3.1. The velocity and viscosity distributions

For long and narrow straight channels in which $h\ll w\ll \ell _{ch}$, we can assume a rectilinear flow, $\boldsymbol {u}=u_{x}(y,z)\boldsymbol {e}_{x}$. Furthermore, the assumption $h\ll w$ implies that $\partial /\partial y\ll \partial /\partial z$, so that the momentum equations (2.1b) together with (2.2) simplify to

(3.1a–d)\begin{equation} \frac{\partial p}{\partial x}=\frac{\partial}{\partial z}\left(\eta(\dot{\gamma})\frac{\partial u_{x}}{\partial z}\right),\quad\frac{\partial p}{\partial y}=0,\quad\frac{\partial p}{\partial z}=0,\quad\mathrm{with\ }\dot{\gamma}=\left|\frac{\partial u_{x}}{\partial z}\right|.\end{equation}

We note that the governing equations (3.1) represent the lubrication equations and can be derived alternatively by introducing two small parameters $\epsilon =h/\ell _{ch}\ll 1$ and $\delta =h/w\ll$1, with the ordering $0<\epsilon \ll \delta \ll 1$, and then formally expanding the velocity and pressure fields in powers of $\epsilon$ and $\delta$, keeping finally the leading-order terms (see e.g. Christov et al. Reference Christov, Cognet, Shidhore and Stone2018). From (3.1b,c), it follows that $p=p(x)$, i.e. the pressure is independent of $y$ and $z$, consistent with the classical lubrication approximation.

Integrating (3.1a) with respect to $z$ and applying the symmetry condition at the midplane yields

(3.2)\begin{equation} \frac{\mathrm{d}p}{\mathrm{d} x}z=\eta(\dot{\gamma})\frac{\partial u_{x}}{\partial z}.\end{equation}

For a straight channel, the pressure gradient is constant and can be expressed as $\mathrm {d}p/\mathrm {d} x=-{\rm \Delta} p/\ell$, leading to

(3.3)\begin{equation} \frac{\partial u_{x}}{\partial z}={-}\frac{{\rm \Delta} p}{\ell}\frac{z}{\eta(\dot{\gamma})},\end{equation}

so that the shear rate is $\dot {\gamma }=z {\rm \Delta} p/ (\eta (\dot {\gamma })\ell ).$ Substituting the latter result into the constitutive equation for $\eta$ results in an implicit nonlinear algebraic equation for the viscosity distribution $\eta$. The Carreau model, (2.3), takes the dimensionless form

(3.4)\begin{equation} \frac{\eta}{\eta_{0}}=\beta+(1-\beta)\left(1+ \left(\frac{CuZ}{\eta/\eta_{0}}\right)^{2}\right)^{(n-1)/2},\end{equation}

where $\beta =\eta _{\infty }/\eta _{0}$, $Z=z/h$ and $Cu$ is the Carreau number (see e.g. Datt et al. Reference Datt, Zhu, Elfring and Pak2015; Shahsavari & McKinley Reference Shahsavari and McKinley2015), defined for convenience based on the pressure drop ${\rm \Delta} p$ rather than on the flow rate $q$ as

(3.5)\begin{equation} Cu=\frac{\lambda{\rm \Delta} p}{\eta_{0}}\frac{h}{\ell}.\end{equation}

Integrating (3.3) again with respect to $z$ and applying the no-slip boundary conditions at the channel walls, we obtain the axial velocity $u_{x}=-({\rm \Delta} p/\ell )\int _{h/2}^{z}z'\eta (z')^{-1}\,\mathrm {d}z'$. Substituting this result into the definition of the volumetric flow rate $q$ through the microchannel, $q=2\int _{-w/2}^{w/2}\int _{0}^{h/2}u_{x}(z)\,\mathrm {d}z\,\mathrm {d} y$, gives the flow rate–pressure drop relation

