## Abstract

We compute the drag on a slender rigid cylinder, of uniform circular cross-section, oscillating in a viscous fluid at small amplitude near a horizontal wall. The cylinder's axis lies at an angle *α* to the horizontal and the cylinder oscillates in a vertical plane normal to either the wall or its own axis. The flow is described using an unsteady slender-body approximation, which we treat both numerically and using an iterative scheme that extends resistive-force theory to account for the leading-order effects of unsteady inertia and the wall. When *α* is small, two independent screening mechanisms are identified which suppress end-effects and produce approximately two-dimensional flow along the majority of the cylinder; however, three-dimensional effects influence the drag at larger tilt angles.

## 1. Introduction

Boundary integral methods, which involve distributing singularities over boundaries of the solution domain, provide a popular and effective means of describing the viscous flow created by the motion of an arbitrarily shaped body. When the body is long and thin, slender-body theory (SBT) offers a further simplification, allowing the flow to be approximated by a one-dimensional distribution of singularities (typically Stokeslets and dipoles) along the body's axis (Hancock 1953). SBT therefore constitutes one form of singularity method, a broader class of approximation techniques capable of describing non-slender shapes, such as biconcave discs and spheroids (Pozrikidis 1989; Shatz 2004). While most previous applications of SBT have been in the quasi-steady regime (describing the motion of flagellae, DNA or sedimenting rods, for example), none (to our knowledge) has considered the regime in which both unsteady inertial and viscous forces are important. Here we adopt such an approach to compute the drag experienced by a micron-sized cantilever oscillating at high frequency (in the 100 Hz–1 MHz range) but small amplitude in the presence of a plane wall, a problem of relevance to the atomic force microscope (AFM) and a variety of other imaging and microelectromechanical systems (MEMS). Previously, we determined the effects of a nearby wall on the drag experienced by an oscillating cylinder assuming the flow was two-dimensional (Clarke *et al*. 2005). Here we wish to determine the conditions under which genuinely three-dimensional effects are significant.

SBT for quasi-steady viscous flow has enjoyed particular success in the study of micro-organism motility (Brennen & Winet 1977). Early theoretical work used resistive-force theory (RFT), a simplification of SBT which yields a local relationship between velocity and force over each segment of the slender body (Gray & Hancock 1955). Unfortunately, RFT is limited by the difficulty of accounting accurately for the motion of neighbouring sections of the body: the long-range character of quasi-steady Stokes flow means that RFT captures only the dominant terms in an asymptotic expansion in powers of (log(1/*ϵ*))^{−1}, where *ϵ*≪1 is the body's aspect ratio. The asymptotic validity of RFT in its various guises (including higher-order corrections) under a range of conditions has been examined in detail (e.g. Tuck 1964; Batchelor 1970; Cox 1970; Tillett 1970; Geer 1976; Keller & Rubinow 1976), and modifications of resistive-force coefficients have been suggested that improve accuracy in certain applications (Lighthill 1976). Further strategies for improving the accuracy of SBT (which is algebraically accurate in *ϵ*) include the use of higher-order singularities in the governing integral equation (Johnson 1980), as well as an asymptotic treatment of the full boundary integral equations (Sellier 1999) that bypasses issues of possible ill-posedness identified by Cade (1994). Furthermore, by choosing suitable distributions of image singularities, SBT is readily extended to account for the presence of nearby boundaries (e.g. Blake 1974; Russell *et al*. 1977; Higdon 1979). While more recent studies have developed SBT to include linearized convective inertial effects (Khayat & Cox 1989; Chadwick 2002), the effects of unsteady inertia in this context (or equivalently of the Darcy term that arises in a Brinkman medium) remain largely unexplored.

