2. Problem formulation

To study the effect of vibrations on the resistance present in the relative movement of parallel plates, consider two surfaces that extend to ${\pm} \infty$ in the $X$ direction and placed a distance $2h$ apart; see figure 1. The lower plate is stationary whilst the upper plate is driven by an external force at a constant prescribed velocity ${U_{top}}$. The external force required to maintain this movement provides a measure of the movement resistance. In an attempt to reduce this resistance travelling-wave vibrations are imposed on the lower plate so that the dimensionless slot geometry is defined by

(2.1a)\begin{gather}{Y_U}(X,t) = 1,\end{gather}

(2.1b)\begin{gather}{Y_L}(X,t) ={-} 1 + A\cos [\alpha (X - ct)],\end{gather}

where the subscripts U and L refer to the upper and lower plates, respectively, and A is the wave amplitude. Furthermore c and $\alpha $ denote the wave phase speed and wavenumber and the half-mean slot opening h has been adopted as the length scale.

Figure 1. A sketch of the flow configuration.

When the system is in a steady state the external force pulling the plate is balanced by the shear force that acts on it. The flow field is determined by solving the Navier–Stokes and continuity equation written the form

(2.2a–c)\begin{equation}\begin{array}{ccccc} & \dfrac{{\partial u}}{{\partial t}} + u\dfrac{{\partial u}}{{\partial X}} + v\dfrac{{\partial u}}{{\partial Y}} ={-} \dfrac{{\partial p}}{{\partial X}} + \dfrac{{{\partial ^2}u}}{{\partial {X^2}}} + \dfrac{{{\partial ^2}u}}{{\partial {Y^2}}},\\ & \quad \dfrac{{\partial v}}{{\partial t}} + u\dfrac{{\partial v}}{{\partial X}} + v\dfrac{{\partial v}}{{\partial Y}} ={-} \dfrac{{\partial p}}{{\partial Y}} + \dfrac{{{\partial ^2}v}}{{\partial {X^2}}} + \dfrac{{{\partial ^2}v}}{{\partial {Y^2}}},\quad \dfrac{{\partial u}}{{\partial X}} + \dfrac{{\partial v}}{{\partial Y}} = 0, \end{array}\end{equation}

where $\boldsymbol{u} = (u,v)$ denotes the velocity vector with components in the $(X,Y)$ directions scaled on ${U_v} = \nu /h$, where $\nu $ denotes the kinematic viscosity. The dimensionless pressure p is scaled on $\rho U_v^2$ and the time t is based on $h/{U_v}$; here $\rho $ denotes the fluid density. The relevant boundary conditions are then

(2.3a–d)\begin{equation}Y = 1:\textrm{ }u = Re,\quad v = 0;\quad Y = {Y_L}(X,t):\textrm{ }u = 0,\quad v = \frac{{\partial {Y_L}}}{{\partial t}},\end{equation}

where

(2.4)\begin{equation}Re \equiv {U_{top}}h/\nu \end{equation}

is the Reynolds number and the normal velocity component at the lower plate is determined by the imposed vibrations. As our interest is in the fluid movement generated by relative movement of the plate as opposed to any pressure gradient that may be present, it is specified that

(2.5)\begin{equation}{\left. {\frac{{\partial p}}{{\partial X}}} \right|_m} = 0,\end{equation}

where subscript m denotes the mean value. The translation of the upper plate in the absence of any vibrations creates a simple Couette flow. In this motion the dimensionless velocity field ${\boldsymbol{v}_0}$, pressure ${p_0}$, pulling force (per unit length and width) ${F_0}$, shear acting on the upper plate ${\tau _0}$ and the flow rate ${Q_0}$ are given by

(2.6a–e)\begin{gather} \begin{gathered}{\boldsymbol{v}_0}(x,y) = [{u_0},{v_0}] = [{\textstyle{1 \over 2}}(1 + Y),\;0],\quad {p_0}(x,y) = \textrm{const}\textrm{.},\quad {F_0} = {\textstyle{1 \over 2}},\\ {\tau _0} = {-} {\textstyle{1 \over 2}},\quad {Q_0} = 1.\end{gathered}\end{gather}

Here ${U_{top}}$ has been adopted as the velocity scale, $\rho U_{top}^2$ as the pressure scale and ${U_{top}}\mu /h$ as the surface force scale; further the flow rate has also been scaled using ${U_{top}}$.

To evaluate the pulling force after the vibrations have been added we integrate the shear stress on the upper plate over a wavelength. The viscous part of the x component of the stress vector ${\sigma _{Xv,U}}$ that acts on the fluid and the external pulling force (per unit length and width) on the plate F are given by

(2.7a,b)\begin{equation}{\sigma _{Xv,U}} = {\left. {\frac{{\partial u}}{{\partial Y}}} \right|_{Y = 1}},\quad F = {\lambda ^{ - 1}}\int_0^\lambda {{{\left. {\frac{{\partial u}}{{\partial Y}}} \right|}_{Y = 1}}\,\textrm{d}X} ;\end{equation}

here the wavelength $\lambda \equiv 2{\rm \pi}/\alpha$. The ultimate effectiveness of the vibrations depends on their potential to reduce the shear stress at the upper plate – this can be achieved by propelling the fluid through the slot thereby reducing its velocity relative to the moving plate. The propulsion can be generated by a combination of viscous and pressure forces that act on the fluid at the vibrating plate. An appreciation of these forces provides a foundation from which more effective forms of vibrations may be developed. The stress vector at the lower plate ${\boldsymbol{\sigma }_L}$ is

(2.8)\begin{equation}{\boldsymbol{\sigma }_L} = \; \left[ {\begin{array}{*{20}{@{}c@{}}} {{\sigma_{X,L}}}&\quad {{\sigma_{Y,L\; }}} \end{array}} \right] = \left[ {\begin{array}{*{20}{@{}c@{}}} {{n_{X,L}}}&\quad {{n_{Y,L}}} \end{array}} \right]\; {\left[ {\begin{array}{*{20}{@{}c@{}}} {2\dfrac{{\partial u}}{{\partial X}} - p}&\quad {\dfrac{{\partial u}}{{\partial Y}} + \dfrac{{\partial v}}{{\partial X}}}\\ {\dfrac{{\partial u}}{{\partial Y}} + \dfrac{{\partial v}}{{\partial X}}}&\quad {2\dfrac{{\partial v}}{{\partial Y}} - p} \end{array}} \right]_{Y = {Y_L}}},\end{equation}

where the normal unit vector ${\boldsymbol{n}_L}$ that points outwards has the form

(2.9a,b)\begin{equation}{\boldsymbol{n}_L} = ({n_{X,L}},{n_{Y,L}}\; ) = {N_L}\left( {\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}},\; - 1} \right),\quad {N_L} = {\left[ {1 + {{\left( {\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}}} \right)}^2}} \right]^{ - 1/2}}.\end{equation}

