4.1 Exact solutions of the $p$ -Poisson equation for one-dimensional configurations

Consider the case where the zeta potential distribution is a function of $x$ only, $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}(x),$ and the electric field is directed along the $\hat{\boldsymbol{x}}$ axis, $\boldsymbol{E}_{\Vert }=E\hat{\boldsymbol{x}}$ . For such configurations, the problem becomes one-dimensional, and (3.6) and (3.7) take the form

(4.1) $$\begin{eqnarray}q=-\left|\frac{\text{d}p}{\text{d}x}\right|^{(1/n)-1}\frac{\text{d}p}{\text{d}x}+\unicode[STIX]{x1D701}(x)E,\end{eqnarray}$$

and

(4.2) $$\begin{eqnarray}\frac{\text{d}}{\text{d}x}\left(\left|\frac{\text{d}p}{\text{d}x}\right|^{(1/n)-1}\frac{\text{d}p}{\text{d}x}\right)=E\frac{\text{d}\unicode[STIX]{x1D701}(x)}{\text{d}x},\end{eqnarray}$$

where $q$ is a volume flux. We prescribe gauge pressure at the inlet and outlet

(4.3) $$\begin{eqnarray}p|_{x=0}=p|_{x=1}=0,\end{eqnarray}$$

or alternatively impose inlet pressure $p|_{x=0}=0$ and a desired volume flux $q$ . Solving (4.2), we obtain

(4.4) $$\begin{eqnarray}p(x)=\int _{0}^{x}|\unicode[STIX]{x1D701}(x)E-q|^{n-1}(\unicode[STIX]{x1D701}(x)E-q)\,\text{d}x,\end{eqnarray}$$

where $q$ may be calculated (in the case of (4.3)) from the equality

(4.5) $$\begin{eqnarray}\int _{0}^{1}|\unicode[STIX]{x1D701}(x)E-q|^{n-1}(\unicode[STIX]{x1D701}(x)E-q)\,\text{d}x=0.\end{eqnarray}$$

We consider a case where the zeta potential acquires a constant positive value, $\unicode[STIX]{x1D701}_{0}$ , before the discontinuity at $x_{0}$ , and vanishes afterwards,

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D701}(x)=\unicode[STIX]{x1D701}_{0}H(x_{0}-x),\end{eqnarray}$$

where $H$ is the Heaviside function. We note that while sharp gradients in zeta potential described by (4.6) lead to local deviations from the lubrication approximation, these deviations are expected to be limited to a narrow region of $O(\unicode[STIX]{x1D716})$ (Brotherton & Davis Reference Brotherton and Davis2004). Substituting (4.6) into (4.5) yields

(4.7) $$\begin{eqnarray}\int _{0}^{x_{0}}|\unicode[STIX]{x1D701}_{0}E-q|^{n-1}(\unicode[STIX]{x1D701}_{0}E-q)\,\text{d}x-\int _{x_{0}}^{1}|q|^{n-1}q\,\text{d}x=0.\end{eqnarray}$$

Assuming that $E$ is positive and $E\unicode[STIX]{x1D701}_{0}\geqslant q$ (since the value $E\unicode[STIX]{x1D701}_{0}$ is the maximal volume flux that can be achieved, when the entire channel has a zeta potential $\unicode[STIX]{x1D701}_{0}$ ), from (4.7) we obtain

(4.8) $$\begin{eqnarray}\frac{q}{E\unicode[STIX]{x1D701}_{0}}=\frac{1}{1+((1-x_{0})/x_{0})^{1/n}}.\end{eqnarray}$$

Substituting (4.8) into (4.4) results in

(4.9) $$\begin{eqnarray}p(x)=\left\{\begin{array}{@{}ll@{}}-q^{n}(x-1) & x>x_{0}\\ (\unicode[STIX]{x1D701}_{0}E-q)^{n}x & x\leqslant x_{0}.\end{array}\right.\end{eqnarray}$$

Figure 2(a) presents the volume flux to slip velocity ratio, $q/E\unicode[STIX]{x1D701}_{0}$ , as a function of $x_{0}$ (the position of the zeta potential discontinuity) for a strongly shear-thinning fluid $n=0.1$ (dashed line), a strongly shear-thickening fluid $n=4$ (dashed–dot line), and a Newtonian fluid (solid line). For a Newtonian fluid and constant $E\unicode[STIX]{x1D701}_{0}$ , the flux $q$ increases linearly with $x_{0}$ , as expected. In the case of strong shear thinning, the flux to slip velocity ratio, $q/E\unicode[STIX]{x1D701}_{0}$ , is approximately zero when the abrupt change in zeta potential occurs before the middle of the channel ( $x_{0}<0.5$ ), while $q/E\unicode[STIX]{x1D701}_{0}$ is approximately 1 in the case where the discontinuity is positioned after the middle of the channel ( $x_{0}>0.5$ ). In contrast, in the case of strong shear thickening, $q/E\unicode[STIX]{x1D701}_{0}$ is weakly dependent on the location of the discontinuity (except near the boundaries).

