This model of experiments on auditory sensory hair cells extends previous work via distributions on a cylindrical pipe of tangentially and normally directed oscillatory point forces, which are modified to achieve no-slip at the wall in two stages. Starting with the pressure and vorticity jumps associated with the oscillatory pressure-driven flow upstream in the pipe, the adjustment of the interior pipe flow from its upstream complex-valued profile to its exit profile is fully included. This is essentially achieved by modifying the steps of the steady case analysis. The flow field oscillates with phase dependent on position, and the level curves of the streamfunction indicate instantaneous particle motion but not streamlines. Thus, an eddy is not indicated by the closed curve that occurs midway through the two half cycles and is due to competing forces between the inflow and outflow, particularly in the second half cycle as the fluid enters the pipe. The wall pressure and wall shear stress also oscillate with the non-uniformities concentrated near the origin, but are relatively damped midway through the two half cycles. Independent of the orifice location, there is a small effect of frequency on the wall pressure and the wall shear stress.
Motivated by physiological experiments on isolated, i.e. in vitro outer hair cells (OHCs) that are thought to contribute to the active process in the cochlea  responsible for the exquisite frequency selectivity and sensitivity of mammalian hearing, Davis et al.  developed a model of a submerged viscous fluid jet impinging on an infinite planar wall and identified non-uniformities in the wall pressure and wall shear stress. This leads to the question of whether modelling the periodic case would be helpful in understanding the experiments by Canlon et al. , Canlon & Brundin , Brundin & Russell  and Rybalchenko & Santos-Sacchi , in which the lateral wall of the OHC was subjected to an oscillatory fluid jet. These experiments were influenced by the suggestion first proposed by Guild  that fluid forces imparted on the lateral wall of the OHC may be the mechanism by which in vivo OHCs are damaged by acoustic trauma such as loud noise ; later, several experiments suggested that pressure within the organ of Corti housing the OHCs could play a role in the active process [9–12]. However, there are wide-ranging and differing views on the actual role of the OHC . The traditional view is that of a travelling wave along the cochlea that then involves the OHCs , but other explanations exist . Despite these differences, there is a need for better understanding of the in vitro and in vivo fluid forces on the lateral wall . So, extending the model developed by Davis et al.  to the oscillatory case should, among other things, provide a more realistic alternative to the uniform pressure assumed in a computational model of the in vitro experiments .
As in Davis et al. , the flow is described by an axisymmetric coordinate system in which dimensions are normalized by the radius (Rp) of a cylinder representing the long tapered end of the stimulating pipette. The flow is then assumed to be generated by oscillatory pressure-driven flow far upstream inside the cylinder with outflow at a distance of h=Hp/Rp from an infinite planar wall. This results in a two parameter family description based on h and , where ω=2πf is the angular frequency with respect to the stimulus frequency f and ν is the kinematic viscosity of the water-based salt solution bathing the OHC. Here, can be regarded as the associated Wommersley number of the flow (not to be confused with the traditional Wommersley number). Since Brundin & Russell  used Rp=7.5–10 μm, f=100–3000 Hz and Hp=15–25 μm, h=1.5–3.3 and . Likewise, Canlon et al.  and also Canlon & Brundin  used Rp=5 μm, f=200 Hz and Hp=10–15 μm, so h=2–3 and ; Rybalchenko & Santos-Sacchi  used Rp=2.5–5 μm, f=1 kHz and Hp=15–20 μm, so h=3–8 and . It is therefore reasonable to consider h and , respectively, in the ranges 1–5 and 0.01–2. As discussed by Davis et al. , the flow in the tapered end of the pipette was both developed and slow enough to ensure that the OHC remained in view under the microscope. This leads to a very low Reynolds number of the order of 10−3, which can be confirmed by the experimental estimate of 10−13 obtained by measuring the induced motion of glass beads . Since the Stokesian assumption remains valid in the oscillatory case, the computational approach adopted here follows Davis et al. , but with modifications based on the associated Wommersley number.
2. Mathematical model
The approach is to generate the whole flow by using distributions on the cylinder (r=1, z>h) of tangentially and normally directed oscillatory Stokeslets, which are modified to achieve no-slip at the wall z=0. Two stages are required: first, the pressure and vorticity jumps associated with the oscillatory pressure-driven flow upstream in the pipe r<1, z>h are forced, and then further oscillatory distributions of zero density far upstream but with a square root density singularity at z=h are added to achieve no-slip on the pipe wall (r=1).
