+1s-iLsinh⁡βh-ysinh⁡βh+sinh⁡βysinh⁡βh+∑l=1∞λls2sis+iL-is-iLsinh⁡βh-ysinh⁡βh+sinh⁡βysinh⁡βhwhere the inversion of the first row leads to a solution of kind (54), while the second and the third rows lead to a linear superposition of solutions of kind (59). The term λ^/s has been moved to the left-hand side and is not discussed, because the inversion is a trivial integration in time.


## 8. Conclusions and Future Perspectives

In this paper a general methodology for achieving the analytical solution of the one-dimensional Stokes’ problem has been illustrated; solutions for constant and periodical velocity laws of the boundaries have been deduced. The present strategy has allowed taking into account the presence of a time-dependent pressure gradient. Moreover, differently from other approaches, the residual theorem allows with a reasonable mathematical effort getting time solutions in closed forms.Solutions for constant and periodical forcing term have been derived and the complete primary motion field for a gravity wave has been achieved. The generalization to generally periodic boundary conditions and pressure gradient has been also discussed.Numerical solutions of the Stokes’ problem, provided by a second-order finite difference scheme, have been employed in order to check the correctness of the analytical solutions, as well as of their evaluations.Thanks to the present approach the extension to two-dimensional problems with nonzero time-dependent pressure gradient is possible and will be the matter of future works. The extension of this strategy to different geometries is also under investigation.



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