Figure 2.

The volume flux and maximal pressure in the case of a step zeta potential distribution for one-dimensional configurations. (a) Comparison of flux to slip velocity ratio, $q/E\unicode[STIX]{x1D701}_{0}$ , as a function of the step location $x_{0}$ for strong shear-thickening (dashed-dot line), strong shear-thinning (dashed line) and Newtonian behaviour (solid line). (b) Maximal pressure obtained from (4.10) as a function of $E\unicode[STIX]{x1D701}_{0}$ for $x_{0}=0.45$ and $n=0.1,1,4$ . Black dashed line describes a strong shear-thinning behaviour which tends to constant value given by (4.11) and shown as red dashed line. Black dashed-dot line corresponds to the case of strong shear thickening, and coincides with asymptotic behaviour given by (4.12) (red dashed-dot line). The black solid line presents a Newtonian case.

It is also interesting to investigate the effect of the parameter $n$ on the resulting maximal pressure in the channel, given by

(4.10) $$\begin{eqnarray}p_{max}\triangleq p_{max}^{ex}=(E\unicode[STIX]{x1D701}_{0})^{n}\frac{1-x_{0}}{(1+((1-x_{0})/x_{0})^{1/n})^{n}}.\end{eqnarray}$$

When $n\ll 1$ , we have

(4.11) $$\begin{eqnarray}p_{max}\sim \left\{\begin{array}{@{}ll@{}}1-x_{0} & x_{0}>0.5\\ x_{0} & x_{0}<0.5,\end{array}\right.\end{eqnarray}$$

independently of value $E\unicode[STIX]{x1D701}_{0}$ , whereas for $n\gg 1$ , we obtain

(4.12) $$\begin{eqnarray}p_{max}\sim (1-x_{0})\left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)^{n}.\end{eqnarray}$$

Figure 2(b) presents a variation of maximal pressure as a function of $E\unicode[STIX]{x1D701}_{0}$ for a specific case of $x_{0}=0.45$ . As expected, for a Newtonian fluid $(n=1)$ we obtain a linear behaviour (black solid line). For small values of $n$ , the maximal pressure (black dashed line) rapidly approaches a constant value given by (4.11), which is solely determined by the location of discontinuity $x_{0}$ (red dashed line). On the other hand, for large values of $n$ the maximal pressure (black dashed-dot line) approaches the asymptotic limit (4.12) (red dashed-dot line). We note that sufficiently small values of $E\unicode[STIX]{x1D701}_{0}$ , corresponding to shear rate less than one, where the viscosity of shear-thinning fluid exceeds the viscosity of the shear-thickening fluid, result in larger maximal pressure of shear-thinning fluid in this region.

4.2 Asymptotic approximation for one-dimensional configurations

The leading- and first-order asymptotic approximations for one-dimensional configurations are

(4.13a ) $$\begin{eqnarray}O(1):\quad \frac{\text{d}^{2}p^{(0)}}{\text{d}x^{2}}=E\frac{\text{d}\unicode[STIX]{x1D701}}{\text{d}x},\end{eqnarray}$$

and

(4.13b ) $$\begin{eqnarray}O(\unicode[STIX]{x1D6FF}):\quad \frac{\text{d}^{2}p^{(1)}}{\text{d}x^{2}}=-\frac{\text{d}}{\text{d}x}\left(\ln \left|\frac{\text{d}p^{(0)}}{\text{d}x}\right|\frac{\text{d}p^{(0)}}{\text{d}x}\right).\end{eqnarray}$$

Equations (4.13a ) and (4.13b ) are readily solved, to attain

(4.14a ) $$\begin{eqnarray}p^{(0)}(x)=\int _{0}^{x}E\unicode[STIX]{x1D701}(x)\,\text{d}x-x\int _{0}^{1}E\unicode[STIX]{x1D701}(x)\,\text{d}x,\end{eqnarray}$$

and

(4.14b ) $$\begin{eqnarray}p^{(1)}(x)=-\int _{0}^{x}\ln \left|\frac{\text{d}p^{(0)}}{\text{d}x}\right|\frac{\text{d}p^{(0)}}{\text{d}x}\,\text{d}x+x\int _{0}^{1}\ln \left|\frac{\text{d}p^{(0)}}{\text{d}x}\right|\frac{\text{d}p^{(0)}}{\text{d}x}\,\text{d}x.\end{eqnarray}$$