The explicit form of the X component of the stress vector can be written as

(2.10)\begin{equation}{\sigma _{X,L}} = {\sigma _{Xv,L}} + {\sigma _{Xp,L}} = {N_L}\left[ {2\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}}{{\left. {\frac{{\partial u}}{{\partial X}}} \right|}_{{Y_L}}} - {{\left. {\left( {\frac{{\partial u}}{{\partial Y}} + \frac{{\partial v}}{{\partial X}}} \right)} \right|}_{{Y_L}}}} \right] - {N_L}{\left. {\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}}p} \right|_{{Y_L}}},\end{equation}

where $({\sigma _{Xv,L}},{\sigma _{Xp,L}})$ denote the viscous and pressure contributions, respectively. The X component of the total force ${F_{X,L}}$ acting on the fluid at the lower plate per unit length is given by

(2.11)\begin{align}{F_{X,L}} &= {F_{Xv,L}} + {F_{Xp,L}} = {\lambda ^{ - 1}}\int_0^\lambda {\left[ {2\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}}\; {{\left. {\frac{{\partial u}}{{\partial X}}} \right|}_{{Y_L}}} - {{\left. {\left( {\frac{{\partial u}}{{\partial Y}} + \frac{{\partial v}}{{\partial X}}} \right)} \right|}_{{Y_L}}}} \right]\,\textrm{d}X}\nonumber\\ &\quad - {\lambda ^{ - 1}}\int_0^\lambda {\frac{{\textrm{d}{Y_L}}}{{\textrm{d}X}}{{ p |}_{{Y_L}}}\,\textrm{d}X} ,\end{align}

where ${F_{Xp,L}}$ and ${F_{Xv,L}}$ denote the pressure and viscous contributions.

In what follows it is helpful to monitor the changes in various quantities that result from the inclusion of vibrations rather than tracking the total values themselves. Accordingly, we express the flow field $\boldsymbol{v}$, the pressure p, the mean flow rate ${Q_{mean}}$, the shear $\tau $ and the pulling force F as

(2.12a,b)\begin{gather}\begin{gathered}\boldsymbol{v}(X,Y) = [u(X,Y),v(X,Y)] = [{\textstyle{1 \over 2}}Re(1 + Y)+\, {u_1}(X,Y),{v_1}(X,Y)],\\ \quad p(X,Y) = C + {p_1}(X,Y),\end{gathered}\end{gather}

(2.12c–e)\begin{gather}{Q_{mean}} = Re + {Q_{1,mean}},\quad \tau \textrm{(}X\textrm{)} ={-} {\textstyle{1 \over 2}}Re + {\tau _1}(X),\quad F = {\textstyle{1 \over 2}}Re + {F_1}.\end{gather}

Then, when there are no vibrations, all the quantities with a subscript 1 are zero. A negative value of ${F_1}$ indicates that the resistance is reduced and if we can identify conditions when $F = 0$ this implies that the pulling force is reduced to zero. Of course, a negative value for F would indicate the need to reverse the direction of this force so that the pulling force would need to be replaced by a braking force to maintain the prescribed plate velocity.

3. The solution method

Having developed our model we next set out the numerical strategy that was used to generate the solutions described in subsequent sections. We begin by noting that the transformations

(3.1)\begin{equation}y = Y,\quad x = X-ct\end{equation}

introduce a frame of reference that moves with the wave phase speed (a Galileo transformation). This leads to the steady problem

(3.2a–c)\begin{equation}\begin{array}{ccccc} & (u - c)\dfrac{{\partial u}}{{\partial x}} + v\dfrac{{\partial u}}{{\partial y}} ={-} \dfrac{{\partial p}}{{\partial x}} + \dfrac{{{\partial ^2}u}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}u}}{{\partial {y^2}}},\\ & \quad (u - c)\dfrac{{\partial v}}{{\partial x}} + v\dfrac{{\partial v}}{{\partial y}} ={-} \dfrac{{\partial p}}{{\partial y}} + \dfrac{{{\partial ^2}v}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}v}}{{\partial {y^2}}},\quad \dfrac{{\partial u}}{{\partial x}} + \dfrac{{\partial v}}{{\partial y}} = 0, \end{array}\end{equation}

subject to

(3.2d–h)\begin{equation}u(1) = Re,\quad v(1) = 0,\quad u[{y_L}(x)] = 0,\quad v[{y_L}(x)]\; ={-} c\frac{{\textrm{d}{y_L}}}{{\textrm{d}x}},\quad {\left. {\frac{{\partial p}}{{\partial x}}} \right|_{mean}} = 0,\end{equation}

with the position of the lower plate given by

(3.2i)\begin{equation}{y_L}(x) ={-} 1 + A\; \textrm{cos}(\alpha x).\end{equation}

The above system was solved numerically by first introducing a stream function $\psi $ defined by

(3.3a,b)\begin{equation}u = \frac{{\partial \psi }}{{\partial y}},\quad v ={-} \frac{{\partial \psi }}{{\partial x}},\end{equation}

which automatically satisfies the continuity equation. The pressure is eliminated by taking the derivative of (3.2a) with respect to y and the derivative of (3.2b) with respect to x and subtracting the results. This reduces the flow problem to the following form:

(3.4a)\begin{gather}\begin{gathered}{\nabla ^2}({\nabla ^2}\psi ) + c\frac{\partial }{{\partial x}}{\nabla ^2}\psi = {N_{uv}}\quad \textrm{where}\;{N_{uv}} \equiv \frac{\partial }{{\partial y}}\left( {\frac{\partial }{{\partial x}}({\widehat {uu}} )+ \frac{\partial }{{\partial y}}({\widehat {uv}} )} \right)\\ \quad - \frac{\partial }{{\partial x}}\left( {\frac{\partial }{{\partial x}}({\widehat {uv}} )+ \frac{\partial }{{\partial y}}({\widehat {vv}} )} \right),\end{gathered}\end{gather}

(3.4b–f)\begin{gather}y = 1:\textrm{ }\frac{{\partial \psi }}{{\partial y}} = Re,\quad \frac{{\partial \psi }}{{\partial x}} = 0;\quad y = {y_L}(x):\textrm{ }\frac{{\partial \psi }}{{\partial y}} = 0,\quad \frac{{\partial \psi }}{{\partial x}} = \; c\frac{{\textrm{d}{y_L}}}{{\textrm{d}x}};\quad {\left. {\frac{{\partial p}}{{\partial x}}} \right|_m} = 0.\end{gather}

The penultimate condition (3.4e) can be written in an alternative way by noting that variations in $\psi $ along the lower plate can be expressed as

(3.5)\begin{equation}\textrm{d}{\psi _L} = {\left. {\left( {\frac{{\partial \psi }}{{\partial x}}\,\textrm{d}x + \frac{{\partial \psi }}{{\partial y}}\,\textrm{d}y} \right)} \right|_{{y_L}(x)}} = c\frac{{\textrm{d}{y_L}}}{{\textrm{d}x}}\,\textrm{d}x.\end{equation}

Integrating (3.5) along this plate yields

(3.6)\begin{equation}{\psi _L}(x) = c[{y_L}(x) - {y_L}({x_0})],\end{equation}

Where the constant of integration has been fixed by assuming that ${\psi _L}({x_0}) = 0$, where ${x_0}$ is some arbitrary point along the plate. The stream function $\psi $ is constant along the upper plate and its value can be determined from the pressure gradient constraint.

