3.1 Longitudinal flow

With respect to the longitudinal flow through the geometry shown in figure 1(a), the velocity $W(X,Y)$ of the unidirectional flow is governed by the balance of viscous and pressure forces while the no-slip boundary conditions apply at both the corrugated bottom wall and the flat top wall, leading to

Here, $\unicode[STIX]{x1D716}=2\unicode[STIX]{x03C0}e/L$ and $H=2\unicode[STIX]{x03C0}h/L$ are a characteristic pattern size and the nominal channel thickness, rendered dimensionless through multiplication with the pattern wavenumber $2\unicode[STIX]{x03C0}/L$ . The $H$ -dependent right-hand side of (3.1a ) is a consequence of the wall-shear-based velocity scale $(\unicode[STIX]{x0394}ph/2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707})$ .

In the following, $\unicode[STIX]{x1D716}$ will be treated as a small parameter, and asymptotic expansions in the form $W=W^{0}+\unicode[STIX]{x1D716}W^{1}+\unicode[STIX]{x1D716}^{2}W^{2}+o(\unicode[STIX]{x1D716}^{2})$ together with the method of domain perturbations (Hinch Reference Hinch1991), which involves the transfer the boundary condition on the curved surface (3.1b ) to $Y=0$ through the Taylor expansion $W(X,\unicode[STIX]{x1D716}g(X))=W(X,0)+\unicode[STIX]{x1D716}gW_{Y}(X,0)+\unicode[STIX]{x1D716}^{2}g^{2}W_{YY}(X,0)/2+o(\unicode[STIX]{x1D716}^{2})$ , will be employed. Here, subscripts denote derivative. Since the highest-order correction evaluated in this study is $W^{2}$ , algebraic expressions for the normalized permeability derived in §§ 3, 3.4 and 4 will contain an $o(\unicode[STIX]{x1D716}^{2})$ error term, even though the same will be indicated explicitly only when important to the discussion. The ‘little o’ asymptotic notation used here may be noted.

3.2 Transverse flow

The transverse flow is solved conveniently in terms of the streamfunction $\unicode[STIX]{x1D713}$ defined through $U=\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}Y$ and $V=-\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}X$ . For Stokes flow with a specified pressure drop between $X=-\unicode[STIX]{x03C0}$ and $X=\unicode[STIX]{x03C0}$ in a channel subject to no slip and no penetration on both the corrugated and planar boundaries of figure 1(a), the following need to be observed:

In the following, $\unicode[STIX]{x1D716}=2\unicode[STIX]{x03C0}e/L$ will be treated as a small parameter, and the asymptotic expansions in the form $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}^{0}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D713}^{1}+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D713}^{2}+o(\unicode[STIX]{x1D716}^{2})$ together with the method of domain perturbations (Hinch Reference Hinch1991), which involves the transfer the boundary conditions on the curved surface ((3.12c ) and (3.12d )) to $Y=0$ through the Taylor expansion $\unicode[STIX]{x1D713}(X,\unicode[STIX]{x1D716}g(X))=\unicode[STIX]{x1D713}(X,0)+\unicode[STIX]{x1D716}g\unicode[STIX]{x1D713}_{Y}(X,0)+\unicode[STIX]{x1D716}^{2}g^{2}\unicode[STIX]{x1D713}_{YY}(X,0)/2+o(\unicode[STIX]{x1D716}^{2})$ and a similar expression for $\unicode[STIX]{x1D713}_{Y}(X,\unicode[STIX]{x1D716}g(X))$ , will be employed. Here, subscripts denote a derivative. Since the highest-order correction evaluated in this study is $\unicode[STIX]{x1D713}^{2}$ , all algebraic expressions derived in the current section will contain an $o(\unicode[STIX]{x1D716}^{2})$ error term. However, for conciseness, this error will not be indicated explicitly in the equations to follow, barring a few exceptions.

3.3 Universal relationship between eigenvalues of the permeability tensor

For isolated surfaces in shear flow represented through the effective slip formalism, a simple algebraic relationship is known to exist between the eigenvalues of the effective slip length tensor (Kamrin et al. Reference Kamrin, Bazant and Stone2010). It is worthwhile investigating if similar relations exist in confined flow between the eigenvalues of the effective permeability tensor.

Unlike isolated surfaces, the permeability of a patterned surface in both principal (eigen-) directions depends on the channel size ( $H$ ) in a manner which may at first appear non-trivial to eliminate (e.g. compare equations (3.8) and (3.18) using respective dependencies) or scale out. However, the situation simplifies considerably when the dependence on $H$ is split into exponentially and algebraically decaying parts, as discussed above. We now derive an interconnecting relationship of comparable simplicity to that applicable between the eigenvalues of the effective slip length tensor (Kamrin et al. Reference Kamrin, Bazant and Stone2010). This relationship only demands the specification of certain global hydrodynamic and metrological characteristics of the channel and the patterned surface.

Appearance of the sums $S_{1}$ and $S_{2}$ in both (3.11) and (3.21) allows for the easy elimination of any one of the two, in order to connect the large- $H$ permeabilities $\tilde{K}^{\Vert }$ and $\tilde{K}^{\bot }$ . We prefer eliminating $S_{1}$ as $S_{2}$ has an intrinsic geometrical (and metrological) significance, as discussed below. Thus, the following relationship between the effective permeability calculated in the two mutually perpendicular principal directions of the flow is obtained:

In the second equality, Parseval’s identity has been used to eliminate the Fourier coefficients in favour of the root-mean-squared spatial average $\langle g^{2}\rangle$ of the topography. In dimensional terms,

where $R_{q}$ is the notation used in the field of surface metrology for root-mean-squared roughness in distance units measured from the mean line (Whitehouse Reference Whitehouse1994). The equations above indicate that, for sufficiently large channel thickness, a universal relationship completely determined by the variance of the topography exists between $K_{\bot }$ and $K_{\Vert }$ and, therefore, it may be sufficient to know only one of the pair $\{K_{\Vert },K_{\bot }\}$ to infer the fluid fluxes parallel and perpendicular to any given direction, provided the $R_{q}$ value of the topography is known from surface metrology (Whitehouse Reference Whitehouse1994). It can be noted that a similar universal relationship is known to exist among the eigenvalues of the effective slip tensor on isolated surfaces (Kamrin et al. Reference Kamrin, Bazant and Stone2010; Asmolov & Vinogradova Reference Asmolov and Vinogradova2012).