We now derive the asymptotic solution corresponding to a step function in zeta potential, equation (4.6), in order to compare the exact and asymptotic results and evaluate the accuracy of the asymptotic solution. We define $x_{0}=0.5$ and substitute (4.6) into (4.13a ), yielding

(4.15) $$\begin{eqnarray}\frac{\text{d}^{2}p^{(0)}}{\text{d}x^{2}}=-2E\unicode[STIX]{x1D701}_{0}\unicode[STIX]{x1D6FF}_{D}(1-2x),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{D}(x)$ is the Dirac delta function. The corresponding solution of (4.15) is

(4.16) $$\begin{eqnarray}p^{(0)}(x)=\frac{1}{2}E\unicode[STIX]{x1D701}_{0}(x+(1-2x)H(2x-1)).\end{eqnarray}$$

Substituting (4.16) into (4.13b ) yields

(4.17) $$\begin{eqnarray}\frac{\text{d}^{2}p^{(1)}}{\text{d}x^{2}}=2E\unicode[STIX]{x1D701}_{0}\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)\unicode[STIX]{x1D6FF}_{D}(1-2x).\end{eqnarray}$$

Figure 3.

Comparison between exact solution and asymptotic approximations in the case of a step zeta potential distribution in one dimension. (a) The resulting pressure distribution calculated from exact (solid lines) and asymptotic (dashed lines) solutions for $n=0.5$ , 1, 1.5, $x_{0}=0.5$ and $E\unicode[STIX]{x1D701}_{0}=4$ . (b) The error in maximal pressure between exact and asymptotic solutions as a function of $n$ , for the case of $x_{0}=0.5$ and $E\unicode[STIX]{x1D701}_{0}=4$ . Black dashed line given by (4.20a ) corresponds to the Newtonian approximation, while black solid line corresponds to the first-order correction (4.20b ). Red dashed line represents a 10 % error. (c,d) Colourmaps of the maximal error as a function of the parameter $n$ and slip velocity $E\unicode[STIX]{x1D701}_{0}$ , for the cases of a Newtonian approximation and a first-order asymptotic approximation, respectively. The cyan dashed line shown in (c) and (d) corresponds to $E\unicode[STIX]{x1D701}_{0}=4$ , shown in (b). As expected, the range of values of $n$ for which error is less than 10 % is significantly increased when the first-order non-Newtonian correction included.

Solving (4.17) before and after the discontinuity, following the same procedure as we illustrated above, and requiring the continuity of the pressure and of the flux at $x_{0}=0.5$ , lead to first-order correction for the pressure distribution

(4.18) $$\begin{eqnarray}p^{(1)}(x)=-\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)(x+(1-2x)H(2x-1)).\end{eqnarray}$$

Combining (4.16) and (4.18), and substituting $x=x_{0}=0.5$ yield a maximal pressure as a function of $E\unicode[STIX]{x1D701}_{0}$ and  $\unicode[STIX]{x1D6FF}$

(4.19) $$\begin{eqnarray}p_{max}^{as}\triangleq \frac{1}{4}E\unicode[STIX]{x1D701}_{0}\left(1-\unicode[STIX]{x1D6FF}\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)\right).\end{eqnarray}$$

Figure 3(a) presents the pressure distribution obtained from exact (solid lines) and asymptotic (dashed lines) solutions for $n=0.5$ , 1, 1.5, $x_{0}=0.5$ and $E\unicode[STIX]{x1D701}_{0}=4$ . Since (4.9) is an exact solution, we define the maximal error of the leading-order and first-order asymptotic solutions

(4.20a ) $$\begin{eqnarray}e^{(0)}=\left|\frac{p_{max}^{ex}-p_{max}^{(0)}}{p_{max}^{ex}}\right|,\end{eqnarray}$$

and

(4.20b ) $$\begin{eqnarray}e^{(0)+(1)}=\left|\frac{p_{max}^{ex}-p_{max}^{as}}{p_{max}^{ex}}\right|,\end{eqnarray}$$

respectively. Figure 3(b) presents the maximal error between the exact solution of the $p$ -Poisson equation and the asymptotic approximation for a range of $n$ values, for the case of $x_{0}=0.5$ and $E\unicode[STIX]{x1D701}_{0}=4$ . The solid black line presents the error of the asymptotic solution which includes the first-order correction, whereas the black dashed line shows the maximal error given by (4.20a ) in the case of a Newtonian approximation (zeroth order only). The red dashed line presents a 10 % error. Clearly, the range of values of $n$ for which the error is less than 10 % is significantly increased when the first-order correction is taken into account.