Thus, consider creeping flow bounded by the wall z=0 and generated by pressure-driven oscillatory flow of period 2π/ω in the finite fixed cylinder r=1, z>h. The velocity and dynamic pressure fields Re[(v,p) e−iωt] are governed by the oscillatory Stokes equations 2.1where μ is the molecular viscosity. Axisymmetric flow allows the velocity representation, with unit vectors, in terms of the stream function ψ that satisfies 2.2The pressure p and the vorticity −r−1L−1ψ are then related by 2.3No-slip and the imposed pipe outlet flow force ψ to satisfy the boundary conditions 2.4 2.5 and 2.6Conditions (2.4) and (2.6) ensure that 2.7which is the far upstream complex amplitude of the pressure-driven flow that creates the oscillatory ‘jet-like’ motion through the orifice at z=h, 0≤r<1. This scaling of the upstream flow to have dimensionless flux 2π e−iωt and mean velocity 2 e−iωt is relevant to the wall pressure and wall stress amplitudes in §4.
(b) Upstream forcing
Following Davis , Stokeslet singularities are distributed along the cylinder wall and modified to accommodate the no-slip condition (2.5). Davis  noted that flow fields around discs can be identified as due to normally and radially distributed Stokeslets. Furthermore, it is shown that a ring of oscillatory force singularities of strength (e−iωt suppressed) at z=0, r=r′ generates the stream function 2.8while a ring of force singularities of strength (e−iωt suppressed) at z=0, r=r′ generates 2.9Alternative forms of these integrals, required here for the identification of discontinuities across the cylinder wall, are obtained by applying Fourier transforms to (2.1), with the force singularity added in each case. They are 2.10and 2.11where , . From (2.10) and (2.11), it is seen that pax and L−1ψrad are continuous at r=r′, but 2.12
The steady limit studied by Davis et al.  can be recovered from these and subsequent expressions by use of the formulae
In the pressure-driven flow (2.7), , , where 2.13and so the required discontinuities can be achieved by constructing 2.14The evaluation of the corresponding pressure field p1(r,z) is achieved by reverting to the forms (2.8) and (2.9), which yield, by use of (2.3), Hence, as required, as , where H(x) denotes the Heaviside unit function.
The no-slip condition (2.5) is evidently satisfied by the stream functions 2.15and 2.16hence, following (2.14), 2.17in which is given by (2.7) and A(q) by (2.13). This arrangement, made possible by the z0 integrals allowing, as in the steady case, use of 2J1(k)−kJ0(k)=kJ2(k), displays the far upstream () flow, together with three contributions arising, respectively, from the prescribed ring singularities, their images directed to achieve zero normal flow at z=0 and the integrals in (2.15) and (2.16).
Note that Ψ1 may be verified to be continuous at z=h up to the third derivative, as expected of a solution of the fourth-order equation (2.2). Also, within the fluid, 2.18and 2.19This completes the first stage of the flow field construction.
(c) No-slip forcing
In the second stage, further distributions of the above Stokeslets are used to cancel the velocity components generated by Ψ1 at the cylinder wall r=1,z>h. According to (2.17), these are 2.20and 2.21which are such that U1(1,z)∼2/z3, W1(1,z)∼1/z2 as , provided is not small.
The additional stream function has the form 2.22with the density functions (C(z0) and D(z0)) expected to vanish at infinity and have square root singularities at the orifice z=h. According to (2.12), the vorticity and pressure discontinuities across the wall are As with flow through a hole in a wall , their edge behaviour is anticipated by setting whose substitution in (2.22) yields, after some manipulation, 2.23where S, Q are Abel transforms, with no obvious physical interpretation. Their important role is as bounded, continuous mappings of the density functions, C,D . The factor q1/2 is inserted to achieve symmetric kernels in the subsequent integral equations.
As in the steady flow, a disadvantage of expressing (2.22) in the same form as (2.17) is that a possible, but disallowed, solution is one that changes the prescribed amplitude of the pipe flow and is such that S(q)=O(q3/2), Q(q)=O(q1/2) as .
The z0 integrals in (2.23) will be evaluated by use of the formulae 2.24but there is a difficulty because neither the form e−k|z−z0|, arising from (2.8) and (2.9) nor the forms and , arising from (2.10) and (2.11) are suitable for direct application of (2.24). The successful strategy is to include the image terms in (2.15) and (2.16) when switching to the alternative forms. Thus, write and Hence, in (2.23), 2.25and 2.26When (2.25) and (2.26) are substituted into (2.23), it follows that the velocity components generated by Ψ2(r,z) at the cylinder wall are given, with Bessel functions evaluated at k unless otherwise stated, by 2.27and 2.28in which the r=1+,1− expressions are equal because kK0I1+kK1I0=1.