Below we present a systematic method for determining the drag on a microcantilever, modelled as a rigid cylinder undergoing prescribed small-amplitude oscillations near a plane wall, using an unsteady version of slender-body theory (USBT) (§2). This is treated both numerically (§2*a*) and iteratively (§2*b*), with the latter method yielding non-local unsteady corrections to RFT. Results in §3 demonstrate how, for a cylinder with its axis parallel to the wall, decreasing the cylinder wall separation and increasing the oscillation frequency both screen the cantilever from three-dimensional effects, which are confined to the ends of the cantilever. We show that three-dimensional flows are more significant when the cylinder is tilted relative to the wall (an important feature of AFM cantilevers). We compare drag predictions with those from an existing two-dimensional model (Clarke *et al*. 2005) and discuss the implications of our results in §4.

## 2. Flow model

We consider a rigid circular cylinder, of length 2*L* and radius *R*, that is tilted at an angle *α* to the horizontal, and which oscillates in a vertical plane either vertically or normal to its axis. The cylinder's axis lies a minimum distance *H* above a plane horizontal wall. The cylinder's maximum speed is =*ω*, where *ω* is the driving frequency and ≪ is the amplitude of oscillation. At small amplitudes, the flow is governed by the linearized Navier–Stokes equations(2.1)where * U*,

*P*,

*ρ*and

*μ*are the fluid velocity, pressure, density and viscosity, respectively. Rescaling time on

*ω*

^{−1}, lengths on

*L*, speeds on , pressure on

*μ*/

*L*and seeking solutions of the form , , (2.1) becomes(2.2a)where . This must be solved subject to the following boundary conditions, expressed in Cartesian coordinates (

*x*

_{1},

*x*

_{2},

*x*

_{3}) (see figure 1):(2.2b)The /

*R*≪1 constraint allows us to linearize the boundary conditions, and so we may take the cylinder surface to be fixed. When the cylinder oscillates normal to its axis ; for vertical motion . The wall lies along

*x*

_{3}=0. The problem is characterized by three dimensionless geometric parameters, the wall cylinder separation

*Δ*=(

*H*−

*R*)/

*L*, the aspect ratio

*ϵ*=

*R*/

*L*≪1 and the tilt angle

*α*. is then parameterized using dimensionless cylindrical polar coordinates (

*s*,

*ϵ*,

*θ*) as(2.3)for

*∈ and where −1≤*

**x***s*≤1 is a coordinate along the cylinder axis. We seek the leading-order flow in the limit

*ϵ*→0 with

*γ*,

*Δ*and

*α*fixed.

In a slender-body formulation, the velocity at a point * X* in the flow may be approximated by the integral (cf. Blake 1974)(2.4)where

*(*

**x***s*) lies on the cylinder axis and .

*S*

_{ij}

^{w}and

*Q*

_{ik}are the three-dimensional oscillatory Stokeslet (accounting for the presence of the wall) and free-space potential dipole, respectively, and is an unknown Stokeslet distribution (scaled on

*μ*

*ω*). The choice of dipole coefficient

*d*

_{k}will be discussed below. Letting

*∈, where*

**X***is prescribed, yields a first-kind integral equation for the Stokeslet distribution.*

**u**The free-space dipole and oscillatory Stokeslet are (Pozrikidis 1989)(2.5a)respectively, where and(2.5b)(2.5c)For *γ*≪1, we recover the quasi-steady Stokeslet(2.6)Pozrikidis (1989) has also determined the three-dimensional oscillating Stokeslet which satisfies no-slip and no-penetration on a plane wall at *x*_{3}=0(2.7)where and **x**^{w} is the image of * x* in the plane

*x*

_{3}=0 (figure 1). The wall-interaction tensor

*Λ*

_{ij}involves integrals over the wall,(2.8a)(2.8b)where , for

*i*,

*k*=1, …, 3 (sum over

*k*) and

*j*=1, 2 with(2.8c)(2.8d)(2.8e)where

*a*

^{2}=

*b*

^{2}+i

*γ*

^{2}, and

*J*

_{0}is a Bessel function of the first kind. Substituting (2.7) into (2.4) yields our USBT description of the flow.