Figures 3(c) and 3(d) show colourmaps of the maximal error as a function of parameter $n$ and slip velocity $E\unicode[STIX]{x1D701}_{0}$ , in the case Newtonian approximation and non-Newtonian correction, respectively. The cyan dashed line in figures 3(c) and 3(d) represents the value of $E\unicode[STIX]{x1D701}_{0}=4$ , which is presented in figure 3(b). As can be inferred from the results of figure 3, the first-order correction significantly reduces the error, and also remains applicable for a wide range of $n$ values. However, this range of asymptotic applicability reduces as $E\unicode[STIX]{x1D701}_{0}$ increases.

5.1 Axisymmetric pressure fields

Assuming a uniform electric field along the $\hat{\boldsymbol{x}}$ axis, $\boldsymbol{E}_{\Vert }=E\hat{\boldsymbol{x}}$ , we consider the case where the gradient of zeta potential in the $\hat{\boldsymbol{x}}$ direction, $\unicode[STIX]{x2202}\unicode[STIX]{x1D701}/\unicode[STIX]{x2202}x$ , is a function of a single argument $r=\sqrt{x^{2}+y^{2}}$ . Therefore, the resulting pressure also depends only on $r$ and is obtained through

(5.1) $$\begin{eqnarray}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(\left|\frac{\text{d}p}{\text{d}r}\right|^{(1/n)-1}\frac{\text{d}p}{\text{d}r}r\right)=E\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}(r).\end{eqnarray}$$

Equation (5.1) is subjected to a symmetry boundary condition

(5.2) $$\begin{eqnarray}\frac{\text{d}p}{\text{d}r}=0\quad \text{at}~r=0,\end{eqnarray}$$

and the pressure is required to decay far from the actuation region,

(5.3) $$\begin{eqnarray}p=0\quad \text{as}~r\rightarrow \infty .\end{eqnarray}$$

Integrating (5.1) with respect to $r$ , we obtain

(5.4) $$\begin{eqnarray}\left|\frac{\text{d}p}{\text{d}r}\right|^{(1/n)-1}\frac{\text{d}p}{\text{d}r}=\frac{E}{r}\left(\int r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}(r)\,\text{d}r+A\right),\end{eqnarray}$$

where $A$ is a constant determined using (5.2)

(5.5) $$\begin{eqnarray}A=-\lim _{r\rightarrow 0}\frac{1}{r}\int r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}(r)\,\text{d}r.\end{eqnarray}$$

From (3.6) and (5.4), it follows that the resulting flow field in this case is two-dimensional, and identical for Newtonian and non-Newtonian fluids independently of the parameter  $n$ .

Hereafter, we focus on configurations where $\unicode[STIX]{x1D701}(r)$ is continuous at the origin, and thus $A=0$ . Integrating once again (5.4) and using (5.3), yields a closed-form expression for the pressure

(5.6) $$\begin{eqnarray}p(r)=E^{n}\int _{\infty }^{r}\frac{1}{r^{n}}\left|\int r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}(r)\,\text{d}r\right|^{n-1}\left(\int r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}x}(r)\,\text{d}r\right)\text{d}r.\end{eqnarray}$$

For a Newtonian fluid, a Gaussian-like pressure

(5.7) $$\begin{eqnarray}p=E\unicode[STIX]{x1D701}_{0}\exp \left(-\frac{x^{2}+y^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)=E\unicode[STIX]{x1D701}_{0}\exp \left(-\frac{r^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D701}_{0}$ are positive constants, may be obtained using the following zeta distribution

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D701}(x,y)=-\frac{\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x1D6FC}^{4}}\left(2x\unicode[STIX]{x1D6FC}+\sqrt{\unicode[STIX]{x03C0}}(-2y^{2}+\unicode[STIX]{x1D6FC}^{2})\exp \left(\frac{x^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\text{erf}\left(\frac{x}{\unicode[STIX]{x1D6FC}}\right)\right)\exp \left(-\frac{x^{2}+y^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right),\end{eqnarray}$$

which is shown in figure 4(a). We note that in this case the limit (5.5) gives $A=0$ .

