(454b) Transition from Rapid to Tranquil Condition in Thin Film Laminar Flow of Viscoplastic Fluid ? Shallow Water Analysis and Numerical Simulation

Viscoplastic fluids are often found in nature in the form of mud, lava, snow avalanches etc. Such kind of fluids are also widely encountered in various engineering applications like different polymeric suspensions, molten metal, liquid chocolate, paint, toothpaste, slurry, and so on. The flow behaviour of viscoplastic fluids can be predicted with reasonable accuracy by Herschel-Bulkley (HB) model, expressed by Eq. (1).

The flow region where shear stress is greater than or equal to yield stress is termed as shear/ yielded zone and the region where shear stress is below yield stress, is termed as plug/ unyielded zone. The surface separating plug and shear zone is called the yield surface. One of the major difficulties in analysing HB model is the attainment of singularity at the yield surface, which is not known a priori. In order to overcome the drawback, different modifications are proposed among which the proposition offered by Papanastasiou [1] is commonly used. The HB model with Papanastasiou regularization parameter m takes the form given by Eq. (2). Equation (2a) reduces to ideal HB model [Eq. (1a)] when m→∞ and predicts the ideal power-law behavior for m = 0.

Several studies are reported on confined and open channel flow of HB fluid. Nevertheless, thin film flow of HB fluids is comparatively less addressed. A unique phenomenon in free surface flow is hydraulic jump which marks the transition from rapid (Fr>1) to tranquil (Fr>1) flow, where Fr is Froude number, the ratio of inertial to gravitational force. A review of past literature shows that planar hydraulic jump in HB fluid has rarely been addressed. There is a study on planar jump [2] but till date to the best of authors’ knowledge, no study is found on thin film planar jump in rectangular channels. In the present study, we investigate laminar planar hydraulic jump of Herschel-Bulkley fluid in a rectangular channel. The investigation comprises of theoretical analysis based on shallow water approximation and numerical simulation using phase-field method in COMSOL Multiphysics. Both theory and simulation have adopted Papanastasiou regularized HB model to avoid singularity during solution.

2. Shallow water analysis

A schematic of flow geometry in presented in Fig. 1. Viscoplastic fluid enters channel ABCDA of length L and height H at a velocity of u_in through a thin vertical slit of height, h_in. Coordinates x and y refer to the direction of fluid flow and the direction perpendicular to flow respectively and u (x,y) and v(x,y) are the corresponding velocity components. The total height of liquid film is h(x) and the thickness of shear zone at any particular x is h_o(x). The velocity increases from u=0 at the wall (y=0) to u=u_m at the yield surface. It assumes a semi-parabolic shape for n=1 that corresponds to Bingham plastic fluid. The plug zone flows at a uniform velocity u=u_m independent of y but changes along x.

Taking cue from literature [3], self-similar velocity profile is considered and the 2-D steady state continuity equation and Navier-Stokes equation are vertically integrated. This gives the equation for steady state free surface profile [Eq. (3)] of HB Papanastasiou regularized fluid.

The equation is solved using numerical integration by Runge-Kutta 4^th order method. The forward integration starting from the inlet provides upstream profile till the critical point where a singularity occurs. Accordingly, the profile downstream of jump is obtained by the backward integration starting near the channel exit. The upstream and downstream profiles are then connected by a Rayleigh shock.

3. Numerical simulation

Numerical simulation is performed for additional insight into the jump physics by exploring the details near the critical point where shallow water theory (SWT) fails. The simulation is done using phase-field method in COMSOL Multiphysics 6.0, a finite element method-based software package. The phase-field method considers free energy minimization principle to track the interface between the phases by diffused interface tracking method in COMSOL. It provides mass conservation and satisfactory efficacy in capturing topological changes of the free surface. We consider air-liquid two phase laminar flow module coupled with the phase-field module. The laminar flow module solves two-dimensional Navier-Stokes equation and continuity equation to estimate the pressure and velocity fields depending on the value of phase-field variable (φ) estimated by solving Cahn-Hilliard equation of phase-field module. φ is directly related to the phase volume factions where the interface is detected by the isocontour of each phase volume fraction is 0.5. φ= -1 for liquid and +1 for air and smoothly varies from (-1) to (+1) at the interface. The boundary conditions for the numerical simulation are depicted in Fig. 1. Grid and time independence studies are done to choose appropriate grid size and time step value.

4.Validation

The steady state free surface profiles obtained from theory and simulation are compared for different input parameters pertaining to practical HB Papanastasiou fluids. The fluids are aqueous suspension of 3-4 (w/v)% bentonite with 0.1-0.3 (w/v)% carboxymethyl cellulose or 0.05-0.15(w/v)% xanthan gum in order to ensure a wide range of yield stress, flow behaviour and consistency index. We obtain a good trend matching of free surface profiles obtained from theory and simulation for all the real fluids considered. Two typical cases are shown in Fig. 2.

5.Results and discussions

After validation, we use theory and simulation to explore different aspect of jump physics and observe the effect of channel dimension, inflow velocity and rheology (yield stress, flow behaviour and consistency index) on jump formation. The results show that yield stress has same effect for all liquids irrespective of n. Further, an increase in yield stress increases the thickness of plug zone and shifts jump towards channel inlet with greater jump strength and film thickness as shown in Fig. 3. Although the effect of Re_in and Fr_in on jump location is similar, these influence the jump strength in opposite ways. An increase in Fr_in and decrease in Re_in increases jump strength and an increase in both shifts jump towards the exit.

We have also used the simulation results to plot phase diagrams in (Fr_in – HB_in) and (Re_in – HB_in) planes for different jump types.

6.Conclusion

The study focuses on laminar planar hydraulic jump during thin film flow of Herschel-Bulkley Papanastasiou regularized fluid through shallow water analysis and numerical simulation. The theory although is based on simplified assumptions of self-similar velocity profile throughout the domain, it gives results close to numerical simulation. We observe the effect of yield stress, flow consistency and behaviour indices on the film profile, jump location and jump strength and present the range of existence of different jump type as phase diagrams.

7. References

1. Papanastasiou, T. C. 1987 Flows of Materials with Yield. Rheol. 31, pp. 385–404.
2. Ugarelli, R. and Federico, V. D. 2007 Transition from supercritical to subcritical regime in free surface flow of yield stress fluids. Res. Lett. 34, L21402.
3. Dhar, M., Das, G. & Das, P. K. 2020 Planar hydraulic jumps in thin film flow. J. of Fluid Mech. 884, 1–26.