The system of equations (3.4)–(3.6) was solved to spectral accuracy by representing the unknowns in terms of Fourier expansions in the streamwise direction so that

(3.7)\begin{equation}\psi (x,y) = \sum\limits_{n ={-} \infty }^{n ={+} \infty } {{\psi ^{(n)}}(y)\,{\textrm{e}^{\textrm{i}\,n\alpha x}}}\end{equation}

for modal functions ${\psi ^{(n)}}(y)$ to be determined. The boundary conditions at the upper plate are

(3.8a,b)\begin{equation}y = 1:\textrm{ }\frac{{\textrm{d}{\psi ^{(0)}}}}{{\textrm{d}y}} = Re,\quad \frac{{\textrm{d}{\psi ^{(n)}}}}{{\textrm{d}y}} = 0,\quad n \ne 0,\end{equation}

with the constraint (3.8a) specifying the mean part of the streamwise velocity field; the requirement (3.8b) states that all the periodic components must be zero on the upper surface. While equations (3.8) provide explicit values for each modal function, they do not provide any coupling owing to the simple geometry. By way of contrast, the boundary conditions at the lower plate couple all the modes through the more complicated surface geometry. These boundary conditions were enforced using the concept of IBC. The modal functions ${\psi ^{(n)}}(y)$ were written as Chebyshev expansions while the flow boundary conditions were imposed as constraints. The overall algorithm is grid-less and is very flexible when changes in wave shape are of interest for it can be adjusted to account for multiple geometries with minimal user intervention. Our computations were carried out to ensure at least five-digit accuracy and this requirement dictated the choice of the number of Fourier modes and Chebyshev polynomials. The algorithm can be used to obtain results to machine accuracy if required. We have deliberately kept our description of the IBC strategy to a summary of the main points in the interest of brevity. Readers who desire a more leisurely and detailed discussion of the numerical method may consult with any of the papers by Szumbarski & Floryan (Reference Szumbarski and Floryan1999), Husain & Floryan (Reference Husain and Floryan2008a,Reference Husain and Floryanb, Reference Husain and Floryan2010) or Husain, Szumbarski & Floryan (Reference Husain, Szumbarski and Floryan2009).

3.1. The peristaltic effect

Before we embark on addressing the principal aim of the present study that concerns the reduction in the resistive forces, we quickly demonstrate the key properties of the peristaltic effect. We remark that vibrations imposed on the lower surface pump the fluid in the direction of the wave motion. The fluid movement exerts a shear stress on the upper plate which then requires an external force ${F_{X,U}}$ to keep it stationary. The sample of results presented in figure 2 demonstrates that ${F_{X,U}}$ increases with growth of $\alpha $ and c, which suggests that short, fast waves potentially provide the most effective form of pumping. Waves with $\alpha \approx 0.5$ provide a local maximum in ${F_{X,U}}$ which can be used to advantage in those applications where the creation of fast waves might prove difficult.

Figure 2. (a) Variations in the external force ${F_{X,U}}/{A^2}$ required to keep the upper plate stationary $(Re = 0)$ as a function of $\alpha $ and c. The dividing red line shows the parameters for which ${F_{X,U}} = 0$ and the grey shading identifies those conditions for which ${F_{X,U}}$ is negative. (b) Some horizontal cuts taken through the results shown in (a) for the two wave speeds $c = 500\;\textrm{and}\;1000$. (c) Some vertical cuts taken through the results in (a) for the four wavenumbers $\alpha = 0.5,1.7,5\textrm{and}\;10$.

4. The reduction of resistance

When an external force is applied to the upper plate it accelerates and tends to pull the fluid along with it. Eventually an equilibrium is achieved in which the resistance generated by shear stresses balances the external force. The imposition of vibrations on the lower plate tends to force the fluid to move in the direction of the wave. This reduces the relative velocity of the fluid with respect to the upper plate and so lessens the shear that acts on that plate. It is likely that there will be a minimum wave velocity required for this effect to come into play and that this least value will be a function of the plate velocity. We shall return to this issue a little later, but first probe the properties of the flow dynamics.

In an effort to understand these dynamics we look at the problem when the waves are long. We develop the detailed form of the long-wave structure in Appendix A and, for the purposes of the present discussion, borrow some of the key results derived there. In the limit $\alpha \to 0$ the streamwise velocity component has the form

(4.1)\begin{align} u & = \dfrac{1}{2}Re(\eta + 1) - \dfrac{3}{4}Re\left[ {1 - \dfrac{{4(4 - {A^2})}}{{(8 + {A^2})[2 - A\; \cos (\xi )]}}} \right](1 - {\eta ^2})\nonumber\\ & \quad - c\dfrac{A}{{2 - A\; \cos (\xi )}}\left[ {\dfrac{3}{2}\cos (\xi ) - \dfrac{{9A}}{{8 + {A^2}}}} \right](1 - {\eta ^2}) + 0(\alpha ), \end{align}

where $\xi \equiv \alpha x$ and $\eta $ is a transformed cross-slot coordinate so that the two plates are given by $\eta ={\pm} 1$.

This expression is comprised of three terms, the first being just the original Couette flow written in the new coordinates. The second term represents a Poiseuille-flow-type correction which we refer to as the ‘groove effect’; we notice that this term does not depend on the wave phase speed but is a function of the groove amplitude so represents the effect of stationary grooves. The last term is proportional to the phase speed c so may be thought of as a ‘wave effect’. We see that the groove effect is always negative; it leads to a retardation of the fluid and thus increases the resistance. On the other hand, the sign of the wave effect term depends on the direction of c; the wave must both travel in the flow direction and be sufficiently fast to overcome the groove effect. The interplay between these two effects can be explored by looking at the force required to maintain the movement of the upper plate. The component of this force that is attributable to vibrations can be written as