Figure 4.

(a) Illustration of two-dimensional zeta potential distribution with an axisymmetric gradient in the $\hat{\boldsymbol{x}}$ direction, described by (5.8). (b) The resulting two-dimensional flow field, due to the zeta potential distribution shown in (a). The streamlines describe stagnation-like flow approaching the origin from both sides of the $\hat{\boldsymbol{x}}$ axis. The flow field is identical for both Newtonian and non-Newtonian fluids as given by (5.11). All calculations were performed using $E=3$ and $\unicode[STIX]{x1D701}_{0}=\unicode[STIX]{x1D6FC}=1$ .

We here examine the effect of non-Newtonian behaviour on the pressure distribution, resulting from the same zeta potential (5.8). Substituting (5.8) into (5.6) provides a closed-form analytical solution for the pressure for a non-Newtonian fluid with arbitrary  $n$

(5.9) $$\begin{eqnarray}p(r)=n^{-1/2(n+1)}\left(\frac{\unicode[STIX]{x1D6FC}}{2}\right)^{1-n}(E\unicode[STIX]{x1D701}_{0})^{n}\unicode[STIX]{x1D6E4}\left(\frac{n+1}{2};\frac{nr^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}(a;x)=\int _{x}^{\infty }\exp (-t)t^{a-1}\,\text{d}t$ is the incomplete Gamma function (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The leading order of the asymptotic solution is simply given in (5.7), while the first order is

(5.10) $$\begin{eqnarray}p^{(1)}(r)=\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\exp \left(-\frac{r^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\left(2+\exp \left(\frac{r^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\text{Ei}\left(-\frac{r^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)-2\ln \left(\frac{2E\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x1D6FC}^{2}}r\exp \left(-\frac{r^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\right)\right),\end{eqnarray}$$

where $\text{Ei}(x)=-\int _{-x}^{\infty }\exp (-t)/t\,\text{d}t$ is the exponential integral function (Abramowitz & Stegun Reference Abramowitz and Stegun1964). The corresponding flow field is readily obtained by differentiating (5.7) and using (5.8)

(5.11) $$\begin{eqnarray}\langle \boldsymbol{u}_{\Vert }\rangle =\frac{E\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x1D6FC}^{2}}\exp \left(\!-\frac{x^{2}+y^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\!\left(\begin{array}{@{}c@{}}\displaystyle \!\left(\!2\left(\!1-\frac{1}{\unicode[STIX]{x1D6FC}}\right)x-\frac{\sqrt{\unicode[STIX]{x03C0}}}{\unicode[STIX]{x1D6FC}^{2}}(-2y^{2}+\unicode[STIX]{x1D6FC}^{2})\exp \left(\frac{x^{2}}{\unicode[STIX]{x1D6FC}^{2}}\right)\text{erf}\left(\frac{x}{\unicode[STIX]{x1D6FC}}\right)\!\right)\hat{\boldsymbol{x}}\\ 2y\hat{\boldsymbol{y}}\end{array}\!\right)\!.\end{eqnarray}$$

Figure 4(b) presents several streamlines given by (5.11) for $E\unicode[STIX]{x1D701}_{0}=3$ . The streamlines describe an incoming fluid flowing along both sides of the $\hat{\boldsymbol{x}}$ axis towards the origin, followed by outgoing flow initially turning in the $\pm \hat{\boldsymbol{y}}$ directions and then curving back in the $\pm \hat{\boldsymbol{x}}$ directions to satisfy mass conservation. The resulting flow field (5.11) is independent of the parameter  $n$ , due to a Neumann-type symmetry boundary condition (5.2) at the origin.

Figure 5.

