Linear response functions of an electrolyte solution in a uniform flow

We study the steady-state response of a dilute monovalent electrolyte solution to an external source with a constant relative velocity with respect to the fluid. The source is taken as a combination of three perturbations: an external force acting on the fluid, an externally imposed ionic chemical potential, and an external charge density. The linear response functions are obtained analytically and can be decoupled into three independent terms, corresponding to (i) fluid flow and pressure, (ii) total ionic number density and current, and (iii) charge density, electrostatic potential, and electric current. It is shown how the uniform flow breaks the equilibrium radial symmetry of the response functions, leading to a distortion of the ionic cloud and electrostatic potential, which deviates from the standard Debye-Hückel result. The potential of a moving charge is underscreened in its direction of motion and overscreened in the opposite direction and normal plane. As a result, an unscreened dipolar electric field and electric currents are induced far from the charged source. We relate our general formalism to several experimental setups, such as colloidal sedimentation.

Figure 1

A schematic illustration of three possible external sources (in red) and their corresponding induced fields (in gray box). All the sources are at rest, while the electrolyte flows with a constant velocity ${\mathit{u}}_0=u_0\widehat{\mathit{z}}$. (a) A thin rod exerting a force density, $\mathit{F}$, in the negative $z$ direction, leading to a pressure, $P$, and velocity field, $\mathit{u}$, in the electrolyte. (b) A tip of a pipette containing the electrolyte with a different ionic chemical potential, $\mu$, causing changes in the total ionic number density, $n$, and number density current, $\mathit{J}$. (c) A colloid with charge density, $q$, producing an electrostatic potential, $\psi$, as well as a charge density, $c$, and electric current, $\mathit{j}$. The partition into sources and consequent fields is described in detail in Sec. 2c. Any surface effects stemming from the boundaries of the sources are neglected.

Figure 2

Response of the hydrodynamic fields, pressure $P_1$ and fluid velocity ${\mathit{u}}_1$, to a point force source at the origin, $\mathit{F}=-\delta \left(\mathit{r}\right)\widehat{\mathit{z}}$ [see Eq. (16)]. All quantities are dimensionless, according to the definitions of Eq. (9). The response is plotted in the $xz$ plane, and can be extended to the entire space due to the azimuthal symmetry around the $z$ axis. (a) Contour plots of equal pressure. The red regions in the upper half-plane correspond to negative values, $P_1<0$, while the blue regions in the bottom half-plane correspond to positive ones, $P_1>0$. Note that the inner area corresponds to larger absolute values of $P_1$ and is left white for clarity sake. (b) Stream lines of the velocity ${\mathit{u}}_1$. The arrows indicate the flow direction.

Figure 3

Response of the number density, $n_1$, and current field, ${\mathit{J}}_1$, to a point source at the origin, $\mu =\delta \left(\mathit{r}\right)$, for the fluid velocity ${\mathit{u}}_0=2\widehat{\mathit{z}}$ [see Eqs. (18) and (19)]. (a) Contour plots of equal number densities $n_1$. The contour $n_1=0$ is drawn by the black line [see Eq. (20)]. Note that the inner (uncolored) areas correspond to even larger absolute values of $n_1$. (b) Stream lines of the number density current, ${\mathit{J}}_1$.

Figure 4

Response of the charge density, $c_1$, and electric current, ${\mathit{j}}_1$, to a negative point charge source at the origin, $q=-\delta \left(\mathit{r}\right)$ for the fluid velocity ${\mathit{u}}_0=2\widehat{\mathit{z}}$ [see Eq. (22)]. (a) Contour plots of equal charge densities $c_1$. Note that the inner (uncolored) area corresponds to even larger absolute values of $c_1$. (b) Stream lines of the electric current ${\mathit{j}}_1$.

Figure 5

Response of of the electrostatic potential, ${\psi }_1$, and electric field, ${\mathit{E}}_1$, to a negative point charge source at the origin $q=-\delta \left(\mathit{r}\right)$ for the fluid velocity ${\mathit{u}}_0=2\widehat{\mathit{z}}$ [see Eqs. (27) and (28)]. (a) Contour plots of equipotential values of ${\psi }_1$. Note that the inner (uncolored) area corresponds to even larger absolute values of ${\psi }_1$. (b) Stream lines of the electric field ${\mathit{E}}_1$.