2. Formulation

Figure 1 shows a schematic of a neutrally buoyant sphere suspended in a Newtonian fluid undergoing plane Couette flow between infinite boundaries separated by a distance $H$. Using the sphere radius $a$ and the velocity scale $\dot {\gamma }a$, $\dot {\gamma }=V_{wall}/H$ being the ambient shear rate, the equations governing the fluid motion, in non-dimensional form, are given by

(2.1a)$$\begin{gather} \nabla^2 \boldsymbol{u}-\boldsymbol{\nabla}p =Re_p\,\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}, \end{gather}$$

(2.1b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0 , \end{gather}$$

where $Re_p=a^2\dot {\gamma }/\nu$ is the particle Reynolds number, and $\nu$ is the kinematic viscosity of the suspending fluid. The unsteady acceleration term has been neglected in (2.1a), and the necessary justification for this is provided at the end of § 4.

For $\lambda \ll 1$, at leading order, the neutrally buoyant sphere translates with the velocity of the plane Couette flow at its centre, while rotating with half the ambient vorticity. In a reference frame centred at the sphere, and translating with this velocity, $\boldsymbol {u}$ still satisfies (2.1a) and (2.1b) with the following boundary conditions:

(2.2a)$$\begin{gather} \boldsymbol{u}={-}\tfrac{1}{2}\boldsymbol{1}_3\wedge \boldsymbol{r} \quad \text{for } \boldsymbol{r}\in S_p, \end{gather}$$

(2.2b)$$\begin{gather}\boldsymbol{u}\rightarrow r_2\boldsymbol{1}_1 \quad \text{for } r_1,r_3\rightarrow \infty\ (r_2\ \text{fixed}), \end{gather}$$

(2.2c)$$\begin{gather}\boldsymbol{u}={-}\lambda^{{-}1}s\boldsymbol{1}_1\quad \text{at}\ r_2={-}s\lambda^{{-}1}\ \text{(lower wall)}, \end{gather}$$

(2.2d)$$\begin{gather}\boldsymbol{u}={-}\lambda^{{-}1}(1-s)\boldsymbol{1}_1 \quad \text{at}\ r_2=(1-s)\lambda^{{-}1}\ \text{(upper wall)}. \end{gather}$$

Here, $S_p$ denotes the surface of the sphere, and $s=d/H$ denotes its (non-dimensional) transverse location within the channel; see figure 1. In terms of the corresponding disturbance fields, $\boldsymbol {u}'=\boldsymbol {u}-\boldsymbol {u}^\infty$, $p'=p-p^\infty$ ($\boldsymbol {u}^\infty = r_2\boldsymbol {1}_1$, $p^\infty$ being an arbitrary constant), the governing equations may be written as

(2.3a)$$\begin{gather} \nabla^2 \boldsymbol{u}'-\boldsymbol{\nabla} p' =Re_p\,(\boldsymbol{u}'\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}'+\boldsymbol{u}'\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u}^\infty+\boldsymbol{u}^\infty\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}'), \end{gather}$$

(2.3b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}'=0, \end{gather}$$

with boundary conditions (2.2a–d) taking the form

(2.4a)$$\begin{gather} \boldsymbol{u}'={-}\tfrac{1}{2}\boldsymbol{1}_3\wedge \boldsymbol{r}- r_2\boldsymbol{1}_1 \quad \text{for } \boldsymbol{r}\in S_p, \end{gather}$$

(2.4b)$$\begin{gather}\boldsymbol{u}'\rightarrow 0 \quad \text{for } r_1,r_3\rightarrow \infty\ (r_2\ \text{fixed}), \end{gather}$$

(2.4c)$$\begin{gather}\boldsymbol{u}'=0\quad \text{at}\ r_2={-}s\lambda^{{-}1}, (1-s)\lambda^{{-}1}. \end{gather}$$

Figure 1.

Schematic of a neutrally buoyant sphere suspended in wall-bounded plane Couette flow.