Similar relationships can be derived for other flow configurations, such as flow through a channel with applied shear on the walls, free-surface flow, flow with two identical topographic patterns on opposite walls and a combination thereof (see § 3.4). If an otherwise identical pattern is phase shifted on the opposite wall, as may arise in practice due to difficulties in alignment, the effect of phase shift does not appear in the large- $H$ theory, but can be resolved through straightforward but somewhat tedious finite- $H$ -theory calculations following the lines of § 3.

3.4 Adaptation of asymptotic findings to other flow configurations

Certain other boundary conditions and flow configurations that arise in practice may be treated by straightforward alterations to the governing equations and boundary conditions. In this section, the normalized permeabilities of certain important configurations will be reported. For example, of relevance to SHS and LIS (Wexler et al. Reference Wexler, Jacobi and Stone2015; Dubov et al. Reference Dubov, Nizkaya, Asmolov and Vinogradova2018), the boundary condition at $y=H$ can either be (a) a prescribed shear with zero applied pressure gradient, or (b) a no shear condition under applied pressure gradient, or a combination thereof, as discussed in Wexler et al. (Reference Wexler, Jacobi and Stone2015) in the context of the shear stability of the lubricants entrapped in the longitudinal grooves of rectangular cross-section in their experiments. For shear flows like (a), the dimensional permeability ( $k$ ) can be defined analogously as before, i.e. the flow rate of a fluid of unit viscosity through the confined passage subject to unit tangential traction at the (solid–fluid or fluid–fluid) interface. The resultant flow for (a) will be indicated by the subscript ‘ $s$ ’, standing for shear, and that for (b) by subscript ‘ $pzs$ ’, standing for pressure-driven flow with ‘zero surface shear’. The resultant normalized permeabilities can then be calculated by steps similar to those described in § 3, differing only in the imposition of the top wall boundary condition. In fact, the expressions for permeability are particularly simple, if in addition, $H$ is large enough to allow the neglect of terms exponentially decaying in $H$ . They are

To address channel height effects completely in longitudinal flow, $S_{1}$ in the above expressions can be replaced by the sum $\sum _{n=1}^{\infty }n\,\text{tanh}(nH)g_{n}^{2}$ . Corresponding expressions containing finite height effects in transverse flow are placed in appendix B. A selection of the above results can be used to assess the extent of shear-driven drainage of the lubricant entrapped by more general groove shapes than the rectangular shape used in Wexler et al. (Reference Wexler, Jacobi and Stone2015), as discussed later in the article. Owing to the unique scale for velocity used in this study, a shear flow bounded only on one side by the patterned surface can be studied by simply observing the limit $H\rightarrow \infty$ in all $H$ -dependent equations reported so far in the article. For such unconfined flows, the pitch-averaged velocity field can be characterized by its effective slip length ( $L_{eff}$ ), which is the distance between the apparent origin of the flow determined from extrapolation of the far-field velocity (as $Y\rightarrow \infty$ ) to the mean line. The quantity $L_{eff}$ is positive/negative according as the apparent origin lies below/above the mean line. For a no-slip topography $L_{eff}$ is negative.

If the above definition is applied to (3.10) and (3.20) in the limit $H\rightarrow \infty$ while setting the constants superscripted with ${\sim}$ to zero, as discussed in their respective following paragraphs, $L_{eff}$ can be found to be given by

Here, $C$ is given by $C^{\Vert }$ calculated by (3.9b ) in longitudinal flow and (3.19a ) in transverse flow. Using (3.9b ) and (3.19a ) in (3.25), the relations $L_{eff,\Vert }=-2S_{1}\unicode[STIX]{x1D716}^{2}$ and $L_{eff,\bot }=-4S_{1}\unicode[STIX]{x1D716}^{2}$ earlier reported in Luchini et al. (Reference Luchini, Manzo and Pozzi1991) and Kamrin et al. (Reference Kamrin, Bazant and Stone2010) are recovered for longitudinal and transverse flows, respectively. In the literature on turbulent drag reduction, a parameter closely related to $L_{eff}$ , named the protrusion height, is also used, which in the current notation will equal $\unicode[STIX]{x1D716}+L_{eff}$ . More specific details on the above three alternate flow configurations (shear driven, pressure driven with zero surface shear and unconfined shear) are discussed later in the article, along with demonstrations of their practical application to experimentally significant settings from the literature in §§ 4.4 and 4.5. It may be noted here that it follows from symmetry considerations that $K_{pzs}$ is also the normalized permeability of a two-dimensional straight channel with identically patterned walls separated by the distance $2H$ .

4.1 Test problems

Both the specific topographies to be studied here are normalized to have $\text{max}(g(x))=1$ , which ensures that the topography is bounded above by $Y=\unicode[STIX]{x1D716}$ . As a result, when the relative roughness $\unicode[STIX]{x1D6FC}$ equals unity, the peak(s) of the topographies will touch the opposing wall. The spectra, sums $S_{1}$ and $S_{2}$ are evaluated in closed form as far as possible using known results about integral forms and infinite series pertinent to the special functions appearing below (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

Figure 2. Surface topography shapes for the two test cases.