(3.6)\begin{equation} q={-}\frac{2({\rm \Delta} p ) w}{\ell}\int_{0}^{h/2}\int_{h/2}^{z}\frac{z'}{\eta(z')}\,\mathrm{d}z'\,\mathrm{d}z,\end{equation}

which requires first solving for the viscosity distribution $\eta$ using (3.4). Defining $Q=q/(wh^{2}/\lambda )$, the flow rate–pressure drop relation (3.6) can be expressed in a non-dimensional form in terms of the Carreau number,

(3.7)\begin{equation} Q={-}2Cu\int_{0}^{1/2}\int_{1/2}^{Z}Z'(\eta(Z')/\eta_{0})^{{-}1}\, \mathrm{d}Z'\,\mathrm{d}Z.\end{equation}

While no general closed-form analytical solution of (3.4) and (3.7) is available, we next provide three asymptotic solutions in the limit of small, intermediate and large values of $Cu$, which allow us to describe analytically almost the entire range of values of the Carreau number.

We note that our approach to calculating the $q{-}{\rm \Delta} p$ relation (3.7) differs from that of Sochi (Reference Sochi2015), which relies on the Weissenberg–Rabinowitsch–Mooney methodology and eliminates the need to find the velocity profile, relating $q$ and ${\rm \Delta} p$ to the a priori unknown wall shear rate $\dot {\gamma }_{w}$ via hypergeometric functions.

3.2. Asymptotic results for small, intermediate and large values of $Cu$

For weak actuation, corresponding to the limit $Cu\ll 1$, the viscosity is $\eta /\eta _{0}=1$ at the leading order, and $Q$ scales linearly with the Carreau number, $Q=Cu/12$. To determine the $q{-}{\rm \Delta} p$ relation at higher orders, we seek the viscosity function in the form $\eta /\eta _{0}=1+Cu^{2}\mathcal {H}_{1}+Cu^{4}\mathcal {H}_{2}+O(Cu^{6})$. Substituting this expansion into (3.4) and solving order by order, we obtain

(3.8a,b)\begin{align} \mathcal{H}_{1}={-}\tfrac{1}{2}(1-n)(1-\beta)Z^{2},\quad \mathcal{H}_{2}={-}\tfrac{1}{8}(1-n)(1-\beta)(1-3n+4(n-1)\beta)Z^{4}. \end{align}

Using (3.8) and approximating $(\eta /\eta _{0})^{-1}$ as $(\eta /\eta _{0})^{-1}=1-Cu^{2}\mathcal {H}_{1}+Cu^{4}(\mathcal {H}_{1}^{2}-\mathcal {H}_{2})$, the flow rate–pressure drop relation (3.7) takes the form

(3.9)\begin{align} Q&=\frac{Cu}{12}+\frac{(1-n)(1-\beta)Cu^{3}}{160} \nonumber\\ & \quad + \frac{(1-n)(1-\beta)(3-5n+6(n-1)\beta)Cu^{5}}{3584}+O(Cu^{7}) \quad\mathrm{for\ } Cu\ll 1. \end{align}

For intermediate values of $Cu$, under the assumption of $\beta =\eta _{\infty }/\eta _{0}\ll 1$, which holds for most shear-thinning fluids, the Carreau model (2.3) reduces to a power-law model with $\eta /\eta _{0}=(CuZ)^{(n-1)/n}$, which has a well-known singularity at $Z=0$ for $0< n<1$. Nevertheless, since the integrand $Z(\eta (Z)/\eta _{0})^{-1}\propto Z^{1/n}$ of (3.7) is a regular function for $Z\in [0,1/2]$, the integration in (3.7) can be calculated, leading to the well-known $q{-}{\rm \Delta} p$ relation (see e.g. Bird et al. (Reference Bird, Armstrong and Hassager1987), p. 176)

(3.10)\begin{equation} Q=\frac{n}{2n+1}\frac{Cu^{1/n}}{2^{(n+1)/n}}\quad\mathrm{for}\ \mathrm{intermediate}\ Cu .\end{equation}