The coefficients *d*_{k} in (2.4) must be chosen to reflect the fact that the prescribed motion at * X*∈ should depend only upon

*.*

**X***, where is the unit vector along the cylinder's axis. In other words, any*

**t***θ*-dependence (see figure 1) about the axis due to the Stokeslets must be cancelled out (to leading order in

*ϵ*) by the dipoles. By expressing a surface point using polar coordinates (2.3),

*θ*-dependence arises from the presence of

*ϵ*cos

*θ*,

*ϵ*sin

*θ*terms in the dipole, Stokeslet and image Stokeslet numerators (2.5

*a*)–(2.5

*c*) and (2.8

*c*)–(2.8

*e*). However, as long as

*Δ*≫

*ϵ*the

*θ*-dependent terms in the image Stokeslets (2.7) and (2.8

*a*)–(2.8

*e*) are negligible. Furthermore, since the

*θ*-dependence about an axial point is a local phenomenon,

*f*

_{j}(

*s*) can be approximated by its value at that axial point. A local analysis for quasi-steady Stokeslets (2.6) and dipoles (Hancock 1953; Higdon 1979) shows that their respective

*θ*-dependence can be cancelled out by setting(2.9)(up to an error which is algebraically small in

*ϵ*). Since this analysis is local about a given axial point, where the oscillatory Stokeslet reduces to (2.6), (2.9) also proves suitable for an unsteady formulation.

In addition to the constraint *Δ*≫*ϵ*, there is also an upper bound to the frequencies which can be explored using (2.4). This can be seen by recalling that boundary integral methods offer an exact representation of the flow through a surface distribution of Stokeslets. Under these circumstances, measures the distance from a point in the flow to a surface point * X*. This differs in value from the distance to the axis point with coordinate (see figure 1) by only an

*O*(

*ϵ*) amount. Therefore, we can expand the surface oscillatory Stokeslets (2.5

*a*)–(2.7) about the cylinder axis, resulting in an axial distribution of Stokeslets at leading-order plus higher-order singularities at

*O*(max(

*ϵ*,

*γϵ*)). We expect

*O*(

*γϵ*) errors due to the neglect of the remaining higher-order singularities (see (2.5

*a*)–(2.5

*c*)). Their contribution will be negligible, leaving the flow well described by (2.4), provided that

*γ*≪

*ϵ*

^{−1}, i.e. when viscous boundary layers are much thicker than the cylinder radius. This also ensures that the dipoles operate effectively via (2.9).

A key quantity of interest is the drag on the cylinder (scaled on *μ**ωL*), which is given by Pozrikidis (1989) as , where *V*_{p} is the volume of the oscillating body and * u* is a flow inside the body which satisfies the surface velocity specified by (2.2

*b*). However, the second drag term is

*O*(

*γ*

^{2}

*ϵ*

^{2}) and so can be safely neglected under our frequency constraint

*γ*≪

*ϵ*

^{−1}.

### (a) Numerical scheme

We solve (2.4) numerically by discretizing the cylinder axis −1≤*s*≤1 into *N* equally sized elements consisting of the closed intervals with mid-point **x**_{m} (*m*=1, …, *N*), and assume that each component of the Stokeslet distribution and prescribed velocity is uniform across each element. We then define(2.10a)(2.10b)where **X**_{m} is a surface point that satisfies **X**_{m}.* t*=

**x**_{m}.

*. Equation (2.4) then produces the linear system (*

**t***α*′,

*β*=1, …, 3

*N*)(2.11)(), where 1≤

*i*,

*j*≤3 and 1≤

*m*,

*M*≤

*N*are given through , and , (for 0≤

*k*′,

*k*″≤

*N*−1 and 1≤

*h*′,

*h*″≤3). Determining involves evaluating the integrals which contain image Stokeslets

*S*

_{ij}

^{w}that are computed by taking derivatives (2.8

*c*)–(2.8

*e*) as specified by (2.8

*a*) and (2.8

*b*) and then numerically integrating over

*b*. The system (2.11) is then solved by Gaussian elimination for the unknown distribution vector

*using*

**F***N*=60 axial elements, with convergence verified using

*N*=100.

