Unsteady and lineal translation of a sphere through a viscoelastic fluid

The unsteady, lineal translation of a solid spherical particle through viscoelastic fluids described by the Johnson-Segalman and Giesekus models is studied analytically. Solutions for the pressure and velocity fields as well as the force on the particle are expanded as a power series in the Weissenberg number. The momentum balance and constitutive equation are solved asymptotically for a steadily translating particle up to second order in the particle velocity, and rescaling of the pressure and velocity in the frequency domain is used to relate the solutions for steady lineal translation to those for unsteady lineal translation. The unsteady force at third order in the particle velocity is then calculated through application of the Lorentz reciprocal theorem, and it is shown that this weakly nonlinear contribution to the force can be expressed as part of a Volterra series. Through a series of examples, it is shown that a truncated representation of this Volterra series, which can be manipulated to describe the velocity in terms of an imposed force, is useful for analyzing specific time-dependent particle motions. Two examples studied using this relationship are the force on a particle suddenly set into motion and the velocity of a particle in response to a suddenly imposed steady force. Additionally, the weakly nonlinear response of a particle captured by a harmonic trap moving lineally through the fluid is computed. This is an analog to active microrheology experiments and can be used to explain how weakly nonlinear responses manifest in active microrheology experiments with spherical probes.

Figure 1

Schematic depicting the flow geometry and boundary conditions for the velocity field.

Figure 2

Projections of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ for a Johnson-Segalman fluid with $b=0.5$ and $\beta =0.5$, with the first harmonic response on the left and the third harmonic response on the right. Solid lines indicate positive values, and dashed lines indicate negative values.

Figure 3

Projections of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ for a Johnson-Segalman fluid. In all cases, $\beta ={10}^{-3}$, while the slip parameter $b$, for the top, middle, and bottom rows, respectively, is 0, 0.5, 1. The first harmonic response is on the left, and the third harmonic is on the right; solid lines indicate positive values, and dashed lines indicate negative values.

Figure 4

Projections of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ for a single-mode Giesekus fluid with $\beta ={10}^{-3}$ and $\alpha =0.5$ (top row), $\alpha =0.3$ (bottom row). The first harmonic response is on the left, and the third harmonic on the right; solid lines indicate positive values, and dashed lines indicate negative values.

Figure 5

Contours of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ for one particular Johnson-Segalman fluid shown in three dimensions on different constant 1-norm surfaces. The top row shows ${\zeta }_3^{\prime }({\omega }_1,{\omega }_2,{\omega }_3)$, and the bottom row shows ${\zeta }_3^{″}({\omega }_1,{\omega }_2,{\omega }_3)$. In both rows, from left to right $\overline{\lambda }{‖\mathit{\omega }‖}_1=0.1,1,10$.

Figure 6

Subregions of the constant $L^1$-norm space ($|{\omega }_1|+|{\omega }_2|+|{\omega }_3|={\omega }^{*}$) used to fully define ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$. Reproduced from Ref. [37] and modified with permission.

Figure 7

Linearly spaced contours of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ at different values of $\overline{\lambda }{‖\mathit{\omega }‖}_1$ for a Johnson-Segalman fluid with $\beta ={10}^{-3}$, and $b=1$ (top row), or $b=0.5$ (bottom row).

Figure 8

Linearly spaced contours of ${\zeta }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ at different values of $\overline{\lambda }{‖\mathit{\omega }‖}_1$ for a Giesekus fluid with $\alpha =0.3$ and $\beta ={10}^{-3}$.

Figure 9

Dimensionless third-order force for $\beta ={10}^{-3}$ over time with a variety of slip parameters as indicated by the legend at the right of the figure.

Figure 10

Total dimensionless force over time following the sudden lineal motion of a particle with both linear and third-order contributions at a variety of Weissenberg numbers and slip parameters with $\beta ={10}^{-3}$.

Figure 11

Dimensionless velocity of a particle over time following a suddenly applied force including both the linear and third-order contributions at a variety of Weissenberg numbers with $b=0$.

Figure 12

Time of undershoot, $t_m/\overline{\lambda }$, as a function of $\mathrm{Wi}$. Note that while the overall trend is similar for both the strong-coupling and weak-coupling limits, the magnitudes are vastly different due to the dependence on $\beta$, with the undershoot occurring at much shorter times in the strong-coupling limit. (a) Strong-coupling limit $(\beta =0.01)$ and (b) Weak-coupling limit $(\beta =0.99)$.

Figure 13

Projections of ${\psi }_3^{*}({\omega }_1,{\omega }_2,{\omega }_3)$ for a fluid with $b=0.5$, $\beta =0.5$, subjected to a single-tone oscillation in a system with trap stiffness $k\overline{\lambda }/\left({\eta }_{0}a\right)=1$ (top row), $k\overline{\lambda }/\left({\eta }_{0}a\right)={10}^{-2}$ (bottom row).