Quantitative model for predicting the imbibition dynamics of viscoelastic fluids in nonuniform microfluidic assays

We develop a mathematical model to quantitatively describe the imbibition dynamics of an elastic non-Newtonian fluid in a conical (nonuniform cross section) microfluidic assay. We consider the simplified Phan-Thien-Tanner viscoelastic model to represent the rheology of the elastic non-Newtonian fluid. Our model accounts for the geometrical features of the fluidic assay, the key parameters affecting the rheological behavior of the fluid, and predicts the imbibition dynamics effectively. By demonstrating the temporal advancement of the filling length in the conical capillary graphically, obtained for pertinent parametric values belonging to their physically permissible range, we report an underlying balance between capillary and viscous forces during imbibition resulting in three distinct regimes of filling. Nonuniformity in the capillary cross section gives rise to an alteration in the viscous force being applied at the contact line (manifested through the alteration in shear rate) during the imbibition process, which upon maintaining a balance with the dominant capillary force results in three different regimes of filling. We believe that the present analysis has a twofold significance. First, this work will enhance the understanding of underlying imbibition dynamics of viscoelastic fluids (most of the biofluids exhibit viscoelastic rheology) in nonuniform fluidic pathways. Second, the developed model is of significant practical relevance for the optimum design of microfluidic assays, primarily used for sample diagnostics in biochemical and biomedical applications.

Figure 1

Schematic diagram depicting the geometric configuration of the fluidic pathway considered in this study for the analysis. The dimensions of the fluidic channel are shown in the schematic. Coordinate system is attached at the center of the channel inlet. Taper angle of conical channel φ is shown at the right.

Figure 2

(a) Plot showing the temporal variation of length of the liquid in the capillary. The solid line represents the results obtained from the present analysis while the markers are used to denote the reported results of Bandopadhyay et al. [27]. The other parameters considered are $\lambda =0.1$ and $\varepsilon =0.1$. A qualitative match between the present and published results is observed from the depicted plot in 2. (b) Plot showing the temporal variation of length of the liquid in the straight capillary. The results obtained from the present analysis pertaining to straight capillary are shown by solid line, and the markers are used to denote the reported results of Bandopadhyay et al. [27]. The other parameters considered are $\lambda =0.01$ and $\varepsilon =0.1$. A fairly accurate match between the present and published results is observed from the depicted plot in 2.

Figure 3

(a) Plot showing the temporal variation of length of the liquid in the capillary, obtained for three different values of taper angle φ corresponding to $\alpha \left(=0.01\right)$, $2\alpha \left(=0.02\right),$ and $3\alpha \left(=0.03\right)$; (b) temporal variation of the velocity, obtained for taper angle φ corresponding to $\alpha$. The other parameters considered for plotting are $\lambda =0.01\mathrm{s},\eta =0.01\mathrm{Pa}\mathrm{s},{\theta }_{\mathrm{s}}=45{}^{\circ }$. The different regimes of imbibition are observed from the depicted variation. Inset shows the zoomed-in view of regime I.

Figure 4

The temporal variation of length of the liquid in the capillary, obtained for different values of contact (static) angle ${\theta }_s={15}^{\circ }$, ${30}^{\circ },{45}^{\circ }$, and ${60}^{\circ }$. The other parameters considered for this plot are $\lambda =0.01\mathrm{s}$, η = 0.01 Pa s, $\phi ={\tan }^{-1}\alpha$. Inset shows the imbibition rate during initial stage. With increasing the magnitude of the static contact angle, the imbibition rate slows down.

Figure 5

Plot of the temporal variation of filling length in the capillary, obtained for different values of $\lambda$ $=0.01$ s, 0.10 s, and 1.0 s, respectively. The other parameters are η = 0.01 Pa s, $\phi ={\tan }^{-1}\alpha ,{\theta }_{\mathrm{s}}={45}^{\circ }$. Inset shows the imbibition rate during initial stage. As the magnitude of relaxation time $\lambda$ becomes less, the filling time becomes larger.

Figure 6

Variation of filling length in the capillary versus time. The depicted plots are obtained for different values of $\eta =0.01\mathrm{Pa}\mathrm{s},0.0125\mathrm{Pa}\mathrm{s}$, and $0.0150\mathrm{Pa}\mathrm{s}$, respectively. The other parameters considered for this plot are $\lambda =0.01\mathrm{s}$, $\phi ={\tan }^{-1}\alpha ,{\theta }_{\mathrm{s}}={45}^{\circ }$. With increasing the magnitude of the apparent viscosity, the filling time becomes higher.

Figure 7

Plot of the scaling analysis of the imbibition dynamics. Filling behavior in three different regimes is shown through the scaling estimate between the length of the liquid column imbibed into the capillary versus time. The depicted plots obtained for different parametric values pertinent to this analysis are $\lambda =0.01\mathrm{s}$, $\phi ={\tan }^{-1}\alpha$, ${\theta }_{\mathrm{s}}={45}^{\circ }$ and $=0.01\mathrm{Pa}\mathrm{s}$.