For strong actuation, corresponding to the limit $Cu\gg 1$, we approximate the viscosity (3.4) as $\eta /\eta _{0}=\beta +(1-\beta )(Cu(\eta /\eta _{0})^{-1}Z)^{n-1}$, so that $\eta /\eta _{0}=\beta$ at the leading order, and $Q$ again scales linearly with the Carreau number, $Q=Cu/ (12\beta )$. To determine high-order solutions, we seek the viscosity function in the form $\eta /\eta _{0}=\beta +Cu^{n-1}\mathcal {H}_{1}+Cu^{2(n-1)}\mathcal {H}_{2}+O(Cu^{3(n-1)}$), satisfying $Cu^{2(n-1)}\ll Cu^{n-1}\ll 1$, which is strictly valid for shear-thinning fluids with $0< n<1$. Substituting this expansion into the approximate expression for viscosity, we obtain

(3.11a,b)\begin{equation} \mathcal{H}_{1}=(1-\beta)\left(\frac{Z}{\beta}\right)^{n-1}, \quad\mathcal{H}_{2}=\frac{(1-n)(1-\beta)^{2}}{\beta} \left(\frac{Z}{\beta}\right)^{2n-2}.\end{equation}

Using (3.11) and approximating $(\eta /\eta _{0})^{-1}$ as $(\eta /\eta _{0})^{-1}=\beta ^{-1}-\beta ^{-2}Cu^{n-1}\mathcal {H}_{1} +\beta ^{-3}Cu^{2(n-1)}(\mathcal {H}_{1}^{2}-\beta \mathcal {H}_{2})$, the flow rate–pressure drop relation (3.7) in the limit $Cu\gg 1$ is

(3.12)\begin{equation} Q=\frac{Cu}{12\beta}-\frac{(1-\beta)Cu^{n}}{2^{n+1}\beta^{n+1}(n+2)} +\frac{n(1-\beta)^{2}Cu^{2n-1}}{2^{2n}\beta^{2n+1}(2n+1)}+O(Cu^{3n-2}) \quad\mathrm{for}\ Cu\gg 1.\end{equation}

We note that for $n=1$, we have $\mathcal {H}_{1}=1-\beta$ and $\mathcal {H}_{2}=0$, and $\eta /\eta _{0}=\beta +Cu^{n-1}\mathcal {H}_{1}+Cu^{2(n-1)}\mathcal {H}_{2}+O(Cu^{3(n-1)}$) reduces to $\eta /\eta _{0}=1$, consistent with the Newtonian limit. In this case, however, the Taylor expansion $(\eta /\eta _{0})^{-1}=\beta ^{-1}-\beta ^{-2}Cu^{n-1}\mathcal {H}_{1} +\beta ^{-3}Cu^{2(n-1)}(\mathcal {H}_{1}^{2}-\beta \mathcal {H}_{2})$, used to calculate (3.7), does not hold other than when $\beta =1$, since $Cu^{n-1}=1$ is no longer a small parameter. Therefore, for $n=1$ (3.12) does not reduce to the Newtonian limit, $Q=Cu/12$, except in the case of $\beta =1$. We also note that $\mathcal {H}_{1}\propto Z^{n-1}$ and $\mathcal {H}_{2}\propto Z^{2(n-1)}$ in (3.11a,b) have a singularity at $Z=0$ for $0< n<1$. Nevertheless, since the first and second terms in the integrand of (3.7) are regular functions for $Z>0$, and the third term $(\mathcal {H}_{1}^{2}-\beta \mathcal {H}_{2})\propto Z^{2n-1}$ has an integrable singularity, (3.7) can be calculated, resulting in (3.12).

