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Flow of a colloidal solution in an orthogonal rheometer

Krishna Kaushik Yanamundra, Chandler C. Benjamin, Kumbakonam Ramamani Rajagopal

TL;DR

This work analyzes the flow of a colloidal solution in an orthogonal rheometer between eccentrically rotating parallel disks using generalized stress-power-law and stress-limiting constitutive relations. It derives and nondimensionalizes a boundary-value problem for pseudo-planar, nearly-viscometric flow and solves the resulting ODEs numerically to explore how material and geometric parameters influence boundary-layer formation and stress distributions. The study reveals boundary layers at low Reynolds numbers across several non-Newtonian classes, including stress-limiting and large-zero-fluidity fluids, and uncovers rich behaviors such as non-monotone and S-type constitutive relations and potential solution multiplicity. Overall, the results provide insights into how microstructural colloidal nonlinearity translates into rheometric responses, informing material design and interpretation of orthogonal rheometer measurements.

Abstract

The flow of a colloidal solution between two parallel disks rotating with the same angular velocity about two non-coincident axes was studied. The problem has been approached from two perspectives, the first wherein the stress is expressed in terms of a power-law of kinematical quantities, and the second wherein we consider a non-standard model where the symmetric part of the velocity gradient is given by a power-law of the stress. For a range of power-law exponents, the class of models are non-invertible. By varying the material and geometric parameters, changes in the flow behaviour at different Reynolds numbers were analysed. We find that pronounced boundary layers develop even at low Reynolds numbers based on the power-law exponents. The new class of stress power-law fluids and fluids that exhibit limiting stress also show the ability to develop boundary layers.

Flow of a colloidal solution in an orthogonal rheometer

TL;DR

This work analyzes the flow of a colloidal solution in an orthogonal rheometer between eccentrically rotating parallel disks using generalized stress-power-law and stress-limiting constitutive relations. It derives and nondimensionalizes a boundary-value problem for pseudo-planar, nearly-viscometric flow and solves the resulting ODEs numerically to explore how material and geometric parameters influence boundary-layer formation and stress distributions. The study reveals boundary layers at low Reynolds numbers across several non-Newtonian classes, including stress-limiting and large-zero-fluidity fluids, and uncovers rich behaviors such as non-monotone and S-type constitutive relations and potential solution multiplicity. Overall, the results provide insights into how microstructural colloidal nonlinearity translates into rheometric responses, informing material design and interpretation of orthogonal rheometer measurements.

Abstract

The flow of a colloidal solution between two parallel disks rotating with the same angular velocity about two non-coincident axes was studied. The problem has been approached from two perspectives, the first wherein the stress is expressed in terms of a power-law of kinematical quantities, and the second wherein we consider a non-standard model where the symmetric part of the velocity gradient is given by a power-law of the stress. For a range of power-law exponents, the class of models are non-invertible. By varying the material and geometric parameters, changes in the flow behaviour at different Reynolds numbers were analysed. We find that pronounced boundary layers develop even at low Reynolds numbers based on the power-law exponents. The new class of stress power-law fluids and fluids that exhibit limiting stress also show the ability to develop boundary layers.

Paper Structure

This paper contains 16 sections, 36 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Constitutive relations
  3. Boundary value problem
  4. Non-dimensionalization
  5. Results
  6. Navier-Stokes
  7. Stress-thinning case
  8. Stress-thickening case
  9. Non-monotone constitutive relation with one inflection point
  10. S-type non-monotone constitutive relation
  11. Stress-limiting case
  12. Fluids with large zero-stress fluidity
  13. Shear-thickening case
  14. Shear-thinning case
  15. Shear-limiting case
  16. ...and 1 more sections

Figures (23)

  • Figure 1: Behaviour of (\ref{['Eqn1']}) for some values of $m$ ($\alpha=1$, $a_1=0$, $a_2=1$, $\beta=1$)
  • Figure 2: Behaviour of (\ref{['Eqn1']}) for some values of $a_1$ ($\alpha=1$, $n=-2$, $a_2=1$, $\beta=1$)
  • Figure 3: Behaviour of $\alpha_g(\|\mathbb{S}\|)$ for (\ref{['Eqn1']}) for some values of $m$ ($\alpha=1$, $a_1=1$, $a_2=0$, $\beta=1$)
  • Figure 4: Schematic diagram of an orthogonal rheometer
  • Figure 5: Solution for a Navier-Stokes Fluid at various Reynolds numbers
  • ...and 18 more figures