(4.2)\begin{align}{F_1} &= {\tilde{F}_0} - \frac{1}{2}Re + 0({\alpha ^2}) \nonumber\\ &= Re\left[ {\frac{{4(2 + {A^2})}}{{(8 + {A^2}){{(4 - {A^2})}^{1/2}}}} - \frac{1}{2}} \right] - c\frac{{6{A^2}}}{{(8 + {A^2}){{(4 - {A^2})}^{1/2}}}} + 0({\alpha ^2}),\end{align}

a result that is noted in equation (A23). The two terms here represent the groove and wave effects, respectively. Reducing the amplitude to zero $(A \to 0)$ eliminates this force which is proportional to ${A^2}$ for small wave amplitudes. When the phase speed is made zero the grooves create an additional resistance $({F_1} > 0)$ but reducing $Re$ results in a peristaltic pumping within a slot with a fixed upper plate which produces a force acting on this plate in the direction of wave propagation. When waves are present and the upper plate is moving, waves propagating upstream $(c < 0)$ increase the resistance encountered by the plate. The use of downstream-propagating waves decreases the resistance but only if the waves are fast enough – the pumping effect must be strong enough to overcome the effects of the grooves. We remark that vibrations with phase speed

(4.3)\begin{equation}c = Re\; {A^{ - 2}}\left[ {\frac{2}{3}(2 + {A^2}) - \frac{1}{{12}}(8 + {A^2}){{(4 - {A^2})}^{1/2}}} \right]\end{equation}

make ${F_1}$ zero and a further increase to

(4.4)\begin{equation}c = \frac{2}{3}Re\; {A^{ - 2}}(2 + {A^2})\end{equation}

brings the total force to zero. Then there is no need for an external force to maintain the specified plate velocity as the vibrations provide sufficient propulsion themselves. Of course, any further increase of c would require the use of a braking force to prevent plate acceleration.

A secondary quantity of interest is the flow rate driven by movement of the upper plate. Part of this flow can be attributed to the vibrations and equals

(4.5)\begin{equation}{Q_{1,mean}} = {\tilde{Q}_{m,0}} - Re + 0({\alpha ^2}) = c\frac{{6{A^2}}}{{8 + {A^2}}} - Re\frac{{3{A^2}}}{{8 + {A^2}}} + 0({\alpha ^2}).\end{equation}

We can see that the elimination of the grooves makes ${Q_{1,mean}}$ zero and that waves propagating upstream $(c < 0)$ always reduce the flow rate. Waves propagating downstream help with propelling the fluid, but their phase speed needs to be at least $c = (1/2)Re$ to neutralize the groove effect.

This preliminary analysis already suggests that the wave speed relative to the fluid velocity may well prove to be important. It is not immediately clear how one should measure the fluid velocity unambiguously as it varies across the slot; consequently, we use the speed of the upper plate as a proxy for the fluid velocity. Thus far we have been concerned with long-wavelength modes and to assess other cases we need to resort to suitable numerical calculations.

4.1. Vibration-induced flow modifications

We have already noted that vibrations produce flow modifications that change the shear stress acting at, and hence the resistance experienced by, the upper plate. We now aim to characterize and understand some of the properties of this phenomenon and start by looking at how the flow is modified with changes in $Re$ and c.

We start by plotting some representative instantaneous vector lines for the velocity modifications ${\boldsymbol{u}_{mod}} = \boldsymbol{u} - Re{\boldsymbol{u}_0}$ at a few typical wavenumbers $\alpha $. The results shown in figure 3 suggest that the qualitative flow topologies appear to change little with variations in $Re$. Vibrations of a small wavenumber (figure 3a,d,g,j) lead to a sloshing pattern with forward movement around the wave troughs and backward movement near the crests. In contrast, at a large wavenumber $\alpha = 10$ (figure 3c,f,i,l) a boundary layer is formed near the vibrating plate and a uniform flow persists above it; we refer to this as a moving-wall regime. Patterns at intermediate values of $\alpha $ appear as a hybrid between the sloshing and the moving-wall structures.

Figure 3. Distributions of the modified streamwise flow field for groove amplitude $A = 0.0025$. The wavenumber $\alpha = 0.1$ (a,d,g,j), 1 (b,e,h,k) and 10 (c,f,i,l) and the results are shown for two wave speeds c = 600 and 5000 and two Reynolds numbers Re = 10 and 1000.

We provide more insight into the velocity modifications by plotting the profiles of this quantity at four streamwise locations. In figure 4(a,d,g,j) we see that at a small wavenumber the modifications are of appreciable size over the entire width of the slot. In contrast, in the case of short waves (figure 4c,f,i,l) the modifications have a somewhat involved x-dependent form near the vibrating plate, but they smoothly morph into an almost x-independent, Couette-like flow outside the boundary layer. These velocity distributions are only slightly modified by variations in $Re$ and their profiles are hardly affected by changes in c although their magnitude is roughly proportional to c.

Figure 4. The distributions of the velocity modification at the four streamwise locations $x/\lambda = 0,0.25,0.5,0.75$ when the groove amplitude $A = 0.0025$. The wavenumber $\alpha = 0.1$ (a,d,g,j), 1 (b,e,h,k) and 10 (c,f,i,l) and the results are shown for two wave speeds c = 600 and 5000 and two Reynolds numbers Re = 10 and 1000.

It is instructive to discuss further the flow modifications at a high wavenumber as they demonstrate explicitly the process of transferring wave propulsion from the lower to the upper plate. Vibrations tend to pump the fluid to the right within the boundary layer whose edge acts rather like a moving wall with an apparent velocity ${U_{app}}$. An estimate of ${U_{app}}$ can be obtained by looking at the value of ${u_{mod}}$ at the edge of the boundary layer (figure 5a) and it seems that ${U_{app}}$ increases proportional to ${\alpha ^2}$ as $\alpha \to \infty $ (figure 5b,c). The flow outside the boundary layer is essentially of Couette type driven by this moving wall in the direction of wave motion and this flow is responsible for transferring the vibration-generated propulsion to the upper plate.

Figure 5. (a) Zoomed-in picture of the velocity distribution for $\alpha = Re = 10\;\textrm{and}\;c = 5000$. Plots of ${U_{app}}$ as a function of $\alpha $ (b) when $c = 1000$ for various Re and (c) when $Re = 500$ for various c. In all cases $\textrm{groove}\;\textrm{amplitude}\;A = 0.0025$.

We now look in detail at the resistance experienced by the upper plate. In order to simplify the presentation, we divide the results into three parts according to the speed of the upper plate. We start with relatively slow motions.

5. Slow plate movement $(Re \le 100)$

The fact that the upper plate moves breaks the symmetry that would otherwise be present between waves that move to the left or to the right. Forward-moving waves, that is, those that move in the same direction as the plate, may well reduce the necessary pulling force required to maintain motion, while backward-moving waves are more likely to increase this force. This speculation is confirmed by the results shown in figure 6.

Figure 6. Variations in the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$ as a function of $\alpha $ and c when (a) $Re = 1$ and (b) $Re = 100$. The grey shading indicates negative values while the red line shows those values that give ${F_1} = 0$. Zones between the blue lines in (b) denote the range of natural frequencies of the Orr–Sommerfeld modes (see Appendix B).