The use of non-Newtonian fluids allows one to significantly increase the pressure resulting from non-uniform EOF by more than two orders of magnitude. (a) Comparison of exact results (solid lines) and asymptotic approximations (dashed lines) for axisymmetric (non-dimensional) pressure distributions, resulting from a two-dimensional zeta potential, equation (5.8), for $n=0.5$ , 1, 1.5. (b) Exact solutions in dimensional form, showing significantly greater pressures for EOF-driven flow of non-Newtonian fluids compared with Newtonian fluids. The different lines correspond to a Newtonian fluid (cyan solid line), 8.33 mM polyacrylic acid (PAA) (purple dashed line) and 1 % carboxymethyl cellulose (CMC) (blue dashed-dot line) solutions. The rheological power-law properties of PAA and CMC solutions are available in table 1. For a zeta potential difference of 75 mV, and an applied electric field of $100~\text{V}~\text{cm}^{-1}$ , a Newtonian fluid yields a maximum pressure of 3 Pa, whereas as PAA and CMC solutions yield pressures of 280 and 1700 Pa, respectively. All calculations were performed using $\tilde{l}=5$  mm,  $\tilde{h}=50~\unicode[STIX]{x03BC}\text{m}$ ,  $\tilde{\unicode[STIX]{x1D701}}^{\ast }=-25$  mV,  $\tilde{E}^{\ast }=100~\text{V}~\text{cm}^{-1}$ , $\tilde{\unicode[STIX]{x1D700}}=7.08\cdot 10^{-10}~\text{F}~\text{m}^{-1}$ , $E\unicode[STIX]{x1D701}_{0}=3$ and $\unicode[STIX]{x1D6FC}=1$ .

In figure 5(a) we compare the resulting exact (solid lines) and asymptotic approximations (dashed lines) for the pressure distribution given by (5.9), (5.7) and (5.10), showing good agreement. Using (4.20a ) and (4.20b ), we also calculated the maximal errors in this case, concluding that in two dimensions the accuracy of the asymptotic solution is approximately the same as in one dimension, yielding error below 10 % for values of $n$ lying between 0.5 and 1.75.

Table 1.

Rheological parameters of polymer solutions at $25\,^{\circ }\text{C}$ . The background aqueous solution has viscosity of $\unicode[STIX]{x1D702}_{N}=1~\text{mPa}~\text{s}$ . PAA parameters are adapted from Paul (Reference Paul2008) and Berli (Reference Berli2010). CMC experimental data is taken from Olivares et al. (Reference Olivares, Vera-Candioti and Berli2009).

Figure 5(b) presents the resulting dimensional pressure distribution of (5.9), where we used three experimental values of $n$ and $\tilde{\unicode[STIX]{x1D707}}_{eff}$ from table 1, corresponding to a Newtonian fluid (cyan solid line), 8.33 mM polyacrylic acid (PAA) (purple dashed line) and 1 % carboxymethyl cellulose (CMC) (blue dashed-dot line) solutions. As can be observed, by leveraging non-Newtonian polymer solutions, the resulting pressure is increased by two to three orders of magnitude as compared to a Newtonian fluid.

5.2 Asymptotic approximation for a circular spot with constant zeta potential

We here study the effect of a spot with non-zero zeta potential on the resulting pressure and flow fields. This simple, yet fully two-dimensional case, is useful in providing physical insight on the effect of the power-law fluid on the flow field. In the following, we adopt the superscripts in and out to distinguish between physical quantities in each one of the two regions. The corresponding zeta potential distribution is given by

(5.12) $$\begin{eqnarray}\unicode[STIX]{x1D701}(r)=\unicode[STIX]{x1D701}_{0}H(r_{0}-r).\end{eqnarray}$$

Here $H$ stands for the Heaviside function and $r_{0}$ is the radius of the spot. Without loss of generality, we assume that the uniform electric field is directed along the $\hat{\boldsymbol{x}}$ axis, $\boldsymbol{E}_{\Vert }=E\hat{\boldsymbol{x}}$ . Substituting (5.12) into (3.14) and taking $p^{(0)}\rightarrow 0$ as $r\rightarrow \infty$ , yields the leading-order solution (Boyko et al. Reference Boyko, Rubin, Gat and Bercovici2015)

(5.13) $$\begin{eqnarray}p^{(0)}(r,\unicode[STIX]{x1D703})=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{3E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{2r}\cos \unicode[STIX]{x1D703} & r>r_{0}\\ \displaystyle \frac{3}{2}E\unicode[STIX]{x1D701}_{0}r\cos \unicode[STIX]{x1D703} & r<r_{0},\end{array}\right.\end{eqnarray}$$

with corresponding flow field

(5.14) $$\begin{eqnarray}\langle \boldsymbol{u}_{\Vert }\rangle ^{(0)}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2}\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r^{2}}(\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{r}}+\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73D}}) & r>r_{0}\\ \displaystyle \frac{1}{2}E\unicode[STIX]{x1D701}_{0}(\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{r}}-\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73D}}) & r<r_{0}.\end{array}\right.\end{eqnarray}$$

Next, we turn to solve the non-Newtonian (first-order) contribution to the pressure. The source term in the first-order equation (3.15) can be expressed using (5.13) as