We present the non-dimensional pressure drop $Cu={\rm \Delta} p\lambda h/(\eta _{0}\ell)$ vs the flow rate $Q=\lambda q/(wh^{2})$ in figure 2($a,b$) for the Carreau model in a straight channel for different values of $n$ and $\beta$. Dots, crosses and circles represent the semi-analytical results, whereas purple solid, red (- -) and cyan dotted lines represent the asymptotic solutions (3.9), (3.10) and (3.12) for small, intermediate and large values of $Cu$, respectively. To obtain the semi-analytical solution for given values of $Cu, n\ \text{and}\ \beta$, we first solved numerically (3.4) using MATLAB's routine fzero for $\eta /\eta _{0}$ as a function of $Z\in [0,1/2]$, where $Z$ was discretized with 201 points. Using the values for $\eta /\eta _{0}$, we integrated twice (3.7), first using MATLAB's routine cumtrapz to find the velocity profile and then using MATLAB's routine trapz to find the flow rate. We performed grid sensitivity tests by increasing the number of grid points to 301 and 401, resulting in a relative error below $10^{-5}$, thus establishing grid independence. We note that alternative numerical approaches to calculating the $q{-}{\rm \Delta} p$ relation of Carreau fluids were presented recently by Sochi (Reference Sochi2015) and Wrobel (Reference Wrobel2020).

Figure 2. Theoretically predicted flow rate–pressure drop relation for the Carreau constitutive model in a straight channel. ($a$) Dimensionless pressure drop $Cu={\rm \Delta} p\lambda h/(\eta _{0}\ell)$ vs flow rate $Q=\lambda q/(wh^{2})$ for $n=0.25,0.5,0.75$ and $\beta =10^{-4}$. ($b$) Dimensionless pressure drop $Cu={\rm \Delta} p\lambda h/(\eta _{0}\ell)$ vs flow rate ${Q=\lambda q/(wh^{2})}$ for $\beta =10^{-4},10^{-3},10^{-2}$ and $n=0.25$. Dots, crosses and circles represent the semi-analytical results obtained by solving numerically (3.4) and (3.7). Purple solid and cyan dotted lines represent the asymptotic solutions (3.9) and (3.12) for $Cu\ll 1$ and $Cu\gg 1$, respectively. Red dashed lines represent the power-law asymptotic solution (3.10) for intermediate values of $Cu$.

As expected, the results in figure 2($a$) show that as the power-law index $n$ characterizing the degree of shear thinning decreases, the $q{-}{\rm \Delta} p$ relation transitions from the power-law to the large-$Cu$ behaviour at smaller values of $Cu$ for fixed values of $\beta$. Similarly, the results in figure 2($b$) indicate that as the viscosity ratio $\beta =\eta _{\infty }/\eta _{0}$ decreases, the $q{-}{\rm \Delta} p$ relation settles on the large-$Cu$ linear behaviour at larger values of $Cu$ for fixed values of $n$.

From figure 2($a,b$), we see that there is excellent agreement between the semi-analytical results and asymptotic solutions over the entire range of values of the Carreau number. For $Cu\ll 1$, the asymptotic solution (3.9) accurately describes the behaviour of the $q{-}{\rm \Delta} p$ relation up to $Cu$ of $O(1)$ and captures well the transition to the power-law dependence ${Q\approx Cu^{1/n}}$, given in (3.10), for larger values of $Cu$. Furthermore, it is evident that the asymptotic solution (3.12) also accurately captures the transition from the power-law dependence (3.10) to the linear scaling $Q\sim Cu/(12\beta)$ for $Cu\gg 1$. We, therefore, conclude that the asymptotic solutions (3.9), (3.10) and (3.12) for small, intermediate and large values of $Cu$, which previously had been given for individual limits, accurately describe almost the entire range of values of the Carreau number. These approximate solutions can be directly compared with the $q{-}{\rm \Delta} p$ experimental data without requiring any additional computations, provided that the rheological parameters are known.