If the waves move more slowly than the fluid, the plate modifications associated with vibrations play a role akin to that of surface roughness. By way of contrast, if the waves are quicker than the fluid this activates peristaltic pumping which decreases the external pulling force. While these comments are somewhat qualitative in nature, we can be more precise as to the divide between slow and fast waves in the case of long wavelengths. The reader is reminded that in the case $\alpha \to 0$ the form of the force correction ${F_1}$ is given by (4.2), and then since this quantity needs to be negative to achieve a force reduction, we can determine the phase speed ${c_b}$ which separates the ‘faster’ and ‘slower’ waves as

(5.1)\begin{equation}{c_b} = \frac{{Re}}{{1 + Re}}{(6{A^2})^{ - 1}}\left[ {4 + {A^2} - \frac{1}{2}(8 + {A^2}){{(4 - {A^2})}^{1/2}}} \right].\end{equation}

The situation is somewhat more involved for waves with $\alpha = 0(1)$. Now an increase in the upper plate velocity Re leads to a discernible dependence of ${c_b}$ on $\alpha $; if we examine the red dividing lines in figure 6 then if Re = 1 the value of ${c_b}$ appears to be virtually independent of $\alpha $ but this is certainly not the case in figure 6(b) for which Re = 100. The size of the resistance reduction correlates reasonably well with the strength of peristaltic effect discussed earlier in the sense that fast, short waves appear to be the most effective. We also point out that there is a local maximum in ${F_1}$ near $\alpha \approx 0.5$ which suggests that if high-velocity waves cannot easily be created, then a good compromise might be achievable using modes of this wavenumber.

More detailed information concerning the dependence of ${F_1}$ on the various parameters can be obtained by looking at the results at a fixed value of c. The results shown in figure 7(a) reveal that ${F_1}$ varies nonmonotonically with $\alpha $ and this quantity has turning points near $\alpha \approx 0.5$ and $\alpha \approx 1.8$. The effectiveness of the vibrations increases as Re falls since momentum inserted at the lower plate is transferred to the upper plate through viscous effects. A reduction in Re can be interpreted as a shortening of the distance between the plates which naturally increases the effectiveness of shear stresses. Sufficiently short-wavelength, fast vibrations can eliminate the required external force completely and may even lead to situations when this force changes direction so that a braking force is needed otherwise the plate would accelerate. If we examine the form of the force ${F_1}$ as a function of c at a fixed value of $\alpha $ we obtain the results sketched in figure 7(b). We see that ${F_1}$ increases with c in a somewhat complicated way when Re is relatively small but at higher values the rise in ${F_1}$ becomes almost exactly proportional to c.

Figure 7. The variations of the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$. (a) Force as a function of $\alpha $ for the three Reynolds numbers Re = 1, 10 and 100 and the two speeds $c = 500\;\textrm{and}\;1000$. The dark shading identifies conditions which require an external braking force while the lighter shading shows those parameters that lead to at least a 50 % reduction in the pulling force. (b) Force as a function of c for the same three values of Re and the three wavenumbers $\alpha = 0.5$ (red), 1.8 (blue) and 30 (green).

We can put a fresh interpretation on the role played by the wavenumber $\alpha $ with reference to the earlier results shown in figures 3 and 4. There we identified three distinct regimes: at small wavenumbers we have a sloshing motion, at large wavenumbers a moving-wall motion and there is an intermediate transitional regime between these extremes. The characteristics of the flow affect the shear exerted on the upper plate, which is the principal factor in achieving the desired resistance reduction. Different processes lead to shear reduction in the sloshing and moving-wall scenarios while the transition regime can be viewed as hybrid of the two; the existence of the three separate cases is evident in figure 7(a).

The underlying Couette flow possesses a collection of natural frequencies which can be determined by methods outlined in Appendix B. These frequencies are associated with travelling waves whose phase speeds ${C_r}$ are known to lie between 0 and Re (Schmid & Henningson Reference Schmid and Henningson2001). Tighter bounds on these speeds have been determined numerically for each $\alpha $ and are superimposed on figure 6. These waves are highly attenuated and can persist only with a continuous addition of energy. It is possible that the vibrations of the lower plate could provoke near-resonances with the natural frequencies and could conceivably drive instabilities (Floryan, Szumbarski & Wu Reference Floryan, Szumbarski and Wu2002). The fact that the phase speed ${C_r}$ of the travelling cannot exceed Re suggests that our imposed vibration waves might be distinguished depending on whether they are fast $(c > Re)$ or slow $(0 < c < Re)$; clearly fast waves cannot interact with the natural frequencies. As $Re$ increases so the natural waves become less attenuated and can interact more effectively with the vibrations thereby restricting the opportunities for resistance reduction. The results displayed in figure 6(a) for $Re = 1$ do not show interactions as the natural modes are highly attenuated but these interactions become noticeable by the stage $Re = 100$ (see figure 6b). We remark that the natural frequencies of the reference Couette flow provide only a guideline as to possible interactions with the vibration modes. This is because the flow modified by vibrations is likely to possess a much more complex spectrum of natural frequencies as evidenced by analogous calculations pertaining to Poiseuille flow in a grooved channel (Asai & Floryan Reference Asai and Floryan2006; Floryan Reference Floryan2007). To the best of our knowledge the stability analysis of Couette flow modified by vibrations has yet to be examined.

5.1. A simple energy budget

Next, we examine the various energy fluxes in the system as the imposition of vibrations requires an expenditure of energy. The energy fluxes are determined by multiplying the x-momentum equation by u, the y-momentum equation by v, adding them together and then integrating over the control volume which extends between the plates in the y direction and over one wavelength in the x direction (Bewley Reference Bewley2009; Jiao & Floryan Reference Jiao and Floryan2021). The resulting balance has the form

(5.2)\begin{align}& \int_0^\lambda {\int_{{y_L}}^1 {\left( {{u^2}\dfrac{{\partial u}}{{\partial x}} + uv\dfrac{{\partial u}}{{\partial y}} + uv\dfrac{{\partial v}}{{\partial x}} + {v^2}\dfrac{{\partial v}}{{\partial x}}} \right)} \,\textrm{d}y\,\textrm{d}x} \nonumber\\ & ={-} \int_0^\lambda {\int_{{y_L}}^1 {\left( {u\dfrac{{\partial p}}{{\partial x}} + v\dfrac{{\partial p}}{{\partial y}}} \right)\,\textrm{d}y\,\textrm{d}x} } + \int_0^\lambda {\int_{{y_L}}^1 {\left( {u\dfrac{{{\partial^2}u}}{{\partial {x^2}}} + u\dfrac{{{\partial^2}u}}{{\partial {y^2}}} + v\dfrac{{{\partial^2}v}}{{\partial {x^2}}} + v\dfrac{{{\partial^2}v}}{{\partial {y^2}}}} \right)\,} \textrm{d}y\,\textrm{d}x.} \end{align}