It is well known that one requires a matched asymptotics expansions approach to solve the Navier–Stokes equations for small but finite $Re_p$ (Proudman & Pearson Reference Proudman and Pearson1957). The matching of an inner expansion, valid in the neighbourhood of the particle, and an outer expansion, valid at distances of order the so-called inertial screening length, is in general necessary to calculate inertial corrections. For an ambient linear shearing flow, the screening length is $a\,Re_p^{-1/2}$ (Saffman Reference Saffman1965), or alternatively $H\,Re_c^{-1/2}$. The latter expression shows that the walls lie in the inner Stokesian region for $Re_c\ll 1$, with fluid inertia being a regular perturbation (see § 3). When $Re_c\gtrsim O(1)$, the inertial screening length is of order the channel width or smaller, and the walls lie in the so-called Oseen region. Fluid inertial effects now constitute a singular perturbation, and one must solve the linearized Navier–Stokes equations, with the solution being a function of $Re_c$ at leading order (see § 4).

3. The inertial lift velocity for $Re_c\ll 1$

In this section, we revisit the inertial lift calculation for $Re_c \ll 1$, a problem first addressed by Ho & Leal (Reference Ho and Leal1974) and Vasseur & Cox (Reference Vasseur and Cox1976). A generalized reciprocal theorem formulation yields the following expression for the inertial lift velocity (Anand & Subramanian Reference Anand and Subramanian2023b):

(3.1)\begin{equation} V_p={-}Re_p\int\boldsymbol{u}^{(2)}\boldsymbol{\cdot}(\boldsymbol{u}^{(1)'}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}^{(1)'}+ \boldsymbol{u}^{(1)'}\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}^\infty+\boldsymbol{u}^\infty \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}^{(1)'})\,{\rm d} V. \end{equation}

In (3.1), problem (1) denotes the one described in § 2, namely a neutrally buoyant sphere suspended in wall-bounded plane Couette flow for small but finite $Re_p$, and accordingly, the disturbance fields $(\boldsymbol {u}',p')$ defined in § 2 are denoted as $(\boldsymbol {u}^{(1)'},p^{(1)'})$ only in this section. Problem (2) describes a simpler test problem, with $(\boldsymbol {u}^{(2)},p^{(2)})$ corresponding to the Stokesian translation of a torque-free sphere between plane parallel walls, under the action of a unit force in the wall-normal direction. The test problem is governed by

(3.2a)$$\begin{gather} \nabla^2 \boldsymbol{u}^{(2)}-\boldsymbol{\nabla} p^{(2)} = 0 , \end{gather}$$

(3.2b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}^{(2)}=0, \end{gather}$$

where $\boldsymbol {u}^{(2)}$ satisfies

(3.3a)$$\begin{gather} \boldsymbol{u}^{(2)}=\boldsymbol{U}_p^{(2)} \quad \text{for } \boldsymbol{r}\in S_p, \end{gather}$$

(3.3b)$$\begin{gather}\boldsymbol{u}^{(2)}\rightarrow 0 \quad \text{for } r_1,r_3 \rightarrow \infty\ (r_2\ \text{fixed}), \end{gather}$$

(3.3c)$$\begin{gather}\boldsymbol{u}^{(2)}=0 \quad \text{for } r_2={-}s\lambda^{{-}1}, (1-s)\lambda^{{-}1}. \end{gather}$$