### (b) Modified RFT

We can also solve (2.4) using techniques which have proved effective in the quasi-steady case (Batchelor 1970; Blake 1974). This approach exploits the existence of an approximately local relationship between the force and velocity at any point along the body axis. The USBT formulation (2.4), on the other hand, takes account of non-local effects arising from the body's extended geometry, as well as the influence of flow inertia. An iterative process can be derived which extracts RFT and its unsteady non-local corrections from (2.4).

To this end, we assume the cylinder lies horizontally and incorporate any complexities arising from finite tilt into the image Stokeslet system. In a cylindrical polar coordinate system (*s*, *r*, *θ*) orientated along the cylinder axis (see figure 1), surface points (*ξ*, *ϵ*, *θ*) and axial points are given by(2.12a)respectively, with image . Hence (for −1≤*ξ*≤1,−1≤*s*≤1),(2.12b)(2.12c)Anticipating a local relationship between force and velocity, we rewrite (2.4)–(2.8*e*) as(2.13a)where(2.13b)(2.13c)(2.13d)*S*_{ij}, *Λ*_{ij} are given in (2.5*a*)–(2.5*c*) and (2.8*a*)–(2.8*e*), respectively. The basis of RFT comes from the fact that the integrands in (2.13*a*)–(2.13*d*) take their largest values close to *ξ* and their integrated contributions (provided *ξ* is not too close to ±1) are(2.14)(*ϵ*≪*δ*≪1) when *i*=*j* and similarly for the (−1, *ξ*−*δ*) intervals. The off-diagonal contributions *i*≠*j* are at most *O*(*ϵ*). Although the Stokeslet integral over [*ξ*−*δ*, *ξ*+*δ*] is *O*(ln *ϵ*), the *θ*-dependent terms make only an *O*(1) contribution and can therefore be cancelled by the *O*(1) dipole contributions, thus leaving *I*_{ij} independent of *θ*.

Over *O*(*δ*) intervals about *ξ*, and so from (2.14), the *T*_{i} will not exhibit any significant *θ* dependence and will make at most an *O*(1) contribution via the outer intervals. This is subdominant to the *O*(ln *ϵ*) terms in *I*_{ij}, which therefore controls the local velocity–force relationship. As the *W*_{i} terms depend upon the cylinder–wall distance, these too will be subdominant to *I*_{ij} and are *θ*-independent provided *Δ*≫*ϵ*. Detailed calculations (see appendix A) reveal(2.15)(no sum over *i*, with and subject to *O*(*ϵ* ln *ϵ*) errors), where(2.16)with *n*_{i}≡(2*δ*_{i1}−1), *e*_{0}≈0.5772 (Euler's constant) and is the exponential integral. Also,(2.17)(no sum over *i*) with the second integral defined in the Cauchy principal value sense. The dominance of the *I*_{ij} term in (2.15) is seen to be *O*(ln *ϵ*), and so by expanding * f* in powers of 1/ln

*ϵ*as(2.18a)the leading-order quasi-steady RFT solution(2.18b)has the first correction(2.18c)(no sum over

*i*in (2.18

*b*) and (2.18

*c*)) with , where we have substituted

*u*

_{i}for

*f*

_{i}in

_{i}using (2.18

*b*). This captures: (i) the non-local force–velocity relationship through integrals over the entire body axis which appear in

_{i}and

*W*

_{i}, (ii) flow inertia through

*γ*-dependence in

*H*

_{i},

_{i}and

*W*

_{i}and (iii) a non-diagonal resistance matrix capturing wall effects via

*W*

_{i}. However,

_{i}depends on differences in

*u*

_{i}, and so for prescribed uniform velocity

_{i}=0. Thus far from the wall, where |

*W*

_{i}|≪1, (2.18

*c*) provides a local velocity–force relationship, but, unlike traditional RFT, this relationship varies with axial position

*s*and frequency

*γ*. The ability of this local relationship to capture finite-length effects can be further explored by examining the limiting behaviour of

*H*

_{i}close to the cylinder ends. For

*ϵ*≪1−

*ξ*≪1, (2.16) takes the limiting form (likewise for

*ξ*→−1); thus,

*γ*controls the magnitude of the logarithmic singularities at the cylinder ends. Higher-order terms (for

*i*>1) can be determined through(2.19)(no summation over

*i*). In the quasi-steady limit (

*γ*≪1),(2.20a)and hence(2.20b)(again, no summation over

*i*) subject to an

*O*(

*γ*) correction, in agreement with Blake (1974).