(5.15) $$\begin{eqnarray}-\unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }(\ln |\unicode[STIX]{x1D735}_{\Vert }p^{(0)}|\unicode[STIX]{x1D735}_{\Vert }p^{(0)})=-\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r^{3}}H(r-r_{0})\cos \unicode[STIX]{x1D703}.\end{eqnarray}$$

Assuming the form $p^{(1)}(r,\unicode[STIX]{x1D703})=f^{(1)}(r)\cos \unicode[STIX]{x1D703}$ , we obtain an ordinary differential equation for $f^{(1)}(r)$

(5.16) $$\begin{eqnarray}\frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r\frac{\text{d}f^{(1)}(r)}{\text{d}r}\right)-\frac{f^{(1)}(r)}{r^{2}}=-\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r^{3}}H(r-r_{0}).\end{eqnarray}$$

Solving (5.16) and requiring regularity, leads to

(5.17) $$\begin{eqnarray}p^{(1)}(r,\unicode[STIX]{x1D703})=\left\{\begin{array}{@{}ll@{}}\displaystyle \left(\frac{a^{out}}{r}+\left(b^{out}-\frac{1}{4}E\unicode[STIX]{x1D701}_{0}\right)r-\frac{1}{4}\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r}\left(2\ln \frac{r_{0}}{r}-1\right)\right)\cos \unicode[STIX]{x1D703} & r>r_{0}\\ b^{in}r\cos \unicode[STIX]{x1D703} & r<r_{0},\end{array}\right.\end{eqnarray}$$

where $a^{out}$ , $b^{in}$ and $b^{out}$ are coefficients to be determined from boundary conditions. Demanding vanishing pressure far from the disk, we find

(5.18) $$\begin{eqnarray}b^{out}=\frac{1}{4}E\unicode[STIX]{x1D701}_{0}.\end{eqnarray}$$

The remaining coefficients $a^{out}$ and $b^{in}$ are determined by requiring a continuity of the pressure and flow field at $r=r_{0}$ , leading to

(5.19a,b ) $$\begin{eqnarray}a^{out}=-\frac{1}{2}E\unicode[STIX]{x1D701}_{0}r_{0}^{2}\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right),\quad b^{in}=-\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\left(\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)-\frac{1}{2}\right).\end{eqnarray}$$

Utilizing (3.13), (5.12) and (5.17)–(5.19) provides a closed-form expressions for the pressure and corresponding flow field at first order

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle p^{(1)}(r,\unicode[STIX]{x1D703})=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{4}\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r}\left(1-2\ln \left(\frac{1}{2}\frac{E\unicode[STIX]{x1D701}_{0}r_{0}}{r}\right)\right)\cos \unicode[STIX]{x1D703} & r>r_{0}\\ \displaystyle \frac{1}{4}E\unicode[STIX]{x1D701}_{0}\left(1-2\ln \left(\frac{1}{2}E\unicode[STIX]{x1D701}_{0}\right)\right)r\cos \unicode[STIX]{x1D703} & r<r_{0},\end{array}\right. & \displaystyle\end{eqnarray}$$

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{u}_{\Vert }\rangle ^{(1)}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{4}\frac{E\unicode[STIX]{x1D701}_{0}r_{0}^{2}}{r^{2}}\left(\left(-1+2\ln \frac{r_{0}}{r}\right)\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{r}}+\left(1+2\ln \frac{r_{0}}{r}\right)\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73D}}\right) & r>r_{0}\\ \displaystyle \frac{1}{4}E\unicode[STIX]{x1D701}_{0}(-\cos \unicode[STIX]{x1D703}\hat{\boldsymbol{r}}+\sin \unicode[STIX]{x1D703}\hat{\unicode[STIX]{x1D73D}}) & r<r_{0}.\end{array}\right.\qquad & \displaystyle\end{eqnarray}$$

Similarly to the one-dimensional case, the maximal pressure of shear-thinning fluids, attained at $r=r_{0}$ , can be either greater or less than the pressure of shear-thickening fluids depending on the value of $E\unicode[STIX]{x1D701}_{0}$ , where the transition occurs at $E\unicode[STIX]{x1D701}_{0cr}\backsimeq 3.29$ .

Figure 6.