To $O(Re_p)$, $\boldsymbol {u}^{(1)'}$ in (3.1) may be replaced by its Stokesian approximation $\boldsymbol {u}_s^{(1)}$, the resulting volume integral being convergent. To infer the dominant scales contributing to this integral, it suffices to use estimates of the disturbance fields pertaining to the interval $1 \ll r \ll \lambda ^{-1}$. Thus, using $\boldsymbol {u}^\infty \sim r$, $\boldsymbol {u}^{(2)}\sim 1/r$, $\boldsymbol {u}^{(1)'}\approx \boldsymbol {u}_s^{(1)} \sim 1/r^2$ yields $\boldsymbol {u}^{(2)}\boldsymbol {\cdot }(\boldsymbol {u}_s^{(1)}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u}^\infty +\boldsymbol {u}^\infty \boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {u}_s^{(1)})\sim 1/r^3$ and $\boldsymbol {u}^{(2)}\boldsymbol {\cdot }(\boldsymbol {u}_s^{(1)}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u}_s^{(1)})\sim 1/r^6$, respectively, for the linear and nonlinear components of the integrand. Since ${\rm d} V \sim r^2 \,{\rm d} r$, the dominant contributions to the integral, due to the nonlinear inertial terms, arise from length scales of $O(a)$. In contrast, the linearized inertial terms appear to lead to a conditionally convergent integral, implying that the dominant contribution must arise from scales intermediate between $a$ and $H$ (or $1$ and $\lambda ^{-1}$, respectively, in non-dimensional terms), with logarithmically smaller contributions from $r \sim O(1)$ and $O(\lambda ^{-1})$. Now, on account of the inversion symmetry of the plane Couette profile, the inertial lift owes its origin entirely to the asymmetry of the sphere location with respect to the walls – in the absence of a symmetry-breaking bifurcation/instability, a neutrally buoyant sphere in unbounded simple shear flow experiences zero lift regardless of $Re_p$. Hence the contributions from $r \ll O(\lambda ^{-1})$, which must involve unbounded-domain expressions for the disturbance fields at leading order, are identically zero, and the dominant contribution due to the linearized inertial terms also arises from scales of $O(H)$. Further, the contribution of the linearized inertial terms may be shown to be larger, by a factor of $O(\lambda ^{-1})$, than that of the nonlinear terms (Anand & Subramanian Reference Anand and Subramanian2023b). This implies that for the purposes of calculating the integral in (3.1), one may replace the neutrally buoyant sphere in problem (1) by a stresslet, and the one in the test problem by a Stokeslet oriented perpendicular to the walls, with the $O(Re_p)$ lift velocity given by the following simplified integral:

(3.4)\begin{equation} V_p ={-}Re_p\int_{V}\boldsymbol{u}_{St}\boldsymbol{\cdot}(\boldsymbol{u}_{str}\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u}^\infty+\boldsymbol{u}^\infty\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u}_{str})\,{\rm d} V, \end{equation}

where

(3.5)$$\begin{gather} \boldsymbol{u}_{St} = \boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{1}_2, \end{gather}$$

(3.6)$$\begin{gather}\boldsymbol{u}_{str} ={-}\frac{20{\rm \pi}\boldsymbol{E}^\infty}{3}\boldsymbol{:}\frac{\partial \boldsymbol{J}}{\partial \boldsymbol{y}}. \end{gather}$$

Here, $\boldsymbol {y}=y_1\boldsymbol {1}_1+s\lambda ^{-1}\boldsymbol {1}_2+y_3\boldsymbol {1}_3$ is the position of the Stokeslet relative to the lower wall, and $\boldsymbol {u}_{St}$ and $\boldsymbol {u}_{str}$ are the bounded-domain Stokeslet and stresslet velocity fields, respectively, with the expression for the second-order tensor $\boldsymbol {J}$ given in Appendix A; $\boldsymbol {E}^\infty =\frac {1}{2}({\boldsymbol 1}_1{\boldsymbol 1}_2+ {\boldsymbol 1}_2{\boldsymbol 1}_1)$ is the rate of strain tensor of the ambient Couette flow. Note that (3.4) corresponds to a dimensional lift velocity of $O(Re_p\,V_{wall}\,\lambda )$.

A more general version of (3.4), pertaining to plane Poiseuille flow, has been evaluated in Anand & Subramanian (Reference Anand and Subramanian2023b), using the convolution theorem to transform the integrations along the flow and vorticity directions to ones in Fourier space. The expression specific to plane Couette flow may be obtained by using $\kappa =1$, $\beta =1$ and $\gamma ''=0$ in (87) of Anand & Subramanian (Reference Anand and Subramanian2023b), and is given by

(3.7)\begin{align} \frac{V_p}{Re_p}={-}\frac{10{\rm \pi}}{3} \int_0^\infty {\rm d} k_\perp''\, \frac{k_\perp'' \exp({-k_\perp'' (27 s+16)})\,I(k_\perp'',s)}{48 {\rm \pi} (\exp({2 k_\perp''})-1) [{-}2 \exp({2 k_\perp''})\, (2 k_\perp''^2+1)+\exp({4 k_\perp''})+1]^2}, \end{align}

where the expression for $I(k_\perp '',s)$ is given in Appendix B. The integral in (3.7) is evaluated readily using Gauss–Legendre quadrature with a suitably large cutoff $K_{max}$ for the upper limit. The choice of $K_{max}$ is crucial close to the walls – as shown in figure 2(a), for any finite $K_{max}$, $V_p$ decreases to zero in the neighbourhood of the walls, the neighbourhood shrinking with increasing $K_{max}$. For $K_{max} = 10\,000$, a numerically converged profile is obtained for $0.001\leq s\leq 0.999$.