4.1. Fit of viscosity data and rheological parameters of the Carreau model

In this section, we compare the predictions of our theoretical analysis with the experimental results of Pipe et al. (Reference Pipe, Majmudar and McKinley2008), who measured the pressure drop of a shear-thinning xanthan gum solution in a slit microchannel for a wide range of flow rates (see their figure 14; unfortunately, there are misprints in the units of pressure in their figure 14, which should be in kPa and not Pa, as appears in table 2 of Pipe et al. (Reference Pipe, Majmudar and McKinley2008)). Using the Weissenberg–Rabinowitsch–Mooney equation (see e.g. Macosko (Reference Macosko1994), p. 240), $\dot {\gamma }_{w}=(\dot {\gamma }_{a}/3)[2+\mathrm {d}\ln \dot {\gamma }_{a}/\mathrm {d}\ln \tau _{w}]$, where $\dot {\gamma }_{a}=6q/(wh^{2})$, $\tau _{w}={\rm \Delta} ph/(2\ell)$ and $\dot {\gamma }_{w}$ is the a priori unknown shear rate. Pipe et al. (Reference Pipe, Majmudar and McKinley2008) converted the $q{-}{\rm \Delta} p$ measurements into viscosity data through $\eta =\tau _{w}/\dot {\gamma }_{w}$ and by fitting the variation of $\ln \dot {\gamma }_{a}$ vs $\ln \tau _{w}$ with a second-order polynomial. In addition, Pipe et al. (Reference Pipe, Majmudar and McKinley2008) provided shear-rate-dependent viscosity data measured in cone-and-plate and parallel-plate rheometers.

Even though Pipe et al. (Reference Pipe, Majmudar and McKinley2008) presented the fit of the Carreau–Yasuda equation to the three viscosity data sets in their figure 17, no resulting rheological parameters were provided in their paper. However, such rheological parameters are essential for quantitative comparison between theory and experiments. Therefore, we reproduce the viscosity data of Pipe et al. (Reference Pipe, Majmudar and McKinley2008) in figure 3 and fit these to the Carreau equation to obtain these parameters; see table 1.

Figure 3. Experimental data from Pipe et al. (Reference Pipe, Majmudar and McKinley2008) and the fitting curves for viscosity as a function of shear rate for aqueous xanthan gum solution. ($a$) Fit of Carreau equation (—) to the complete set of parallel-plate ($\boldsymbol {\circ }$, red), cone-and-plate ($\boldsymbol {\times }$, grey) and microchannel ($\blacktriangle$) data. ($b$) Fit of Carreau equation only to microchannel data with the constrained value of $\eta _{0}$ that matches the rheometer data at low shear rates. For clarity, the greyed-out regions in ($a$,$b$) indicate the range of shear rates corresponding to the microchannel data. The rheological parameters obtained from the fitting are summarized in table 1.

Table 1. Rheological parameters obtained from fitting the viscosity dependence on the shear rate to the Carreau equation based on the complete data set (upper row) and based solely on microchannel data with the constrained value of $\eta _{0}$ (lower row).

The viscosity data from Pipe et al. (Reference Pipe, Majmudar and McKinley2008) for aqueous xanthan gum solution, obtained from microchannel $q{-}{\rm \Delta} p$ measurements and with cone-and-plate and parallel-plate rheometers, are shown in figure 3($a$). The black line represents the fit of the Carreau equation (2.3) to the three data sets obtained using MATLAB's nonlinear least-squares routine lsqcurvefit (release R2020b, Mathworks, USA); the corresponding rheological parameters are provided in the upper row of table 1. It can be seen from figure 3($a$) that cone-and-plate and parallel-plate rheometers provide viscosity data only for low and intermediate shear rates but are unable to capture the behaviour at high shear rates. This limitation of standard rheometers clearly highlights the importance of the microchannel flow measurements that allow us to assess the viscosity at intermediate and high shear rates. It is evident from figure 3($a$) that for intermediate shear rates there is a significant discrepancy between the microchannel viscosity data and cone-and-plate and parallel-plate rheometer viscosity data, which significantly affects the values of the obtained rheological parameters. As we are interested in comparing the microchannel $q{-}{\rm \Delta} p$ measurements with our theory, in figure 3($b$) we present the fit of the Carreau model only to the microchannel rheological data and provide the resulting parameters in the lower row of table 1. While no microchannel rheological data are available for low shear rates, we expect the zero-shear-rate viscosity $\eta _{0}$ to attain the value obtained from rheometer measurements, i.e. $\eta _{0}=11.9$ Pa s, and thus in the fitting we constrain the zero-shear-rate viscosity to this value.