When oscillations are high frequency (1≪*γ*≪*ϵ*^{−1}), the (viscous) contributions to the oscillatory Stokeslet (2.5*b*) and (2.5*c*) are exponentially small except within *O*(*γ*^{−1}) distances of the singularity (*ϵ*≪*γ*^{−1}≪1)(2.21)(). Hence, the *T*_{i} terms in (2.13*a*)–(2.13*d*) are algebraically small in *γ*^{−1}, leading (assuming the wall is sufficiently distant) to a stronger local force–velocity relationship(2.22)(no sum over *i*). As will be demonstrated in §3, high-frequency RFT (HFRFT (2.22)) captures *γ*-screening and end-effects in the Stokeslet profile without the need for the iteration scheme (2.18*a*)–(2.18*c*) and (2.19), which converges slowly in the large-*γ* limit.

### (c) Two-dimensional models

Under certain circumstances we expect the flow along much of the cylinder to become effectively two-dimensional. For this reason we summarize some relevant results for two-dimensional flow (Clarke *et al*. 2005). Here, lengths are scaled on cylinder radius, making *Δ*/*ϵ* and *γϵ* the natural separation and frequency parameters, respectively. The regime *Δ*, *γ*=*O*(1) then corresponds to the large-separation, low-frequency regime in the two-dimensional problem, for which the drag per unit length on a non-tilted circular cylinder oscillating normally to a wall, derived by applying the method of reflections to two-dimensional unsteady Stokeslets, is(2.23)where *τ*=Δ*γ* and *K*_{0} and *K*_{1} are modified Bessel functions andwith , and . This takes us between the classical unsteady result of Stokes (1851) when vorticity diffuses distances much smaller than the wall–cylinder separation (*τ*≫1)(2.24)and the quasi-steady limit of Jeffrey & Onishi (1981) in which vorticity diffuses distances much greater than the separation distance (*τ*≪1)(2.25)with . These results provide useful validation of the RFT approach (see below). For example, when , (2.24) reduces to ; the same limit may be obtained from (2.22) by expanding *H*_{i} in (2.16) for *γ*≫1 while staying more than an *O*(1/*γ*) distance from each end.

## 3. Results

We now use USBT to examine (in §3*a*) finite-length effects as a function of oscillation frequency *γ* and wall separation distance *Δ* when the tilt angle *α*=0. We then inspect Stokeslet distributions (§3*b*) to judge the accuracy of modified RFT against USBT computations. Drag curves are presented in §3*c*, computed for a range of *γ* and *Δ*; §3*d* examines the influence of tilting the cantilever as the wall is approached.

### (a) Screening of three-dimensional effects

The role of end-effects as the frequency and separation distance are changed is explored in figures 2 and 3, where computations highlight the *Δ* and *γ*-dependence of the Stokeslet distributions. In figure 2*a*, the separation distance is large (*Δ*=100), making the wall's presence largely irrelevant when *γ*=0.1, 5. Vorticity diffuses *O*(*γ*^{−1}) distances with respect to the cylinder's length and, consequently, we expect finite-length effects to cover similar distances. This is confirmed in figure 2*a*, where for *γ*^{−1}=10 vorticity diffuses distances greater than the cylinder length and the Stokeslet profile is everywhere non-uniform. However, when *γ*^{−1}=0.2, not only do we see a decrease in the strength of the profiles, but finite-length effects are indeed restricted to *O*(*γ*^{−1}) distances of the ends. The two-dimensional model (2.24) then provides an accurate picture of the Stokeslet profile over much of the cylinder; we shall refer to this as *γ*-screening. Agreement with (2.24) provides useful validation of the USBT computations.