Figure 2.

(a) Lift velocity profiles for a sphere in plane Couette flow, for $Re_c\ll 1$, obtained using (3.7) and the indicated values of $K_{max}$. The wall asymptotes calculated using (3.8) are denoted by dotted horizontal lines. (b) Lift velocity profile for a sphere in plane Couette flow for $Re_c\ll 1$ and $K_{max}=10\,000$ compared against the data obtained from table 4 in Ho & Leal (Reference Ho and Leal1974) and from digitizing figure 3 in Vasseur & Cox (Reference Vasseur and Cox1976); the pair of wall asymptotes appears as horizontal dashed lines.

In the neighbourhood of the walls ($s = 0,1$), the primary contribution to the integral in (3.7) comes from $k_\perp$ of $O(s^{-1})$ or $O((1-s)^{-1})$, so the dominant length scales are of order the small separation between the sphere and the wall. This also implies that the limiting wall value obtained below may be derived as the far-field limit of the single-wall problem (Cherukat & Mclaughlin Reference Cherukat and Mclaughlin1994). Considering the wall at $s = 0$, for instance, and using a rescaled wavenumber $k_w=k_\perp '' s$, gives

(3.8)\begin{equation} V_p^{wall} = \lim_{s \rightarrow 0} V_p = \frac{5\,Re_p}{72} \int_0^\infty {\rm d} k_w\exp({-2 k_w})\,k_w (3 k_w^2 -2 k_w+3), \end{equation}

which may be evaluated analytically and gives $55\,Re_p/576$; the $O(s)$ correction to this asymptote involves scales of $O(H)$. It is important to note here that the actual lift velocity must go to zero at the wall on account of the diverging lubrication resistance. Thus the finite wall value obtained above must be interpreted as corresponding to the intermediate asymptotic interval $\lambda \ll s\ll 1$ (for small $Re_c$); this aspect is implicit in the connection to the single-wall problem mentioned above (see Anand & Subramanian Reference Anand and Subramanian2023b).

Figure 2(b) compares the lift velocity profile obtained from (3.7), with $K_{max} = 10\,000$, against profiles extracted from Ho & Leal (Reference Ho and Leal1974) and Vasseur & Cox (Reference Vasseur and Cox1976). Agreement with the Ho & Leal (Reference Ho and Leal1974) profile is poor in general, especially near the walls. In contrast, a near-exact match is obtained with the Vasseur & Cox (Reference Vasseur and Cox1976) profile throughout the channel. The wall asymptote above is also shown, and is consistent with the limiting values of our profile and the one in Vasseur & Cox (Reference Vasseur and Cox1976); note that the aforementioned asymptote was also mentioned in Vasseur & Cox (Reference Vasseur and Cox1976), albeit without an explanation. Finally, as is evident from the profiles shown, the only equilibrium for small $Re_c$ is the stable one at the channel centreline.

4. The inertial lift velocity for $Re_c\gtrsim O(1)$

For a general shearing flow, the primary contribution to the inertial lift for $Re_c \gtrsim O(1)$ comes from scales of $O(H\,Re_c^{-1/2})$. For plane Couette flow, however, as already pointed out, the lift arises solely due to the asymmetric interaction of the sphere with the boundaries, and the relevant scales are $O(H)$ regardless of $Re_c$. Now, for $Re_p\ll 1$, either $H$ or $H\,Re_c^{-({1}/{2})}$ is much larger than $O(a)$, and determining the lift for $Re_c \gtrsim O(1)$, at leading order, requires solving the linearized Navier–Stokes equations, with the neutrally buoyant sphere treated as the same point-singularity (a stresslet) as in § 3. It is appropriate to use $H$ as the relevant length scale by defining $\boldsymbol {r}=\lambda ^{-1} \boldsymbol {R}$, with the rescalings $\boldsymbol {u}'=\lambda ^2 \boldsymbol {U}$ and $p'=\lambda ^3 P$ for the velocity and pressure fields based on the Stokesian rates of decay for a stresslet. From (2.3a,b), $\boldsymbol {U}$ and $P$ are seen to satisfy the equations