(d) Problem statement
The no-slip conditions (2.4) are now applied in the form by use of the formulae and the k-derivative of the latter to obtain, after substitution of (2.20), (2.21), (2.27) and (2.28), a pair of integral equations. Then, seeking to mimic the disc and hole-in-a-wall calculations [18,19], in which integrals from 0 to of , yield multiples of δ(t−s), it is noted that −2k2(d/dk)(K1I1) and 2(d/dk)(k2K0I0) both as and The corresponding result involving ksY 0(ks)Y 0(kq) must include the delta function because of the similar behaviour of the integrands as . Working from integrals [21, see 6.671], it may be shown, via a lengthy calculation, that Whence the transformed no-slip conditions are identified as a pair of Fredholm integral equations of the second kind, namely 2.29and 2.30in which the kernels are given by 2.31 2.32 2.33and 2.34and the forcing functions are given by 2.35and 2.36Evidently, the system is symmetric, that is, any unbiased discretization gives a linear system of equations whose matrix of coefficients is symmetric.
(e) Wall pressure and stress
With S and Q determined, the pressure and stress on the plane wall can be readily evaluated, since the restriction z<h<z0 allows use of the simpler expressions (2.8) and (2.9). From (2.19) and (2.23), the wall pressure is 2.37and the wall stress is, from (2.18) and (2.23), 2.38
3. Numerical method
As in the steady case , the pair of Fredholm integral equations of the second kind, (2.29) and (2.30), was solved via the Neumann method in which the forcing terms were used to initiate the iteration. Thus, for J quadrature points and weights λi and wi, respectively, the solution c=[S,Q]t was obtained from successive approximations of . The infinite integrals in the kernels and forcing terms, i.e. G and F involve the product of a well-behaved function involving modified Bessel functions and two Bessel functions of the first or second kind. These were calculated at Gauss–Laguerre quadrature points . The streamfunction was plotted over [0,6]×[0,6] with uniform grid spacing of 0.2 for both r and z, except for 0.02 when 0.8<r<1.2; the same 77 collocation points for r were used in the plots of the wall pressure and wall shear stress. In addition, a simplified version of IIPBF for dealing with infinite integration of a product of a function and a Bessel function was used for the pressure at r=0.
Computations of densities S and Q, wall pressure and wall shear stress were first checked with the steady case for and J=16. It was found that due to a typographical error, the pressure previously obtained for the steady case  was slightly underestimated for h=1 and h=2. It was also noted that the dominant contribution to the wall pressure and wall shear stress came from the upstream influence with the correction, as expected, provided by the no-slip forcing. Using J=64, the flow field described by Ψ1+Ψ2 computed from (2.17) and (2.23) in the first half cycle, i.e. at times ωt=0,π/4,π/2 and 3π/4 for all pairs of and h=1,3 is shown in figures 1 and 2. In the second half cycle, the flow direction is reversed. Note that an eddy is not indicated by the closed curve in the instantaneous streamfunction that appears at approximately midway through the first and second half cycle. The plots give good overall descriptions of the flow cycle, but, being constructed without imposition of the pipe and formed by interpolation, indicate non-existent leakage. The wall pressure and wall shear stress are shown in figures 3 and 4, respectively, for h=1−5, but with to emphasize the differences in amplitudes.
A model of an oscillatory submerged viscous jet impinging on an infinite planar wall is developed. It extends the steady case of Davis et al.  by involving a periodic version of Stokeslets imposed along the pipe wall and the plane wall. A more detailed description of the flow field is thus presented, in order to display the periodic streamfunction through instantaneous level curves during a half cycle. A closed curve, but not an eddy, is generated at approximately the level of the orifice and is weaker or absent when the orifice is closer to the wall or the stimulus frequency is lower and occurs midway through the first and second half cycle. As in the steady case , the upstream flow is adjusted due to the periodic equivalent of the Papkovich–Fadle eigenfunctions that contribute to a biharmonic streamfunction. The closed curve results from competing forces between the inflow and outflow as the fluid enters the pipe, which becomes more accentuated when the orifice is further from the planar wall, especially at higher frequencies. It is believed that this kind of flow field has not been observed before. The wall pressure and wall shear stress also oscillate a full cycle from the steady-state position with the non-uniformities concentrated in the immediate impinging zone of the plane wall, but are relatively damped midway through the first and second half cycle. Independent of h, there appears to be a mild effect of frequency on the wall pressure, but a weak if not negligible one on the wall shear stress.
In the context of the larger picture of cochlear mechanics, it is argued that the relative movement of the reticular lamina and the basilar membrane that, respectively, lie above and below the OHCs causes fluid displacement within the thin fluid-filled space of Nuel compartment encasing the OHCs . Such fluid displacement occurs on a cycle-by-cycle basis, and therefore its effect on the lateral wall of the OHC may be critical in understanding what might occur in extreme events such as acoustic trauma. It is hoped that this work as well as the steady case will be used to develop a biophysical model of membrane mechanoporation induced by the non-uniformities in the wall pressure and wall shear stress that occur in the OHC experiments.
We thank Prof. W. E. Brownell and Prof. I. J. Russell, FRS, for their long-standing interest in this problem.
- Received May 3, 2013.
- Accepted June 18, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.