Two-dimensional asymptotic approximations for the velocity and pressure fields of power-law fluids in the presence of a spot with constant zeta potential. (a,b) The pressure distribution (colourmap) in the first quadrant for $n=0.5$ and $n=1.5$ corresponding to shear-thinning and shear-thickening behaviour, respectively, showing that the isobars of shear-thinning fluid are more spaced compared with those of a shear-thickening fluid. Isobars $p^{as}=0.75$ presented in (c) and contours of velocity magnitude $|\langle \boldsymbol{u}_{\Vert }\rangle ^{as}|=0.4$ presented in (d) correspond to $n=0.5$ (black dashed line), $n=1$ (black solid line) and $n=1.5$ (black dashed-dot line). Note that contours in (d) are more closely spaced than the contours in (c), thus indicating a weak dependence on $n$ for the shear-thickening fluid. (e,f) The distance from the origin along the $\hat{\boldsymbol{x}}$ axis, where the pressure and the velocity magnitude drops to 20 % of its maximum value, as a function of $n$ , respectively. The solid part of the line in (e) and (f) corresponds to the range of $n$ lying between 0.5 and 1.75, where the asymptotic approximation is assumed to be valid, while the dashed line corresponds to the region that may involve significant errors. It follows from (c–f) that the pressure distribution is strongly dependent on $n$ for $n<1$ and weakly dependent on $n$ for $n>1$ as compared to the velocity magnitude, which shows only a weak dependence on $n$ . All calculations were performed using $E\unicode[STIX]{x1D701}_{0}=4$ and $r_{0}=1$ .

Figure 6 summarizes the effect of non-Newtonian behaviour on the resulting pressure and velocity magnitude in the Hele-Shaw cell for the problem considered. Figures 6(a) and 6(b) present the pressure distribution in the first quadrant for shear-thinning and shear-thickening fluids, respectively. We note that, in order to provide clear comparison between the cases, we chose to present the result for $n=0.5$ and $n=1.5$ , which for the previous two-dimensional problem considered (see figure 5 a) bracketed the range of $n$ values for which the error was less 10 %. However, since the current case has no analytical solution, the precise accuracy of the solution at these $n$ values is unknown. The shear-thinning fluid (figure 6 a) is characterized by greater spacing in the isobars, compared to the shear-thickening fluid (figure 6 b). For further clarification, figures 6(c) and 6(d) present the same isobar and equivelocity line, for three different values of the parameter  $n$ . This is also shown quantitatively in figure 6(e), showing the distance from the origin over which the pressure decays to 20 % of its peak value, as a function of  $n$ . The solid part of the line corresponds to the range of $n$ lying between 0.5 and 1.75, where the asymptotic approximation is assumed to be valid (according to the previous estimations), whereas the dashed region may involve significant errors. Figure 6(f) presents a complementary view, indicating that the velocity magnitude decays more slowly for larger values of  $n$ .

Figure 7.

Effect of different non-Newtonian behaviour on the resulting streamlines (in the first quadrant) due to a positive zeta potential, $\unicode[STIX]{x1D701}_{0}/2$ , inside the spot of radius $r_{0}$ and negative zeta potential, $-\unicode[STIX]{x1D701}_{0}/2$ , outside the spot. In the Newtonian case (solid lines) the streamlines bend around the spot without entering it. The streamlines of the shear-thinning fluid ( $n=0.5$ , dashed lines) are diverted towards the disk, while the streamlines of the shear-thickening fluid ( $n=1.5$ , dashed-dot lines) divert away from it. All calculations were performed using $E\unicode[STIX]{x1D701}_{0}=4$ and $r_{0}=1$ .

To highlight the effect of different non-Newtonian behaviour on the resulting flow field, we take advantage of the uniform flow field in the inner region and of the fact that by adding a constant value to the zeta potential the resulting pressure remains unaffected. Adding a bias value of $-\unicode[STIX]{x1D701}_{0}/2$ to the zeta potential everywhere in the domain leads to zero net flow in the spot for the Newtonian fluid and the resulting streamlines curve around the spot without penetrating it. Figure 7 presents the resulting streamlines for shear-thinning, Newtonian and shear-thickening fluids. The incoming black dashed streamlines, which represent shear-thinning behaviour, are diverted towards the disk penetrating it, due to negative net flow inside the spot. On the other hand, the incoming black dashed-dot streamlines, which represent shear-thinning behaviour, are diverted away from the disk, owing to positive net flow inside the spot. This is consistent with (5.21), which predicts a positive velocity field, $(1/4)|\unicode[STIX]{x1D6FF}|E\unicode[STIX]{x1D701}_{0}\hat{\boldsymbol{x}}$ , for shear-thickening fluids and a negative velocity field, $-(1/4)|\unicode[STIX]{x1D6FF}|E\unicode[STIX]{x1D701}_{0}\hat{\boldsymbol{x}}$ , for shear-thinning fluids inside the spot.