Topography A: phase-modulated sinusoids. As seen in figure 2(a), a family of phase-modulated sinusoids can be parameterized by a dimensionless shape factor $a$ as follows:

Note that a base shift has been applied to the basic phase-modulated cosine function $\cos (a\cos X)$ to ensure $\langle g\rangle =0$ . In addition, the denominator provides a scale shift that ensures that the peak value of $g_{A}$ does not exceed unity, so that the value of $\unicode[STIX]{x1D716}$ is indeed representative of the characteristic pattern size. Equation (4.1) exemplifies a reasonably complex but smooth topography which can have a number of local and global extrema tuneable with the parameter $a$ . The three $a$ values shown are $a=\unicode[STIX]{x03C0}/2,5\unicode[STIX]{x03C0}/4,5\unicode[STIX]{x03C0}/2$ . For $a=5\unicode[STIX]{x03C0}/4$ , eight extrema (four global minima, two local maxima and two global maxima) and four zero crossings occur over one wavelength of topography, although the variance of  $0.225$ for $g^{A}(X)$ with $a=5\unicode[STIX]{x03C0}/4$ (as calculated from (4.2c ) using Parseval’s identity) is the lowest of the three profiles shown. For $a=5\unicode[STIX]{x03C0}/2$ , twelve extrema (six global maxima, four global minima, two additional local minima) and twelve zero crossings occur per pattern pitch, and the variance is also the largest ( $0.49$ ). The relatively less complex (in terms of types and number extrema) profile for $a=\unicode[STIX]{x03C0}/2$ has four extrema (two maxima and minima) with four zero crossings but the second highest variance of $0.449$ among the three $a$ values shown. The asymptotic variance for large $a$ calculated from (4.2c ) is $1/2$ , but is reached through very slow oscillations. Although, a phase-modulated sinusoid in time is an important waveform in signal processing and optics (Dunlop Reference Dunlop2017), we are unaware of any study utilizing its spatial version in the study of fluid flow phenomena. The Fourier coefficients of Topography A are available in closed form, thanks to Jacobi–Anger expansions (Abramowitz & Stegun Reference Abramowitz and Stegun1964). Topography A can be represented by

A closed-form expression for $S_{1}^{A}=\sum _{n=1}^{\infty }|ng_{n}^{A}|^{2}$ appears not to be available, although p. 532 of Watson (Reference Watson1995) equates this sum to an iterated integral. It was found to be convenient to approximate $S_{1}^{A}$ to at least four-digit accuracy numerically by truncation to a sufficiently large number of terms.

Topography B: triangular grooves. The triangular-wave topography (figure 2 b) which is represented by the wall shape given by (4.3a ), and on which periodicity is imposed by (4.3b ), is a non-smooth topography possessing ‘corners’. The permeability for this topography has a spatial and spectral description that allows the convenience of closed forms, as revealed by the following results:

The closed forms given by (4.3e ) and (4.3f ) for the sums required for the large- $H$ exponentially accurate calculation of the normalized hydraulic permeability of the triangular-grooved surface are obtained using the properties of Riemann zeta functions (Abramowitz & Stegun Reference Abramowitz and Stegun1964), leading to

Here, $\unicode[STIX]{x1D701}$  stands for the Riemann zeta function for which precise enumeration/tabulation and numerical implementations are readily available and $\unicode[STIX]{x1D701}(3)=1.20205\cdots \,$ is also known as Apéry’s constant (Sloane Reference Sloane2019). Equation (4.4b ) is an important closed-form result from this study applicable for the friction factors of the oft-studied triangular topography (Taylor Reference Taylor1971; Dey, Saha & Chakraborty Reference Dey, Saha and Chakraborty2018).

4.2 Numerical assessment of analytical results

The expressions for permeability provided in § 3 are expected to be accurate only for sufficiently small values of $\unicode[STIX]{x1D716}$ . For numerical assessment of the errors arising from the asymptotic approximation, a finite-element-method-based solution of the governing equations and boundary conditions discussed in § 3 is obtained using the commercial software COMSOL Multiphysics^© (COMSOL AB, Sweden). The permeability values are calculated on the basis of the numerically calculated flow rate. Independence of the calculated permeability values from the sizes of the unstructured triangular mesh used in the simulation is tested for the largest reported $\unicode[STIX]{x1D716}$ ( $\unicode[STIX]{x1D716}=5$ ) and for $\unicode[STIX]{x1D6FC}=0.99$ ( $99\,\%$ channel blockage in transverse flow) up to four-digit accuracy. In transverse flow, mass conservation was ensured between the three cross-sections at $x=-\unicode[STIX]{x03C0},0,\unicode[STIX]{x03C0}$ up to at least four-digit accuracy, and the flow rate through the $x=0$ plane is used for permeability evaluation. Any infinite sums required by the theoretical model, for which a closed form is not reported in this article, are evaluated by numerical truncation to at least four-digit accuracy.

Figure 3. Variation of effective permeability with dimensionless characteristic pattern size ( $\unicode[STIX]{x1D716}$ ) in longitudinal (a,c,e) and transverse flow (b,d,f) for different values of $a$ and $H=\unicode[STIX]{x03C0}/2$ in Topography A. The solid black line, blue dashed line and black dotted line indicate the asymptotic predictions from finite channel height theory (3.8)/(3.18), exponentially accurate theory for the finite channel height effect (3.11), (3.21) and $O(1/H)$ theory (only the bracketed terms of (3.11), (3.21)). The red square symbols are data from fully resolved numerical simulations employing the finite-element method. The range of $\unicode[STIX]{x1D716}$ values shown in each figure corresponds to ${<}5\,\%$ numerical error by the finite height theory.

Figure 4. Variation of effective permeability with dimensionless characteristic pattern size ( $\unicode[STIX]{x1D716}$ ) in longitudinal (a) and transverse flow (b) for $a=\unicode[STIX]{x03C0}/2$ and $H=\unicode[STIX]{x03C0}/4$ in Topography A, as predicted by fully resolved simulations (symbols) and asymptotic theories. The colour/pattern of lines with respect to different asymptotic theories have the same meaning as in figure 3. The range of $\unicode[STIX]{x1D716}$ values shown in each panel corresponds to ${<}5\,\%$ numerical error by the finite height theory. The peaks of the pattern touch the planar top wall ( $\unicode[STIX]{x1D6FC}=1$ ) at the rightmost value of abscissa in (a).