(4.1a)$$\begin{gather} \nabla^2 \boldsymbol{U}-\boldsymbol{\nabla}P =Re_c\,U_2\boldsymbol{1}_1+Re_c\,R_2\, \frac{\partial \boldsymbol{U}}{\partial R_1}-\frac{20{\rm \pi}}{3}\,\boldsymbol{E}^\infty \boldsymbol{\cdot}\boldsymbol{\nabla}\delta(\boldsymbol{R}), \end{gather}$$

(4.1b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{U}=0, \end{gather}$$

with

(4.2a)$$\begin{gather} \boldsymbol{U} \sim{-}\frac{5}{2}\,\frac{\boldsymbol{R}(\boldsymbol{E}^\infty\boldsymbol{:}\boldsymbol{R}\boldsymbol{R})}{R^5} \quad\text{for } \boldsymbol{R}\rightarrow 0, \end{gather}$$

(4.2b)$$\begin{gather}\boldsymbol{U}=0 \quad \text{at } R_2={-}s, 1-s. \end{gather}$$

The neutrally buoyant sphere appears as a stresslet forcing, this being the final term on the right-hand side of (4.1a), with $\boldsymbol {E}^\infty$ being the rate of strain tensor of the ambient flow as before. Relation (4.2a) is the requirement of matching with the Stokesian field in the inner region, and defined earlier in (3.6). Following Schonberg & Hinch (Reference Schonberg and Hinch1989), we define a partial Fourier transform

(4.3)\begin{equation} \hat{f}(k_1,R_2,k_3)=\int_{-\infty}^\infty\int_{-\infty}^\infty f(\boldsymbol{R})\exp({\iota (k_1 R_1+ k_3 R_3)})\,{\rm d} R_1 \,{\rm d} R_3, \end{equation}

which yields the following coupled ODEs for the transformed pressure and normal velocity fields:

(4.4a)$$\begin{gather} \frac{{\rm d}^2\hat{P}}{{\rm d} R_2^2}-k_\perp^2\hat{P}=2\iota k_1\,Re_c\,\hat{U}_2, \end{gather}$$

(4.4b)$$\begin{gather}\frac{{\rm d}^2\hat{U}_2}{{\rm d} R_2^2}-k_\perp^2\hat{U}_2=\frac{{\rm d}\hat{P}}{{\rm d} R_2}-\iota k_1\,Re_c\, R_2\hat{U}_2, \end{gather}$$

where $k_\perp ^2=k_1^2+k_3^2$, with the matching condition

(4.5)\begin{equation} \hat{U}_2 \sim{-}\frac{5{\rm \pi}\iota k_1\,|R_2|\exp({-k_\perp\,|R_2|})}{3} \quad \text{for } k_1,k_3\to\infty \text{ and } R_2\rightarrow 0, \end{equation}

and the wall boundary conditions

(4.6)\begin{equation} \hat{U}_2=\frac{{\rm d}\hat{U}_2}{{\rm d} R_2}=0\quad \text{at } R_2={-}s, 1-s. \end{equation}

The boundary conditions above must be supplemented by the following jump conditions, which relate the limiting values of the transformed fields above and below the location ($R_2 = 0$) of the stresslet forcing (Anand & Subramanian Reference Anand and Subramanian2023b):

(4.7a)$$\begin{gather} \hat{P}^+(k_1,0,k_3)-\hat{P}^-(k_1,0,k_3) ={-}\frac{20{\rm \pi}\iota k_1}{3}, \end{gather}$$

(4.7b)$$\begin{gather}\frac{{\rm d}\hat{P}^+}{{\rm d} R_2}(k_1,0,k_3)=\frac{{\rm d}\hat{P}^-}{{\rm d} R_2}(k_1,0,k_3), \end{gather}$$

(4.7c)$$\begin{gather}\hat{U}_2^+(k_1,0,k_3)=\hat{U}_2^-(k_1,0,k_3), \end{gather}$$

(4.7d)$$\begin{gather}\frac{{\rm d}\hat{U}_2^+}{{\rm d} R_2}(k_1,0,k_3)-\frac{{\rm d}\hat{U}_2^-}{{\rm d} R_2}(k_1,0,k_3) ={-}\frac{10{\rm \pi}\iota k_1}{3}. \end{gather}$$