*Δ*-dependence is explored in figure 2*b*, where the separation distance between the cylinder and the wall is decreased to reveal another mechanism by which end-effects are suppressed. We expect finite-length effects to be confined to within an *O*(*γ*^{−1}) distance or an *O*(*Δ*) distance (when *Δ*≪*γ*^{−1}) of the cylinder ends. Therefore, when *γ*≪1 and *Δ*=10 (figure 2*b*), the finite-length character of the cylinder is evident along its entire length and the quasi-steady two-dimensional approximation (2.25) proves inadequate. However, when *Δ*=0.1, the Stokeslet distribution increases in strength and becomes uniform away from the ends, where it is approximated well by (2.25) (we call this *Δ*-screening). It is also instructive to compare these quasi-steady predictions to others in the literature, summarized by Brennen & Winet (1977). In the large-*Δ* limit, the leading-order effects of the finite length of the cylinder are captured by (Brennen & Winet 1977)(3.1)which predicts a uniform Stokeslet distribution that is closer than (2.25) to the computed result when *Δ*=10 (figure 2*b*). For small *Δ*, (2.25) reduces to a simpler form (Brennen & Winet 1977)(3.2)which works well when *Δ*=0.1 (figure 2*b*), providing further validation of the numerical scheme.

Figure 3 demonstrates the transition between various two-dimensional models through *γ*- and *Δ*-screening. For *Δ*=*γ*=0.1, vorticity diffuses much further from the cylinder than the separation distance and so, in this case, the quasi-steady model (2.25) is appropriate. However, when *γ*=20, vorticity diffuses distances comparable to *Δ*=0.1 and so unsteady inertia is important in the wall interaction, with (2.23) accurately reproducing the uniform portion of the Stokeslet distribution. Finally, when *γ*=100 vorticity is confined very close to the cylinder surface, leaving wall interactions governed by weak inertial effects. Under these circumstances, Stokes' (1851) classical result (2.24) works well.

### (b) Modified RFT models

Figures 4 and 5 assess the performance of modified RFT and its higher-order corrections (2.18*a*)–(2.18*c*) and (2.19) in capturing the non-uniform Stokeslet distribution. Comparisons are made with the USBT computations (2.11) at both moderate and high frequencies for a cylinder far from the wall. When *γ*=1 (figure 4*a*,*b*), while the leading-order quasi-steady RFT profile is uniform, it can be modified to capture the distribution's non-uniform character using just two corrections, with only small deviations from USBT very close to the ends of the cylinder. However, when the frequency is increased to *γ*=100 (figure 5), convergence of the modified RFT iterative scheme is slower. Under these circumstances, the high-frequency expression (2.22) (which is equivalent to (2.24) away from the ends of the cylinder) is an effective and direct approximation.

### (c) Drag

Integrating the Stokeslet distribution along the length of the cylinder yields the drag. This is plotted in figure 6 under differing sets of conditions, against modified RFT predictions (2.18*a*–*c*) and the two-dimensional approximation (2.23). Figure 6*a*,*b* illustrates the effect of varying *Δ*: we see an increase in drag amplitude and a decrease in its phase at smaller *Δ*. When *γ* is decreased both the drag amplitude and phase decrease (figure 6*c*,*d*), as is also the case when the slenderness ratio is reduced (figure 6*e*,*f*).

At large separation distances and low frequencies (figure 6*a*,*b*, *γ*=0.1), there is no screening of finite-length effects and so the three-dimensional drag compares unfavourably with the two-dimensional result (2.23). In particular, for two-dimensional flow we observe a continuing change in drag with *Δ* (due to Stokes' paradox), which saturates via weak inertial effects only at very large *Δ*, whereas the algebraic far-field decay of the three-dimensional steady flow causes the drag to approach a limiting value more quickly as *Δ* increases. However, as *Δ* is decreased the two-dimensional drag prediction becomes more effective through *Δ*-screening, an effect enhanced when *γ*=10.