Figure 5. Variation of effective permeability with dimensionless characteristic pattern size ( $\unicode[STIX]{x1D716}$ ) in longitudinal (a,c) and transverse flow (b,d) for different values of $H$ in Topography B, as predicted by fully resolved simulations (symbols) and asymptotic theories. The colour/pattern of lines with respect to different asymptotic theories have the same meaning as in figure 3. The range of $\unicode[STIX]{x1D716}$ values shown in each figure corresponds to ${<}5\,\%$ numerical error by the finite height theory, except for $H=\unicode[STIX]{x03C0}/2$ (a) in longitudinal flow, where the numerical error is ${<}1.67\,\%$ up to the rightmost value of the abscissa where the peaks of the pattern touch the planar top wall ( $\unicode[STIX]{x1D6FC}=1$ ).

In figures 3–5 the asymptotic predictions for permeability are compared against the corresponding numerical predictions for a range of $\unicode[STIX]{x1D716}$ values. The range of $\unicode[STIX]{x1D716}$ shown in the abscissae of this figures correspond to either ${<}5\,\%$ error committed relative to the numerical predictions by the arbitrary- $H$ theory, or the pattern peaks touching the opposite wall ( $\unicode[STIX]{x1D6FC}=1$ ), depending on which of these two limits is reached at a smaller $\unicode[STIX]{x1D716}$ . The latter situation occurs only for figures 4(a) and 5(a). Based on the above definition, the largest value of $\unicode[STIX]{x1D716}$ appearing in all other figures will be designated as $\unicode[STIX]{x1D716}_{max}^{5\,\%}$ .

In figures 3–5, whenever the arbitrary $H$ prediction (shown as solid black lines) is indistinguishable up to four-digit accuracy from the exponentially accurate large $H$ predictions, which have simpler analytical forms, only the latter (blue dashed curves) is shown. Also, shown as dotted curves, whenever numerically comparable predictions are obtained, are the approximations obtained from removing even the algebraically $H$ -decaying terms, i.e. using only the bracketed terms of (3.11) and (3.21) to estimate the permeability. A dotted curve thus signifies the best permeability prediction that can be obtained by using only the existing domain-perturbation-based theoretical models (Luchini et al. Reference Luchini, Manzo and Pozzi1991; Kamrin et al. Reference Kamrin, Bazant and Stone2010) for arbitrary periodic surface topographies, which are all limited to shear flows. It is clear that, in none of figures 3–5, can the shear-flow model be applied at levels of accuracy comparable to the large- $H$ and finite- $H$ models, although its performance is superior in transverse flow. This can be rationalized from the algebraic decay of errors in the expressions to the right of the second equal sign in (3.9b ) and (3.19a ). The shear-flow model is therefore excluded from subsequent discussions, although it is shown in the figures. Lubrication theory predictions of $K$ (not shown) for thin channels which equal the arithmetic and harmonic means over one period of the function of $(1+\unicode[STIX]{x1D6FC}g)^{3}$ in longitudinal and transverse directions, respectively, were found to be exceedingly inaccurate even for the smallest $H$ studied in figures 3–5, and are therefore kept out of consideration. The numerical errors (not shown) were monotonic with $\unicode[STIX]{x1D716}$ for all situations studied in figures 3–5.

Figures 3 and 4 pertain to wall Topography A and show the effect of parameter $a$ and channel size $H$ , respectively. Figure 5 shows the effect of channel size $H$ with wall Topography B. While for sufficiently small $\unicode[STIX]{x1D716}$ the asymptotic theories are expectedly accurate, qualitatively different patterns emerge in the behaviour of both the permeability and the error of the asymptotic prediction, depending on the functional form of the topography and the orientation of the topography with respect to the mean flow.

For longitudinal flow, $\unicode[STIX]{x1D716}_{max}^{5\,\%}$ increases with miniaturization (decreasing the dimensionless nominal channel size $H$ ) for both topographies. For transverse flow, the corresponding $\unicode[STIX]{x1D716}_{max}$ decreases with miniaturization for both the topographies. The latter behaviour can be expected since, in transverse flow, a decrease of $H$ implies the creation of more significant blockages in the flow passage, and as $\unicode[STIX]{x1D716}$ approaches $H$ ( $\unicode[STIX]{x1D6FC}\rightarrow 1$ ) the flow passage is completely blocked.

Figure 3 along with figure 2(a) suggest that more complex Topography A shapes (with a larger number of extrema) have a steeper drop in the numerically calculated permeability values with $\unicode[STIX]{x1D716}$ and also pose a stronger challenge to asymptotic theories in the sense that $\unicode[STIX]{x1D716}_{max}^{5\,\%}$ is smaller for $a=5\unicode[STIX]{x03C0}/2$ than $a=5\unicode[STIX]{x03C0}/4$ . Qualitatively, this finding may be attributed to the appearance of strong velocity gradients not amenable to asymptotic treatment. Strong gradients can be expected from both the larger variance ( $0.49$ versus $0.225$ for $a=5\unicode[STIX]{x03C0}/4$ ) as well as the more complex shape of the $a=5\unicode[STIX]{x03C0}/2$ topography (figure 7 c). The narrow wall-proximal gorges for $a=5\unicode[STIX]{x03C0}/2$ create large gradients in longitudinal flow; in transverse flow the tortuous path of near wall fluid particles is also a reason for the appearance of large gradients. The larger variance of the topography with $a=5\unicode[STIX]{x03C0}/2$ also signifies a larger distance between peaks and troughs and a larger ‘sidewall’ perimeter of liquid–solid contact per pitch. Despite its relatively simple shape, the topography with $a=\unicode[STIX]{x03C0}/2$ has nearly twice the variance ( $0.449$ ) of the topography with $a=5\unicode[STIX]{x03C0}/4$ ( $0.225$ ). The $\unicode[STIX]{x1D716}_{max}^{5\,\%}$ values for $a=\unicode[STIX]{x03C0}/2$ and $a=5\unicode[STIX]{x03C0}/4$ are broadly of comparable magnitude, regardless of the flow orientation. This finding can be interpreted in terms of an interplay between the comparable simplicity in shape of the $a=\unicode[STIX]{x03C0}/2$ and its larger variance. The former is expected to make the asymptotic approximations accurate, while the latter is expected to disrupt the accuracy.