Here, the superscripts ‘$-$’ and ‘$+$’ pertain to the intervals $-s\leq R_2<0$ and $0< R_2\leq (1-s)$, respectively. In the matching region, $\boldsymbol {U}$ must reduce to the sum of a singular stresslet contribution at leading order and a uniform flow in the transverse direction that arises due to inertia. Since the sphere is force-free, it must be convected by the latter uniform flow. The lift velocity may be determined via an inverse Fourier transform after removing (for numerical convenience) the normal component of the stresslet contribution:

(4.8)\begin{equation} V_p=\frac{Re_p}{4{\rm \pi}^2\,Re_c}\,{\rm Re}\left\{\int_{-\infty}^\infty\int_{-\infty}^\infty \hat{U}_2^\pm(k_1,0,k_3;Re_c)\,{\rm d} k_1\,{\rm d} k_3\right\}, \end{equation}

where ${\rm Re}\{\ \cdot\ \}$ denotes the real part of a complex-valued function, and automatically achieves the removal of the purely imaginary stresslet contribution (see (4.5)). As indicated in (4.8), one may use either $\hat {U}_2^-$ or $\hat {U}_2^+$ on account of continuity; see (4.7c). The ODEs (4.4a,b) along with the boundary conditions (4.6) and jump conditions (4.7a–d) are solved using the shooting method described in Appendix A of Schmid, Henningson & Jankowski (Reference Schmid, Henningson and Jankowski2002). After computing $\hat {U}_2$, the inverse Fourier transform (4.8) is evaluated using Gauss–Legendre quadrature in a truncated domain that is a circle of a large but finite radius $K_m$. Note that the dimensional lift velocity corresponding to (4.8) may be written formally in the form $Re_p\, V_{wall}\,\lambda \,F(Re_c)$, where the function $F(Re_c)$ includes the $Re_c$ dependence of the Fourier integral; a plot of $V_p$ against $Re_c$ (not shown here), for a fixed $s$, shows that $F(Re_c)$ is bounded by $Re_c^{-\delta (s)}$, with $\delta (s) \sim 1.7\unicode{x2013}1.8$ for the range of $Re_c$ examined.

The lift velocity scaling for $Re_c \gtrsim O(1)$ above, and the one given in § 3.4 for $Re_c \ll 1$, may now be used to justify the quasi-steady approximation used in the formulation. Neglect of the unsteady term in (2.1a) requires that the time for momentum to diffuse, over scales contributing dominantly to the lift, be much smaller than the $O(H/V_p^*)$ time scale for configurational change, $V_p^*$ here being the dimensional lift velocity. The dominant scales are $O(H)$ regardless of $Re_c$, and the momentum diffusion time scale is therefore $O(H^2/\nu )$. Using $V_p^* \sim O(Re_p\,V_{wall}\,\lambda )$ for $Re_c \ll 1$, the configurational time scale is $O(\lambda ^{-1}\,Re_p^{-1}\,H/V_{wall})$, and the ratio of the aforementioned time scales is $\lambda \,Re_p\,Re_c$, which is the order of magnitude of the unsteady acceleration (${\partial \boldsymbol {u}'}/{\partial t}$) in relation to the viscous terms. The other terms in the inertial acceleration are larger, being only $O(Re_c)$ smaller than the viscous terms. For large $Re_c$, $V_p^* \sim O(Re_p\,Re_c^{-\delta (s)}\,V_{wall}\,\lambda )$, and the resulting ratio of the unsteady acceleration to the viscous terms is $O(\lambda \,Re_p\,Re_c^{1-\delta (s)})$, which must again be small compared to unity for the quasi-steady assumption to hold. Note that the unsteady acceleration continues to be small in relation to the other inertial terms, and will also remain small compared to the viscous terms when $\lambda \,Re_p\,Re_c^{1-\delta (s)} \ll 1$, or when $\lambda \ll O(Re_c^{-(2-\delta (s))/3})$. For the $\delta$ values quoted above, the minimum value of $Re_c^{-(2-\delta (s))/3}$ is $0.49$ (for $Re_c=1200$, just prior to transition), and the requirement above is satisfied readily in the small-$\lambda$ limit. This implies that the unsteady acceleration remains subdominant over the entire range of $Re_c$ examined.