Figure 6*c*,*d* further emphasizes the drag's frequency dependence. At small *γ*, vorticity diffuses distances comparable with the cylinder length and end-effects once again exert a strong influence over the drag, causing it to diverge from the two-dimensional approximation. However, the quasi-steady approximation (3.1) that captures leading-order end-effects agrees with the predicted drag as *γ*→0. As *γ* is increased, finite-length effects decay more rapidly away from the ends and this induces two-dimensionality in the flow along much of the cylinder length, leading to improved agreement with the two-dimensional approximation. In figure 6*e*,*f*, we observe the expected breakdown of the two-dimensional approximation as the cylinder is made less slender.

Figure 6 also demonstrates the accuracy of modified RFT. Figure 6*a*,*c* demonstrates that for *γ*=0.1 the first RFT correction proves suitable over a wide range of *Δ*, but achieves greatest accuracy at smaller separations. The iterative scheme struggles at high frequencies irrespective of the separation distance (figure 6*a*,*b*, *γ*=10), where four corrective terms are still not sufficient to reach agreement with the USBT computations. The dependence of asymptotic accuracy upon *γ* is emphasized further in figure 6*c*,*d*, where fewer corrective terms are seen to be required when *γ* is small. Since RFT is based on an expansion in powers of log 1/*ϵ*, the number of corrections required is relatively insensitive to the value *ϵ* (figure 6*e*,*f*). In summary, modified RFT works well in situations where the two-dimensional model breaks down, specifically at low and moderate frequencies when separation distances are large.

### (d) Tilt

So far we have considered a cylinder with its axis parallel to a plane wall. However, USBT allows us to tilt the cylinder relative to the wall. The variation in normal drag for a cylinder driven normal to its axis at fixed *γ* and varying *Δ* is plotted in figure 7*a*,*b* for several inclination angles. As the cylinder is brought closer to the wall, the drag increases and the phase decreases, more so at low tilt angles where the influence of the wall is greatest. At large *Δ*, the normal drag is insensitive to the angle of tilt, since the wall induces negligible axial flows. Only for *Δ*≲20 (with *γ*=0.1) do wall interactions influence the normal drag at larger tilt angles. The effects of tilt are also reduced by increasing the frequency of oscillation, which results in wall interactions being governed by weaker inviscid effects. Figure 7*c*,*d* shows that for *γ*≳50 (with *Δ*=0.1) the normal drag is largely independent of inclination angle, and in this limit *γ*-screening means that the two-dimensional drag result (2.23) works well (applied in a frame of reference in which the cylinder axis is horizontal).

When the cylinder is tilted but driven vertically, we can make meaningful comparisons with two-dimensional models over a wider range of *Δ*. Although at large *Δ* wall interactions are weak, vertical driving generates non-negligible axial flows which affect the vertical drag, particularly at large tilt angles (figure 8), and two-dimensional models are a poor approximation. When the separation distance is decreased, however, the effects of *Δ*-screening lead to improved agreement between two-dimensional results and USBT computations, although at severe tilt (*α*=*π*/4) full convergence is never quite reached.

## 4. Summary and conclusions

We have used an approximation based on unsteady slender-body theory (USBT) to describe the flow generated by oscillations of a finite-length cylinder in the presence of a plane wall. We assumed throughout that oscillation amplitudes were sufficiently small to allow linearization of the Navier–Stokes equations and boundary conditions. The flow was modelled using an axial distribution of three-dimensional oscillatory Stokeslets and dipoles (plus appropriate images) in a way which, to leading order in the slenderness parameter *ϵ*, satisfied no-slip and no-penetration on the cylinder surface and the wall. The singularity distribution was determined both numerically and by perturbing about the local velocity–force relationship (a modification of RFT). The slender-body approximation holds for *Δ*≪*ϵ* and *γ*≪*ϵ*^{−1} (figure 9), breaking down either when the wall separation distance or viscous boundary layers are comparable to the cylinder radius.