4.2. Finite-length and finite-width effects

Before making a quantitative comparison between theory and experiments, we summarize in table 2 the values of the physical parameters of the experimental system (microchannel A) of Pipe et al. (Reference Pipe, Majmudar and McKinley2008) used for the $q{-}{\rm \Delta} p$ measurements of the shear-thinning xanthan gum solution. These values match well the assumptions of our theoretical analysis. In particular, we observe that $\epsilon =1.94\times 10^{-3}$ and $\delta =7.94\times 10^{-3}$, which satisfy $\epsilon \ll 1$, $\delta \ll 1$ and $\epsilon <\delta \ll 1$, although not strictly $\epsilon \ll \delta \ll 1$.

Table 2. Values of the physical parameters for the experimental system (microchannel A) of Pipe et al. (Reference Pipe, Majmudar and McKinley2008), which measured the pressure drop ${\rm \Delta} p$ as a function of the flow rate $q$ for the shear-thinning xanthan gum solution. The values of $Cu$ are calculated taking $\eta _{0}$ and $\lambda$ from the lower row of table 1, corresponding to the microchannel viscosity data.

We note that the finite-length effect generated by entrance/exit flow disturbances, as commonly encountered in capillary rheometry, may affect the comparison between the theory and experiments and require some corrections (Macosko Reference Macosko1994). However, the experimental system of Pipe et al. (Reference Pipe, Majmudar and McKinley2008), which is schematically shown in figure 1, was designed to avoid such an effect by placing the pressure sensors far from the channel entrances and exits. Specifically, the distance between the inlet and the first sensor was $\ell _{in}=2.025$ mm. As pointed out by Pipe et al. (Reference Pipe, Majmudar and McKinley2008), $\ell _{in}$ is equivalent to $42 d_{h}$, where $d_{h}=2hw/(h+w)\approx 48.2\ \mathrm {\mu }$m is the hydraulic diameter, and is significantly larger than the entrance length needed for fully developed flow at low Reynolds numbers. The distance between the last sensor and the outlet was $\ell _{out}=4.325$ mm, which is even larger than $\ell _{in}$. Therefore, at both points where the sensors measured the pressure, the flow was fully developed, and thus no correction is required.

Beyond the finite-length effect, an effect of the lateral sidewalls may also affect the comparison between the theory and experiments. However, as we show here, this effect is also negligible since $h/w\ll 1$. For small and large values of $Cu$, the viscosity is approximately constant, $\eta _{0}$ and $\eta _{\infty }$, respectively, and the fluid behaves as Newtonian. To assess the finite-width effect for $Cu\ll 1$ and $Cu\gg 1$, we can therefore use the solution for the $q{-}{\rm \Delta} p$ relation of the pressure-driven flow of a Newtonian fluid with viscosity $\eta _{0}$ in a three-dimensional channel, which is well approximated as $12\eta _{0}\ell q/(wh^{3}{\rm \Delta} p)=1-0.627(h/w)$, yielding less than 10 $\%$ error for $h/w\leq 0.7$ (Stone Reference Stone2007). We have $h/w=7.94\times 10^{-3}$, and thus effect of the lateral sidewalls is negligible for $Cu\ll 1$ and $Cu\gg 1$. For intermediate values of $Cu$, the fluid approximately behaves as a power-law fluid. Therefore, to assess the finite-width effect in this limit, we can use figure 5 from Middleman (Reference Middleman1965) showing the parameter $S_{p}=q/wh^{2}(2\eta _{0}\lambda ^{n-1}\ell /(h{\rm \Delta} p))^{1/n}=Q(2/Cu)^{1/n}$ as a function of $n$ for different values of $w/h$, obtained for the pressure-driven flow of power-law fluids in a three-dimensional channel. This figure shows that as the ratio $w/h$ increases, $S_{p}$ rapidly approaches the infinite-width limit ${S_{p}=Q(2/Cu)^{1/n}=n/(4n+2)}$, given in (3.10). For example, for $n=0.5$ and $w/h=4$ it follows that $S_{p}\approx 0.09$, whereas the infinite-width limit yields $S_{p}=0.125$. In the experiments of Pipe et al. (Reference Pipe, Majmudar and McKinley2008), $w/h\approx 126\gg 1$, and thus the effect of the lateral sidewalls is negligible for intermediate values of $Cu$ as well.