To understand the role of the shape of the topography, two distinct forms of Topography A were chosen with the variance controlled strictly to its asymptotic value $1/2$ . The $a$ values of these topographies can be obtained by numerically finding the roots of (4.2c ) with the right-hand side set to $1/2$ . The two smallest roots are $a=\{a_{1},a_{2}\}$ with $a_{1}=5.68$ and $a_{2}=8.12$ . It was found that a stronger $\unicode[STIX]{x1D716}$ -dependence of the numerically calculated permeability as well as a larger degree of inaccuracy (as estimated by $\unicode[STIX]{x1D716}_{max}^{5\,\%}$ ) in asymptotic prediction occurred for the more complex topography ( $a_{2}=8.12$ ). Thus, in addition to the role of the root-mean-square (r.m.s.) height (or variance), the specific $a$ -dependent shape of ‘Topography A’ plays an important and independent role in determining the degree of (in)accuracy of the asymptotic predictions. The ‘complexity’ of the $a$ -dependent shape reflects in the number of zero crossings, depth, width and number of gorges and hills. For a more ‘complex’ shape obtained with a larger $a$ , the asymptotic theory has the prescribed numerical accuracy of $5\,\%$ up to a smaller $\unicode[STIX]{x1D716}$ value in comparison to that for a shape with smaller $a$ , even when both have the same $R_{q}$ roughness.

There are observable distinctions between longitudinal and transverse flows across all topographies. At $H$ is lowered progressively, the large- $H$ trend of permeability decreasing with pattern amplitude ( $\unicode[STIX]{x1D716}$ ) transitions into an increase with $\unicode[STIX]{x1D716}$ in longitudinal flow, whereas the trend of decrease of permeability with $\unicode[STIX]{x1D716}$ occurs at all $H$ in transverse flow. In general, the asymptotic approaches are more accurate for longitudinal flow than transverse flow. Further, the numerical accuracy of the predictions worsens with decreasing channel size in transverse flow, while the accuracy improves in longitudinal flow on reducing the channel size. In transverse flow, the asymptotic theory obviously encounters more difficulties as it has to cope with a stronger decrease of permeability with $\unicode[STIX]{x1D716}$ , as signified by the complete flow blockage at $\unicode[STIX]{x1D716}=H$ . If $H$ is small enough, contact with the top wall ( $\unicode[STIX]{x1D6FC}=1$ ) poses no significant challenge to numerical accuracy of the arbitrary- $H$ theory in longitudinal flow, e.g. the error is ${<}1.67\,\%$ for a triangular wave at $H=\unicode[STIX]{x03C0}/2$ (figure 5 a) and ${<}5\,\%$ for the phase-modulated cosine at $H=\unicode[STIX]{x03C0}/4$ (figure 4 a). This observation will be useful in studying an application of the theory to the shear stability of lubricant-infused textured surfaces later in the article.

Between the two topographies, comparison of height effects in figures 3 and 4 reveals that, in the smoother topography (Topography A), the onset of the large- $H$ behaviour occurs at a smaller height, whereas the cornered peaks of a triangular pattern (Topography B) exert their influence up to larger distances, requiring larger $H$ values for a reasonable accuracy from the large- $H$ theory in comparison to the smoother Topography A.

Figure 5 provides verification that the presence of corners in the triangular profile does not preclude the asymptotic theory from providing numerically accurate permeability predictions for a triangular profile for sufficiently small $\unicode[STIX]{x1D716}$ values. As per the steps of our theoretical model shown in § 3, correct effective permeability prediction necessarily requires correct effective slip prediction. The findings in figure 5 therefore contradict the directive of Kamrin et al. (Reference Kamrin, Bazant and Stone2010) on the avoidance of ‘surfaces with corners’ in using domain perturbation for prediction of effective slip, as discussed at the beginning of current section. This finding can be anticipated theoretically from the convergence of $S_{1}$ , as it follows that $S_{1}\propto L_{eff}$ , when $L_{eff}$ is calculated either by (5.1) of Kamrin et al. (Reference Kamrin, Bazant and Stone2010) or equation (3.25). The latter can be recalled to be obtained in the limit $H\rightarrow \infty$ of the current theoretical model.

Interestingly, the effective permeability with the triangular profile is accurately predicted, despite the fact that the series in certain intermediate results pertaining to the local flow field such as $W^{1}$ from (3.4a ) and $U^{1}$ from (3.14f ) (up to $e.s.t.(H)$ ) are actually divergent.

Another consequence of the smoothness of the test topographies is corroborated by the numerical comparisons. If the sums required for the expression for permeability are truncated to the same number of (sufficiently large) modes, the predictions for $C^{\infty }$ smooth Topography A was found to provide higher numerical accuracy than those for $C^{0}$ smooth Topography B, which can be anticipated from the faster convergence gained from the faster asymptotic decay of the Fourier coefficients (Canuto et al. Reference Canuto, Hussaini, Quarteroni and Zang1987) of Topography A.

4.3 Extension of the range of accuracy of the asymptotic approximations

For the usability of asymptotic approximations, ‘how large an $\unicode[STIX]{x1D716}$ ’ can be admitted by a nominally small- $\unicode[STIX]{x1D716}$ theory is an important question (Dyke Reference Dyke1984; Dewangan & Datta Reference Dewangan and Datta2019). Polynomial form approximations (even with higher orders) do not often provide numerically encouraging answers to this question, which can be attributed to convergence limitations imposed by the singularities in the complex $\unicode[STIX]{x1D716}$ -plane possessed by the functions these polynomials expand in Taylor series. Although obtaining an extended accuracy solution requires a knowledge of a sufficiently large number of terms of the polynomial which cannot be afforded here, we note that for any $a+b\unicode[STIX]{x1D716}^{2}+o(\unicode[STIX]{x1D716}^{2})$ -form prediction in this article, an alternative form avoiding polynomials and attempting to anticipate singularities can be heuristically chosen as the asymptotically equivalent $(1,2)$ Padè (rational) approximant form $a^{2}/(a-b\unicode[STIX]{x1D716}^{2})+o(\unicode[STIX]{x1D716}^{2})$ . In the typical situation, except for small $H$ , where the permeability decreases with $\unicode[STIX]{x1D716}$ , there is also a large- $\unicode[STIX]{x1D716}$ tendency in the numerically evaluated scaled permeability value to saturate, which is broadly consistent with the infinite groove depth effective slip predictions from the literature obtained by the Wiener–Hopf method (Hocking Reference Hocking1976; Luchini et al. Reference Luchini, Manzo and Pozzi1991). This behaviour can be anticipated to be captured better by a rational approximant due to the presence of a denominator. Thus, the rational approximations accessible to the current theoretical model were subject to numerical experimentation.