We can write this result in a more convenient way by integrating each term by parts, simplifying using continuity and then imposing the requisite streamwise periodicity properties. This process leads to the result that

(5.3a)\begin{equation}Re{\lambda ^{ - 1}}\int_0^\lambda {{{\left( {\frac{{\partial u}}{{\partial y}}} \right)}_{y = 1}}\,\textrm{d}x} = {P_\varepsilon } - {P_{wv}} - {P_k} - {P_{wp}},\end{equation}

where the left-hand side of this balance denotes the external power (per unit length of the slot) required to drive the upper plate. The four terms on the right-hand side are defined by

(5.3b,c)\begin{gather}\begin{gathered}{P_\varepsilon } \equiv {\lambda ^{ - 1}}\int_0^\lambda {\int_{ - 1}^1 {\left[ {{{\left( {\frac{{\partial u}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial u}}{{\partial y}}} \right)}^2} + {{\left( {\frac{{\partial v}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial v}}{{\partial y}}} \right)}^2}} \right]\,\textrm{d}y\,\textrm{d}x} } ,\\ {P_{wv}} \equiv {\lambda ^{ - 1}}\int_0^\lambda {{{\left( {v\frac{{\partial u}}{{\partial x}}} \right)}_{y = {y_L}}}\,\textrm{d}x} ,\end{gathered}\end{gather}

(5.3d,e)\begin{gather}{P_k} \equiv {\lambda ^{ - 1}}\int_0^\lambda {{{\left[ {v\left( {\frac{1}{2}{v^2}} \right)} \right]}_{y = {y_L}}}\,\textrm{d}x} \quad \textrm{and}\quad {P_{wp}} \equiv {\lambda ^{ - 1}}\int_0^\lambda {{{(vp)}_{y = {y_L}}}\,\textrm{d}x} ,\end{gather}

with ${P_\varepsilon }$ representing the rate of energy dissipation, ${P_{wv}}$ denoting the rate of work done by viscous forces at the vibrating plate, ${P_k}$ representing the rate of injection of kinetic energy into the flow at the vibrating plate and ${P_{wp}}$ standing for the rate of work done by pressure forces at the vibrating plate. We point out that ${P_k} \equiv 0$ for sinusoidal vibrations and that the application of this balance to a smooth slot leads to the simple expression for the external power, i.e. ${\textstyle{1 \over 2}}R{e^2}\textrm{ }( = {P_\varepsilon })$ with ${P_{wv}},{P_k}$ and ${P_{wp}}$ all zero. This enables us to derive an expression for the change in the external energy required to generate the vibrations which is given by ${P_{wv}} + \; {P_{wp}}$. The introduction of vibrations changes the velocity field thereby increasing dissipation by $\Delta Dis = {P_\varepsilon } - {\textstyle{1 \over 2}}R{e^2}$ and moderating the external power $\Delta {P_{we}}\textrm{ }( = \Delta Dis - {P_{wv}} - \; {P_{wp}})$ necessary to maintain the plate motion. Rearranging terms in (5.3a) we get

(5.3f)\begin{equation}{P_\varepsilon } = \; Re\; {\lambda ^{ - 1}}\int_0^\lambda {{{\left( {\frac{{\partial u}}{{\partial y}}} \right)}_{y = 1}}\,\textrm{d}x} + {P_{wv}} + {P_k} + {P_{wp}},\end{equation}

which shows how to reduce dissipation to achieve net energy savings. Sample forms of the pressure work term ${P_{wp}}$ are shown in figure 8(a,e) as a function of $\alpha $ and $c$, and equivalent results for ${P_{wv}}$ and the dissipation $\Delta Dis$ are presented in figures 8(b,f) and 8(c,g), respectively. Inspection of the form of $\Delta {P_{wr}}$ required by the upper plate suggests that a reduction in power requirements is necessary for sufficiently fast forward-moving waves with larger reductions obtained for shorter waves (see figure 8d,h). It would seem that ${P_{wp}}$ represents the dominant energy flux and most of it is consumed by the increase of dissipation with only small fraction that can be ascribed to a reduction in the power required to maintain the plate movement. So, it can be inferred that the vibration configurations considered within the scope of this analysis do not produce net energy saving.

Figure 8. Components of the energy budget at the two Reynolds numbers $Re = 1$ and $Re = 100$. Work done (a,e) by pressure at the lower plate ${P_{wp}}$ and (b,f) by viscous forces at the lower plate ${P_{wv}}$. (c,g) The increase of dissipation above its reference value $\Delta Dis$ and (d,h) the change in power $\Delta {P_{we}}$ required to drive the upper plate. All results correspond to a groove amplitude $A = 0.01$; solid (dashed) lines denote positive (negative) values. The red lines indicate zero values and the zones between the blue lines shows the range of natural frequencies of the Orr–Sommerfeld modes.

6. Intermediate plate speeds $({100 < Re < 1000} )$

We have seen that an increase in the plate velocity much above about $Re = 100$ requires faster waves to achieve a reasonable reduction in the resistance. Once we reach this regime, the waves may approach a near resonance with the natural flow frequencies and this plausibly could result in a significant deterioration of the system performance. This problem can be circumvented by working with waves that are faster than the plate velocity; we have already noted that such waves will then be outside the range within which the natural flow frequencies must lie. This allows us to divide the waves into two classes: waves with $c > Re$ are subsequently referred to as supercritical as they cannot interact with the natural frequencies; those waves with $0 < c < Re$ will be deemed to be subcritical and interaction may be possible.

The form of ${F_1}$ as a function of c and $\alpha $ is illustrated in figure 9. Supercritical waves appear to reduce the magnitude of ${F_1}$ although the functional from of ${F_1}(\alpha ,c)$ is very reminiscent of the results found for the slow-plate problem. In contrast, subcritical waves lead to a complex structure in which zones of resistance reduction are interspersed with regions of parameter space in which the resistance increases; this is qualitatively different behaviour from that found at smaller Re.

Figure 9. The variations in the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$ as a function of $\alpha $ and c for the three Reynolds numbers (a) $Re = 250$, (b) $Re = 500$ and (c) $Re = 700$. The grey shading indicates regions of parameter space corresponding to a reduction in resistance and the red lines mark conditions for which ${F_1} = 0$. The zones between the blue lines denotes the range of natural frequencies of the Orr–Sommerfeld modes. The black dots identify conditions used in the later figures 12–16.