4.3. A quantitative comparison between theory and experiments

We present in figure 4 a comparison of our semi-analytical and asymptotic solutions in dimensional form with the experimental measurements of Pipe et al. (Reference Pipe, Majmudar and McKinley2008) for the flow rate–pressure drop relation. We observe that when using the rheological parameters from a fit of the Carreau viscosity model to the three data sets shown in figure 3($a$), our theory (red crosses) significantly overpredicts the experimental pressure drop (black triangles) for intermediate flow rates. We attribute this discrepancy to an apparent difference between the microchannel viscosity data and the cone-and-plate and parallel-plate rheometer viscosity data for intermediate shear rates (see figure 3$a$), which, as mentioned before, influences the values of the Carreau rheological parameters obtained from the fitting to all data sets. Hence, since we compare with microchannel $q{-}{\rm \Delta} p$ measurements, we expect to obtain a much better agreement with experiments when using the rheological parameters corresponding to the fit of the Carreau viscosity model only to the microchannel data shown in figure 3($b$).

Figure 4. Comparison between our theory and the experimental data from Pipe et al. (Reference Pipe, Majmudar and McKinley2008) for the flow rate–pressure drop relation of xanthan gum solution. Triangles represent the experimental data and red crosses represent the theoretical results with the rheological parameters based on viscosity fitting to all data sets (table 1, upper row). Black dots represent the theoretical results, and red and cyan dashed lines represent the asymptotic solutions (3.10) and (3.12), with the rheological parameters based on viscosity fitting to the microchannel data set (table 1, lower row). A shaded trust region, based on taking $\ell =6.3 \pm 1.6$ mm due to the axial size 0.8 mm of each pressure sensor that measured ${\rm \Delta} p$ in the experiments of Pipe et al. (Reference Pipe, Majmudar and McKinley2008), is included.

Black dots in figure 4 represent our semi-analytical solution, and red and cyan dashed lines represent the asymptotic approximations (3.10) and (3.12) with the rheological parameters from fitting the Carreau viscosity model to the microchannel data. Clearly, there is good agreement between our theoretical predictions and the experimental results. In particular, our theory captures the change in slope for high flow rates observed in experiments. As the asymptotic solutions (3.10) and (3.12) indicate, this change in the slope is associated with the transition from the power-law to the large-$Cu$ behaviour with the linear leading-order $q{-}{\rm \Delta} p$ relation, and cannot be predicted using a simple power-law model. Furthermore, we do not observe the transition from the low-$Cu$ to the power-law behaviour in the experimental data shown in figure 4, since the microchannel $q{-}{\rm \Delta} p$ measurements correspond to intermediate and large values of Carreau number, as follows from table 2. This is consistent with the microchannel viscosity data shown in figure 3($b$), which indicates that the data are within the intermediate and high ranges of shear rates, as highlighted by the greyed-out region, and never reach the low-shear-rate viscosity plateau.

The theoretical results shown in figure 4 systematically overpredict the experimental pressure drop. One reason for this discrepancy is inherent uncertainty in the length $\ell$ over which the pressure drop measurements are performed, due to the finite axial size of 0.8 mm of each pressure sensor in the experiments of Pipe et al. (Reference Pipe, Majmudar and McKinley2008). We therefore added a shaded trust region, based on taking $\ell =6.3\pm 1.6$ mm, and obtained a much better agreement between theory and experiments. Finally, it is important to note that, unfortunately, Pipe et al. (Reference Pipe, Majmudar and McKinley2008) did not provide error bars for the $q{-}{\rm \Delta} p$ measurements, thus making it difficult to draw conclusions about the relative error between the theory and experiments. Nevertheless, based on the microchannel viscosity data shown in figure 3($b$), which are obtained from the $q{-}{\rm \Delta} p$ measurements using a procedure briefly outlined in § 4.1 and characterized by large error bars at intermediate shear rates, one might also expect significant error bars in $q{-}{\rm \Delta} p$ data for the intermediate flow rates.