Figure 6. Demonstration of range extension through use of the ( $1,2$ ) Padè approximant (dashed lines) and polynomial form (solid lines) of the asymptotic predictions for triangular grooves with $H=2\unicode[STIX]{x03C0}$ . The symbols are from fully resolved simulations. The range of $\unicode[STIX]{x1D716}$ values shown corresponds to ${<}5\,\%$ numerical error by the Padè ( $1,2$ ) approximant. The double arrow indicates the location where $5\,\%$ error is incurred by the polynomial form approximation.

Figure 7. (a) Shows a schematic of a pitch-averaged gas/lubricant cushion model of effective slip for hydrodynamic characteristics of LIS/SHS. (b) Is an adapted reproduction of figures 1(a) and 2(c) of Wexler et al. (Reference Wexler, Jacobi and Stone2015) showing their flow cell (above) and the rectangular groove containing the lubricant (below). (c) Is the cross-section of a conceived complex-shaped lubricant groove under experimental conditions otherwise identical to Wexler et al. (Reference Wexler, Jacobi and Stone2015). In (a), the stripes are transverse to the period-averaged velocity $\langle U\rangle$ . Panel (b) is reproduced from Wexler et al. (Reference Wexler, Jacobi and Stone2015) with permission from the American Physical Society.

Figure 6 for triangular grooves with $H=2\unicode[STIX]{x03C0}$ in transverse flow over Topography B is an example of extending the $\unicode[STIX]{x1D716}$ -range of validity using rational approximations, where approximately $50\,\%$ lower permeability values (from ${\sim}0.8$ to ${\sim}0.4$ ) or permeability at nearly three times larger $\unicode[STIX]{x1D716}$ (up to $\unicode[STIX]{x1D716}\simeq 5$ ) can be predicted with $5\,\%$ accuracy using the Padè approximant corresponding to the two-term polynomial approximation given by (3.18). The polynomial approximation produces unphysical negative results at $\unicode[STIX]{x1D716}\simeq 5$ where the Padè approximant reaches five per cent error. Qualitatively similar improvements were offered by the Padè approximant for both the topographies, whenever permeability decreased with $\unicode[STIX]{x1D716}$ in either longitudinal or transverse flow. For permeability increasing with $\unicode[STIX]{x1D716}$ where there is no saturation effect at large $\unicode[STIX]{x1D716}$ , polynomial (here parabolic) approximation was expectedly (although marginally) superior until $\unicode[STIX]{x1D6FC}=1$ is reached.

4.4 Application to slippery lubricant impregnated surfaces

Biologically inspired patterned surfaces infused with a lubricant (LIS) have gained attention for a diversity of applications for their much-desirable liquid repellency, self-cleaning, self-healing, anti-icing, dropwise-condensation-promoting properties (Wong et al. Reference Wong, Kang, Tang, Smythe, Hatton, Grinthal and Aizenberg2011), in addition to a possible capability to reduce drag (Solomon et al. Reference Solomon, Chen, Rapoport, Helal, McKinley, Chiang and Varanasi2018). Two distinct applications of the developed theory to LIS will be discussed. Capacity to represent the role of a complex texture analytically is a connecting feature of the applications. For simplified modelling of these multi-phase systems, a simplification used here will involve giving prescribed shapes to fluid–fluid interfaces, which helps us maintain analytical tractability. In principle, more-resolved multi-phase model(s) of flows over LIS may, however, be constructed numerically or in the long-wave limit (Qi & Ng Reference Qi and Ng2015; Sun & Ng Reference Sun and Ng2017).

Roughness effects on the lubricant cushion model. Consider a submerged complex topography underneath the interface between a lubricant (with viscosity $\unicode[STIX]{x1D707}_{2}$ ) and the working fluid (with viscosity $\unicode[STIX]{x1D707}_{1}$ ) as shown in figure 7(a). Here, a procedure to incorporate the effects of the topography into the pitch-averaged velocity field of the working fluid will be shown, based on the gas/lubricant cushion model (Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015; Dubov et al. Reference Dubov, Nizkaya, Asmolov and Vinogradova2018). In principle, the interface may touch the peaks of the topography in figure 7(a) without invalidating the subsequent description. The macroscale (period-averaged) construction for the gas/lubricant cushion model (Vinogradova & Belyaev Reference Vinogradova and Belyaev2011; Nizkaya et al. Reference Nizkaya, Asmolov, Zhou, Schmid and Vinogradova2015; Dubov et al. Reference Dubov, Nizkaya, Asmolov and Vinogradova2018) for the effective slip length $B_{eff}$ of the LIS referred to the mean line of the topography is shown in figure 7(a), assuming a case favourable for drag reduction ( $\unicode[STIX]{x1D707}_{1}/\unicode[STIX]{x1D707}_{2}>1$ ). In figure 7(a), stripes are transverse to the period-averaged velocity $\langle U\rangle$ although in principle the tensorial slip length (Bazant & Vinogradova Reference Bazant and Vinogradova2008) formalism can be used to generalize this description. Note that $B_{eff}$ must be distinguished clearly from $L_{eff}$ . The latter gives the location of the effective no-slip plane for the submerged topography with respect to the mean line. The magnitude of the negative-signed $L_{eff}$ is shown in figure 7(a). As per the construction of the ‘gas/lubricant cushion model’ shown in figure 7(a), the continuity of stress and the fact that the effective no-slip plane is shifted upward by a distance $L_{eff}$ due to the complex topography allows us the following adaptation of the traditional ‘gas cushion model’ to calculate the scaled effective slip length $B_{eff}$ :

Equation (3.25) of the current work can provide the $L_{eff}$ required in (4.5), thus allowing the effect of topographies encapsulated by the lubricant to be evaluated under a gas-cushion model-based effective slip formalism. It can be noted, however, approaches to improving the local resolution of the traditional gas cushion model employed here has been proposed, which require specification of a distribution of the slip length over the gas–liquid interface (Dubov et al. Reference Dubov, Nizkaya, Asmolov and Vinogradova2018).