The subcritical waves generally lead to an increase in the resistance. A reduction in resistance may be possible using waves with velocities towards the upper limit of the subcritical range. The cross-over point where there is no change in the resistance $({F_1} = 0)$ is a function of $\alpha $ and is delineated by the red lines in figure 9. There are also small isolated regions of negative ${F_1}$ centred near $\alpha \approx 0.8$ and $c/Re \approx 0.3$ and these zones identify additional conditions under which force reduction is possible. These regions appear to expand as $Re$ increases. There are also other regions of parameter space in which the opposite effect occurs and an anomalously large increase in resistance is seen. These zones seem to be centred around $\alpha \approx 0.8$ and $c/Re \approx 0.7$ and, again, they spread out with an increase of $Re$ with most of the expansion occurring in the $\alpha $ direction.

Cuts taken through the contour plots at a fixed c within the force-reducing zone are shown in figure 10(a). These results demonstrate that ${F_1}$ varies as function of $\alpha $ in a somewhat complicated fashion. We see that the magnitude of the correction force has a local maximum near $\alpha \approx 0.5$ and a local minimum around $\alpha \approx 1.8$; this is not dissimilar to what was seen for slow plate movements (figure 7). The effectiveness of the vibrations diminishes with Re, but this can be compensated for by a concomitant increase in the wave speed. The influence of short waves appears to grow approximately proportional to ${\alpha ^2}$; see figure 10(a). The form of the correction force as a function of c at fixed $\alpha $ is shown in figure 10(b). We conclude that the magnitude of ${F_1}$ increases in a non-simple manner for slow (subcritical) waves with this growth becoming nearly proportional to c for supercritical modes.

Figure 10. Variations of the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$. (a) The correction as a function of $\alpha $ for the Reynolds numbers $Re = 250$, $500$ and $700$ and wave speeds $c = 1500$ and $2500$. (b) The correction as a function of c for the same Reynolds numbers and wavenumbers $\alpha = 0.5$, $1.8$ and 30.

Supercritical waves are associated with energy fluxes whose variations with c and $\alpha $ are qualitatively not dissimilar to those seen relating to the slow plate movements (figures 8 and 11). Subcritical waves lead to intricate and quite different variations as illustrated in figure 11 – we shall look at these in some detail. Vibrations do the work for waves with velocities near the upper limit of the subcritical range, with the borderline ${P_{wp}} = 0$ being a function of $\alpha $ and being marked red in figure 11(a). There are small parameter regions where ${P_{wp}} > 0$ and these are centred around $\alpha \approx 0.8$ where the vibrations also do the work. The flow appears to perform most of the work in the remaining part of the parameter space displayed in figure 11(a) with the maximum of this effort occurring near $\alpha \approx 0.8$, $c \approx 0.7$. Variations in the power required to maintain the plate movement show similar areas of decreased external power as well as regions where a reduction in this power is possible (figure 11d). The variations in the viscous work are quite regular (figure 11b) with this work increasing with both c and $\alpha $. Rapid changes in the dissipation (see figure 11c) appear at those parts of the parameter conditions corresponding to the islands identified in figure 11(a); outside this zone dissipation increases monotonically with an increase of both c and $\alpha $. There is no net energy savings associated with such waves.

Figure 11. (a) The work done by the pressure at the lower plate ${P_{wp}}$ and (b) the work done by viscous forces at the lower plate ${P_{wv}}$. (c) The increase in dissipation above its reference value and (d) the change in power $\Delta {P_{we}}$ required to drive the upper plate. Solid (dashed) lines represent positive (negative) values. The red lines denote zero contours and the zones between the blue lines indicate the range of natural frequencies of the Orr–Sommerfeld modes. All results given are for $Re = 500$ and $A = 0.01$. The grey shading identifies the region of ${P_{wp}} > 0$ in (a) and $\Delta {P_{we}} < 0$ in (d).

We next look in detail at the variations in the various forces and energies as functions of the wave speed and focus on the case $Re = 500$ for definiteness. We summarize the main findings in figure 12. The key quantities, being the pressure component of the force on the lower plate ${F_{Xp,L}}$ (2.11), the work done by the pressure at the lower plate ${P_{wp}}$ (5.3), the force correction ${F_1}$ and the change in power $\Delta {P_{we}}$, change in smooth monotonic ways for supercritical waves but behave in a rather complicated manner when the waves are subcritical. These complications simplify somewhat for shorter waves as then the corresponding natural frequencies are highly attenuated.

Figure 12. (a) The variation in the x component of the pressure force ${F_{Xp,L}}$ that acts on the fluid at the lower plate. (b) The pressure work done at the lower plate ${P_{wp}}$; (c) the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$ and (d) the change of the power $\Delta {P_{we}}$ required to drive the upper plate. All results relate to $Re = 500$, a groove amplitude $A = 0.01\; $ and the five wavenumbers $\alpha = 0.5$, 1, 2, 3 and 5.

To probe further into the system dynamics for subcritical waves, we reconfigure the data from figure 12 and discuss the results for fixed values of the wavenumber $\alpha $; see figure 13. The results confirm significant intricacies in the flow response for long waves. The waves with $\alpha = 0.5$ in figure 13(a) include a cross-section that cuts through an island of decreased ${F_1}$ in figure 9(b) as well as a zone of significant increase in ${F_1}$. This cut also contains regions of positive and negative pressure work (figure 11a). Our results demonstrate a change in the direction of the pressure force as c increases; this force opposes the fluid movement for $0 < c < 0.25\; Re$, then weakly encourages the fluid movement and then opposes the fluid movement again when $0.5\; Re < c < 0.75\; Re$. Any further increase in c just results in an ever-increasing force that promotes the fluid movement. The external force applied to the upper plate increases when the pressure force opposes the fluid movement and decreases when it is supporting (figures 12c and 13a) and changes in the external power display similar variation (figures 12d and 13a).

Figure 13. The x component of the pressure force ${F_{Xp,L}}$, the pressure work done at the lower plate ${P_{wp}}$, the normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$ and the change of the power $\Delta {P_{we}}$ required to drive the upper plate all plotted as functions of $c/Re$. All results relate to $Re = 500$, a groove amplitude $A = 0.01$ and prescribed wavenumbers $\alpha = $ (a) 0.5, (b) 1, (c) 2 and (d) 5.

A slight increase in the wavenumber to $\alpha = 1$ largely preserves the qualitative character of variations of all the quantities depicted in figure 13(a). A further increase to $\alpha = 2$ decreases the amplitude of oscillation in ${F_1}$ and $\Delta {P_{we}}$ almost tenfold (figure 13c) and growth in both ${P_{wp}}$ and ${F_{Xp,L}}$ is noted once $c > 0.4Re$. With $\alpha = 5$ (figure 13d) the growing trend in ${P_{wp}}$ and ${F_{Xp,L}}$ sets it at smaller values of c.