When the cylinder is parallel to the wall, two mechanisms suppress finite-length effects and leave the flow two-dimensional along much of the cylinder's length. Under such circumstances, we can exploit an extensive range of asymptotic drag approximations for two-dimensional flow (Clarke *et al*. 2005). Increasing the frequency of oscillation *γ* or decreasing the separation distance between the cylinder and the wall *Δ* both limit the range of end-effects, and the Stokeslet distribution away from the ends rapidly attains the uniform value given by two-dimensional flow models. When vorticity diffuses distances much greater than *Δ*, and we can appeal to a quasi-steady two-dimensional drag (2.25). When , we can use a result which exploits unsteady image Stokeslets (2.23). For , wall interactions are inviscid and weak, allowing us to exploit an unbounded drag result (2.24). Flows at higher frequencies () and lower separations () can be described using additional two-dimensional models due to Clarke *et al*. (2005) (see figure 6). Conversely, USBT can describe the genuinely three-dimensional flow which occurs when , i.e. when the viscous length-scale and the wall separation distance are both comparable to, or larger than, the cylinder length. Therefore, used in conjunction, USBT and two-dimensional models are able comprehensively to cover (*γ*, *Δ*)-parameter space.

When the cylinder is tilted and driven normal to its axis, large separation distances and high frequencies render the drag largely insensitive to the angle of inclination, due to the weak character of wall interactions (figure 7). Alternatively, when the cylinder is tilted but driven vertically, axial flows result in drag differences at different angles of inclination even in the absence of wall interactions (figure 8). At small separations, however, there is good agreement between the two-dimensional prediction (2.23) and USBT computations, provided that the angle of inclination is not too great. We also developed a modified form of RFT, which exploits the approximately uniform local relationship between velocity and force in the USBT equation (2.4) that arises at leading order under quasi-steady conditions. Higher-order corrections in 1/ln *ϵ*, computed iteratively, account for non-local geometry and the effects of flow inertia. For *γ*=*O*(1), just one or two corrective terms in the RFT expansion are generally sufficient for convergence to the USBT computations for sufficiently slender cylinders. It was shown how interaction with the wall leads to off-diagonal elements in the cylinder's resistance matrix (through *W*_{i} in (2.18*c*)). In the low-frequency limit (*γ*≪1), the quasi-steady SBT formulation of Blake (1974) is recovered. In the high-frequency limit, convergence of the modified RFT is slow; however, *γ*-screening means that the high-frequency limit of RFT (2.22) (which extends Stokes' classical result (2.24) to capture end-effects) works effectively. Some concerns have been expressed regarding the strict legitimacy of asymptotic or numerical solutions to the integral equations, which describe an SBT approximation to the flow (Cade 1994). We believe that the comparisons made here with independently derived two-dimensional results, as well as work done by others comparing slender-body predictions with full boundary-element computations (e.g. Pozrikidis & Farrow 2003), demonstrate the practical effectiveness of USBT.

The approximations presented here describing the drag experienced by a slender oscillating microcantilever have applications to various fields, including the AFM (particularly when operating in a liquid environment at the lower end of its frequency range) and MEMS technologies. Our calculations provide justification for the use of simpler two-dimensional models in many practically relevant circumstances (depending, for example, on the level of cantilever tilt), and they provide a relatively straightforward mechanism (particularly through our modification of RFT) for capturing genuinely three-dimensional effects if necessary. Furthermore, when coupled to models of cantilever flexibility, the simplifications offered by modified RFT or by two-dimensional approximations offer considerable computational benefits, for example in computing driven or thermal spectra; further work in this area will be reported elsewhere.

## Acknowledgments

This work was supported by EPSRC grant GR/R88991.

## Footnotes

- Received June 1, 2005.
- Accepted October 31, 2005.

- © 2006 The Royal Society