In figure 7(a), a provision is maintained for a film of liquid overlying the tips of the topography, which has been found experimentally to be desirable for drag reduction (Solomon et al. Reference Solomon, Chen, Rapoport, Helal, McKinley, Chiang and Varanasi2018), but not necessarily for shear stability of the lubricating fluid (Wexler et al. Reference Wexler, Jacobi and Stone2015).

Shear stability of lubricant entrapped by complex groove shapes. Stability under shear is a perceived weakness of lubricant-infused patterned surfaces that needs to be addressed, if they are to be used in dynamic fluid environments, e.g. for antibacterial/anti-fouling protection in the human circulatory system or on marine surfaces. The next application is inspired by the study of shear stability of lubricant fluid in patterned surfaces with infused lubricants conducted by Wexler et al. (Reference Wexler, Jacobi and Stone2015). In this study, for the experimental configuration of longitudinal flow (‘worst-case scenario’ for shear stability, to quote the authors) shown in figure 7(b), it is demonstrated that the lubricant (in the regime $\unicode[STIX]{x1D707}_{1}/\unicode[STIX]{x1D707}_{2}\ll 1$ ) combats the drainage effects of shear $\unicode[STIX]{x1D70F}_{xy}$ from the overlying fluid through a pressure-driven back flow arising from the Laplace pressure at the interface with surface tension $\unicode[STIX]{x1D6FE}$ as a result of which an equilibrium length $L_{\infty }$ of the lubricant is always adherent. According to Wexler et al. (Reference Wexler, Jacobi and Stone2015), the larger the value of $L_{\infty }$ the more shear stable is the surface. Equation (2) of Wexler et al. (Reference Wexler, Jacobi and Stone2015) provides an equation for $L_{\infty }$ applicable to rectangular grooves. If the derivation of equation (2) as discussed in section 2 of the supplementary material for Wexler et al. (Reference Wexler, Jacobi and Stone2015) is retraced, an essential component of their model involves evaluating the permeability for the fully developed combined shear and back pressure driven flow in a channel of rectangular cross-section. From § 3.4 of the current study, the permeability of the component longitudinal shear-driven ( $s$ ) and free-surface pressure-driven ( $pzs$ ) flows can be evaluated for more general shapes, using the finite- $H$ generalization of (3.24b ) and (3.24c ) which, as discussed earlier, involves replacing the sum $S_{1}$ appearing in this equation by $\sum _{n=1}^{\infty }n\,\text{coth}(n)g_{n}^{2}$ .

Given the generality of certain results (specifically the finite- $H$ versions of (3.24b ) and (3.24c ) from § 3.4) of this study on longitudinal shear-driven ( $s$ ) and free-surface pressure-driven ( $pzs$ ) flows, in this section we outline the procedure to generalize their result to groove shapes that are more complex than rectangular.

A plausible situation where more complex topographical patterns given by a function $g(x)$ are used to contain the lubricant can be envisaged as shown on figure 7(c), where the lubricant film thickness measured from the mean line is given by $H$ . The physical requirement of pinning the interface to the tips of the topography (encapsulation being considered undesirable for shear stability) and the geometrical requirement of the side ‘walls’ in addition to the bottom ‘wall’ of a complete ‘groove’ being formed by specifying the single function $g(x)$ together with the need for $2\unicode[STIX]{x03C0}$ -periodic extension enforce the following restrictions on the groove shape and size: $\unicode[STIX]{x1D6FC}=1$ (or $\unicode[STIX]{x1D716}=H$ ) and $g(\pm \unicode[STIX]{x03C0})=\text{max}(g(x))=1$ . It is assumed here that $g(X)$ is so scaled that $\unicode[STIX]{x1D6FC}=1$ corresponds to tips touching the interface. These restrictions are also indicated in figure 7(c).

For such a groove, otherwise retracing the derivation of equation (2) of Wexler et al. (Reference Wexler, Jacobi and Stone2015) leads to the following proposed generalization with the new definitions written in the notation of the current work:

Recollecting Wexler et al. (Reference Wexler, Jacobi and Stone2015), the constant of proportionality is $A=(h\unicode[STIX]{x1D6FE})/(\unicode[STIX]{x1D70F}_{yx}r_{min})$ , where $r_{min}$ is the minimum transverse radius of curvature on the upstream edge of the lubricant layer (shown for rectangular grooves in figure 7 b). If the quantity $K_{pzs\Vert }/K_{s\Vert }$ is evaluated using (3.24b ) and (3.24c ), it may be inferred that a complex groove topography with a large $R_{q}$ measure (leading to larger $S_{1}$ ) for the ‘roughness’ appears to be desirable for increasing $L_{\infty }$ , which in turn offers improved shear stability.

It should be noted, however, that numerical comparisons in § 4 have indicated that the small $-\unicode[STIX]{x1D716}$ theory underlying the permeability predictions is numerically accurate even at $\unicode[STIX]{x1D6FC}=1$ for sufficiently shallow (small- $H$ ) grooves. For example, for $H=\unicode[STIX]{x03C0}/2$ , the triangular grooves in figure 5(a), the accuracy at $\unicode[STIX]{x1D6FC}=1$ is $1.67\,\%$ . Incidentally, the value of $H$ for the rectangular-wave topography fabricated by Wexler et al. (Reference Wexler, Jacobi and Stone2015), which is shown in panel (d) of figure 1 of their study, is approximately $2.8$ using the data provided in that work. It is worthwhile, however, to recall that the domain perturbation approach, despite its applicability to complex groove shapes similar to that shown in figure 7(c), does not specialize meaningfully to the rectangular topography, because of its jump discontinuity (Kamrin et al. Reference Kamrin, Bazant and Stone2010).

Figure 8. Trapezoidal and scalloped topographies. The function $g(x)$ is shown by blue lines, which overlie the grey-shaded solid zone.