To explain the observed variations in forces and power, we present in figure 14 streamwise profiles of the pressure at the lower plate ${p_L}$ together with the x component of the pressure force ${\sigma _{xp,L}}$. We fix the wavenumber $\alpha = 1$ and consider two ranges of wave speed. When $c = 0$ the groove is stationary, and the minimum pressure occurs slightly downstream from the peaks with the maximum positioned slightly near the troughs. The distribution of the force ${\sigma _{xp,L}}$ shows that it is directed upstream and so is opposing the fluid movement. As the wave speed increases, so the motion drives the pressure minimum downstream with the size of the pressure variations remaining virtually unchanged until $c \approx 0.1Re$ but these tend to grow thereafter with the greatest amplitude occurring when $c \approx 0.3Re$. The evolution of ${\sigma _{xp,L}}$ follows a broadly similar pattern as illustrated in figure 14(c,d). Once the wave speed exceeds $c \approx 0.45Re$ the pressure distribution shifts by a half-period with respect to the wave and any further increase in c leads only to an increase in the pressure amplitude but with no change in the position of the pressure peaks. The profile of ${\sigma _{xp,L}}$ also shows the formation of a regular pattern characterized by an increasing amplitude of oscillation. We saw in figure 13(b) that for $Re = 500$ and $\alpha = 1$ there are ranges of c over which the pressure force opposes fluid movement interspersed with regimes over which it promotes it. This suggests that it is the combination of the size of pressure variations in conjunction with the position of the pressure pattern relative to the wave which together determine whether the vibration-created force reduces or increases the movement resistance.

Figure 14. The distributions of the surface pressure ${p_L}$ at the lower plate (a,b) and the x component of the pressure force acting on the fluid at the lower plate ${\sigma _{xp,L}}$ (c,d). All results are for $\alpha = 1$, $Re = 500$, $A = 0.01$.

To complement the results displayed in figure 14 we investigate how the properties of ${p_L}$ and ${\sigma _{xp,L}}$ behave as functions of the wavenumber for a fixed wave speed c. We chose to focus on the case $c = 0.3Re$ and this is motivated by the observation that this cross-section in figure 9(b) includes regions in which the overall resistance grows and others in which it decreases. It is seen that the pressure maxima for the long waves are located near the wave crests and minima near the troughs. There is a half-wavelength phase shift in the pressure distribution when $\alpha $ increases much beyond about $\alpha = 1.5$ and any further increase in $\alpha $ leads only to an increase in the size of the pressure variations. The forms of ${\sigma _{xp,L}}$ illustrate the effects of both a phase change and a monotonic increase of amplitude when $\alpha $ increases beyond $\alpha = 1.5$. We remark that the results summarized in figure 9(b) suggest that when $Re = 500$ and $c = 0.3Re$, the force correction is positive for longish $(\alpha < 0.33)$ or shortish $(\alpha > 1.35)$ waves but is negative otherwise. At the lower speed $c = 0.1Re$ we note from figure 9(b) that the resistance always increases irrespective of the value of $\alpha $. At this relatively slow speed the results in figure 15 demonstrate that the maximum in the surface pressure when $\alpha $ is small tends to be located near wave troughs but it shifts downstream towards the crests as $\alpha $ increases. There is no discernible phase shift such as that associated with the waves at the higher speed $c = 0.3\; Re$. We also note that the distribution of ${\sigma _{xp,L}}$ is characterized by a large amplitude which seems to increase with $\alpha $ up to the point when $\alpha \approx 1$ and that the pressure force always opposes the fluid movement.

Figure 15. The distributions of the surface pressure at the lower plate ${p_L}$ (a,c) and the x component of the pressure force acting on the fluid at the lower plate ${\sigma _{xp,L}}$ (b,d). All results are for $\; Re = 500$ and $A = 0.01$.

7. Fast plate speeds $(Re \ge 1000)$

In order to explore the characteristics of the flow when the upper plate moves much more quickly, we focus on the two Reynolds numbers Re = 1000 and 2000. We find that the overall properties of the force correction term ${F_1}$ remain somewhat like those already seen in as much that short, fast waves seem to be the most effective in reducing the resistance. It, however, becomes increasingly difficult to achieve a significant reduction as high wave speeds are required to achieve supercritical performance. This brings into play the importance of understanding the flow modulations caused by subcritical waves which, it should be stressed, though slower than the speed of the upper plate, are by no means slow in any absolute sense. The complexities of the system response to such waves are illustrated in figure 16 which shows islands of reduced values of ${F_1}$ surrounded by regions in which the resistance increases.

Figure 16. The normalized force correction ${F_{norm}} = ({F_1}/Re{F_0}{A^2})$ as a function of $\alpha $ and c for the two Reynolds numbers (a) $Re = 1000$ and (b) $Re = 2000$. The grey shading identifies resistance-reducing conditions while the red lines show the parameters for which ${F_1} = 0$. The regions between the blue lines denote the range of natural frequencies of the Orr–Sommerfeld modes.

If we look at the results summarized figure 16 but at a fixed value of c that lies solidly in the region where ${F_1} < 0$ we obtain profiles of the type shown in figure 17(a). It is clear that ${F_1}$ has a local maximum around $\alpha \approx 0.5$, not dissimilar to the case of smaller $Re$ (see figures 7 and 10), but the local minimum moves to $\alpha \approx 2.2$. The overall effectiveness of vibrations decreases but this can be compensated for by reducing the wavelength and increasing the phase speed. Results at a fixed value of $\alpha $, see figure 17(b), show that ${F_1}$ increases in a somewhat intricate way for slow waves but the growth becomes nearly proportional to c for sufficiently fast modes.

Figure 17. The normalized force correction ${F_{norm}} = {F_1}/Re{F_0}{A^2}$. (a) This quantity plotted as a function of $\alpha $ for the two Reynolds numbers used in figure 16 and the two wave speeds $c = 3000$ and $c = 5000$. (b) The normalized force as a function of c for the wavenumbers $\alpha = 0.5$, 1.8 and 30.

We conclude our brief look at the fast-wave problem by examining the various energy terms relevant to the vibrations. The distributions in figure 18 show significant complexity for subcritical waves owing to phase changes between the wave and pressure patterns. These structures are somewhat reminiscent of those relating to flows and are like those when $100 \le Re < 1000$ so are not discussed further here. There is no net energy saving achieved by these waves.

Figure 18. (a) The work done by the pressure at the lower plate ${P_{wp}}$ and (b) the work done by viscous forces at the lower plate ${P_{wv}}$. (c) The increase of dissipation above its reference value and (d) the change in power $\Delta {P_{we}}$ required to drive the upper plate. Solid (dashed) lines represent positive (negative) values. The red lines denote zero contours and the zones between the blue lines indicate the range of natural frequencies of the Orr–Sommerfeld modes. All results given are for $Re = 2000$ and $A = 0.01$. The grey shading identifies the region of ${P_{wp}} > 0$ in (a) and $\Delta {P_{we}} < 0$ in (d).