4.5 Other surface topographies and experimental comparison

In this section, we study certain distinctive topographies arising in applications and perform a comparison with experimental data from the literature. The closed-form results presented here are obtained using known results on the infinite series pertinent to the special functions appearing below (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

Topographies presenting a cusp have attracted the attention of several researchers (Richardson Reference Richardson1973; Bechert & Bartenwerfer Reference Bechert and Bartenwerfer1989), partly because of their importance in the design of turbulent-drag-reduction riblets. Geometrically, and in terms of local flow resolution, cusps are interesting in comparison to corners (such as in Topography B) as the left- and right-side derivatives are not finite. The scalloped pattern obtained by joining semicircles shown in figure 8(b) is in fact a common motif in engineering and biology (Imani & Elbestawi Reference Imani and Elbestawi2001; McCarthy, Ring & Rana Reference McCarthy, Ring and Rana2010). The current spectral approach is very convenient to access the oft-used semicircle-based scalloped pattern, as shown through the evaluation of the parameters for closed-form large- $H$ permeability prediction in the second row of table 1. Unlike the current study, the cusped patterns obtained through conformal-mapping approaches in the literature (Richardson Reference Richardson1973; Bechert & Bartenwerfer Reference Bechert and Bartenwerfer1989) involve approximations to the semi-circle, necessitated by the specific mapping functions.

Converged predictions are provided by the current theory even for cusped profiles (but not for profiles with jumps). The results of table 1 concerning the trapezoidal and scalloped topographies shown in figure 8 are in closed form, except the numerically summed $S_{1}$ for the scalloped profile. The asymptotic validity of the permeability predictions from table 1 were also numerically tested against finite-element predictions, although not shown here for brevity.

The first row of table 1 presents parameters for trapezoidal grooves parameterized by the distance $a$ (figure 8 a). Lecoq et al. (Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) have performed experiments on the effect of trapezoidal wall corrugations on the drag on an approaching sphere, while Bechert & Bartenwerfer (Reference Bechert and Bartenwerfer1989) have studied the protrusion length of trapezoidal grooves in shear flow through conformal mapping. Trapezoidal grooves often arise as inevitable consequences of fabrication and machining processes (Rangsten et al. Reference Rangsten, Hedlund, Katardjiev and Bäcklund1998; Lecoq et al. Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004), even when square ( $a=\unicode[STIX]{x03C0}/2$ ) or triangular grooves ( $a=0$ ) are targeted. For the domain perturbation approach to give bounded permeability predictions, the value of $a$ must be restricted in the trapezoidal surface profile to $0\leqslant a<\unicode[STIX]{x03C0}/2$ , again due to its non-resolution of jump discontinuities (Kamrin et al. Reference Kamrin, Bazant and Stone2010).

We have evaluated the sum $S_{1}$ required for the trapezoidal grooves shown in figure 8(a) analytically, including an asymptotic form avoiding special functions. The sum $S_{1}$ is given by

In the above, $\text{Li}$ is the polylogarithm function. The second equality obtained with Padè approximants, although nominally valid for small  $a$ , gives ${<}1\,\%$ error for $0\leqslant a<1.355$ , i.e. over $86\,\%$ of the domain of definition $0\leqslant a<\unicode[STIX]{x03C0}/2$ .

Lecoq et al. (Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) have performed experiments inquiring about the location of the effective no-slip plane when a sphere approaches a precisely fabricated corrugated surface. More recent experiments with a similar configuration are available (Mongruel et al. Reference Mongruel, Chastel, Asmolov and Vinogradova2013), but for rectangular grooves, the domain perturbation approach of the current model fails to give a bounded prediction, as discussed elsewhere in the article and the literature (Kamrin et al. Reference Kamrin, Bazant and Stone2010; Asmolov et al. Reference Asmolov, Zhou, Schmid and Vinogradova2013b ). Although the current study mainly concerns confined flows, the location of the effective no-slip plane is a by-product of the current study, as per (3.25). The experimental amplitude to pitch ratios of the trapezoidal corrugations used in Lecoq et al. (Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) nominally appear too large for asymptotic treatment with the shear-flow limit of the current domain-perturbation theory. However, as an heuristic approach, if rational approximations are used to the two-term protrusion length polynomials and the smallest amplitude ( $\unicode[STIX]{x1D716}=1.14$ ) ‘Topography ${\mathcal{B}}$ ’ discussed in tables 1 and 3 of that work is studied, a prediction $L_{eff}=0.51$ is obtained following the procedure for interpreting their squeezed-film rather than shear-flow experimental data discussed in this study, whereas $L_{eff}=0.47\pm 0.02$ is reported experimentally. Thus, considering the error bounds, an error magnitude of $4.3\,\%{-}12.8\,\%$ between the rational approximation variant of the current theory and the experiment of Lecoq et al. (Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) is obtained. The semi-analytical model used by (Lecoq et al. Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) to model their experimental data lacks a closed form, unlike the current model, but because of the inclusion of $\unicode[STIX]{x1D716}$ orders higher than two and the subsequent range extension through Euler transformation (Dyke Reference Dyke1984), it may be better suited for the specific large pattern size to pitch ratio used in their experiments, which might explain its better accuracy ( $\pm 4.3\,\%$ ) than the current model.

Note that ‘Topography ${\mathcal{B}}$ ’ of Lecoq et al. (Reference Lecoq, Anthore, Cichocki, Szymczak and Feuillebois2004) is not related to the Topography B of the current work. Note also, excepting the scalloped topography, that all topographies studied as special cases in the current study are normalized for convenience to have $\text{max}(g(x))=1$ .

It can be pointed out in the current context that Bechert & Bartenwerfer (Reference Bechert and Bartenwerfer1989) use a slight generalization of the trapezoidal topography used here which has unequal flat zone length on its peak and valley and evaluate the protrusion height using Schwartz–Christoffel transformations. Closed-form results for $S_{1}$ and $S_{2}$ (not shown here for brevity) and therefore the protrusion height and large- $H$ permeability can also be obtained for the same in a straightforward manner from the procedure used to calculate the trapezoidal topography entries in table 1.