Velocity-Only Continuous Data Assimilation and Exponential Synchronization for Two-Dimensional Non-Newtonian Micropolar Fluid Flows

Abstract

A velocity-only continuous data assimilation algorithm is formulated for a two-dimensional incompressible micropolar fluid whose Cauchy stress follows a regularized shear-thinning power law. The reference state consists of a solenoidal velocity and a scalar microrotation, whereas the observer is supplied only with coarse velocity measurements through a linear interpolant. No angular-velocity data are imposed. The error equations retain the nonlinear stress difference and the complete velocity–microrotation coupling. Strong monotonicity on an absorbing strain set, the interpolant approximation inequality, and two-dimensional interpolation estimates yields an explicit differential inequality for a coupled error energy. If the observational spacing is small enough to absorb the interpolation defect and the nudging gain exceeds the reference-gradient production rate, both velocity and microrotation converge exponentially to the reference solution. A twelve-shell Fourier–Galerkin comparison system illustrates the gain–resolution tradeoff. For the baseline shear-thinning index p=1.6, observing the first three shells gives a critical gain of 0.375; gains 0.50, 0.80, and 1.50 produce decay exponents 0.107, 0.300, and 0.407, respectively. The analysis isolates the mechanism by which unobserved microrotation synchronizes through dissipative coupling to the nudged velocity field.

Introduction

The text presents a velocity-only data assimilation method for two-dimensional shear-thinning micropolar fluids. Micropolar fluid theory extends classical fluid mechanics by giving fluid particles an independent microrotation, making it suitable for fluids with rigid microstructure such as suspensions and polymeric liquids.

The paper builds on established theories of micropolar fluids, generalized Newtonian fluids, and continuous data assimilation. Its central question is whether coarse measurements of velocity alone can reconstruct both velocity and microrotation, even though the rotational field is not directly observed.

The model consists of coupled equations for velocity u and microrotation ω, with a power-law stress
S(A)=2ν0(δ2+?A?2)(p−2)/2A,1<p≤2,
where p<2 represents shear-thinning behavior. The regularization parameter δ>0 avoids singular viscosity at zero strain.

The proposed observer modifies only the velocity equation by adding an AOT-type nudging term,
−μIh(v−u),
where μ is the nudging gain and h is the observation scale. No artificial feedback is introduced into the microrotation equation. The key idea is that the coupling between velocity and microrotation, together with rotational diffusion and damping, allows the unobserved spin field to be reconstructed from velocity observations.

For the synchronization analysis, the errors are defined as
e=v−u,η=ξ−ω.
Subtracting the reference and observer systems produces coupled error equations. The analysis relies on monotonicity of the nonlinear stress operator, uniform bounds on the strain after trajectories enter an absorbing set, and the dissipative micropolar coupling. In particular, retaining the coupling in the form
κ?curl?e−2η?22
preserves its nonnegative contribution to the energy estimate rather than estimating the two fields separately.

A major point is that, for 1<p<2, the effective coercivity of the stress depends on the maximum admissible strain. Therefore, the synchronization condition cannot simply use the zero-shear viscosity ν0. Instead, it introduces an effective viscosity νG determined by the absorbing-set bound G. This affects the required relationship between observation resolution h and nudging strength μ. When p=2, the model becomes Newtonian and νG becomes independent of G, recovering the familiar AOT-type scaling.

Main contributions

• Velocity-only observation: the observer measures and nudges velocity while leaving microrotation completely unobserved.

• Preservation of micropolar coupling: the velocity–spin terms are combined into a nonnegative square, strengthening the energy estimate.

• Treatment of shear thinning: the analysis explicitly incorporates the strain-dependent effective viscosity νG.

• Exponential synchronization: the intended theorem provides an explicit gain–resolution condition under which both velocity and microrotation errors decay exponentially.

• Numerical validation: a Fourier–Galerkin comparison is used to test the theoretical synchronization behavior.

Conclusion

A velocity-only continuous data assimilation scheme has been established for two-dimensional regularized shear-thinning micropolar flow. The observer uses coarse velocity measurements and no microrotation data. Monotonicity of the nonlinear stress, the interpolant approximation property, and the exact micropolar cross-dissipation yield a closed coupled error estimate. When the observation scale resolves the interpolation defect and the gain exceeds the uniform convective-production coefficient, the velocity and spin errors satisfy E(t)?E(0) e^(-2?t). The result explains complete-state recovery as a consequence of velocity feedback, reciprocal curl coupling, spin diffusion, and angular damping. The Fourier–Galerkin comparison confirms two design principles. Observations must include every productive low shell, and increasing gain beyond velocity locking cannot overcome the intrinsic microrotation timescale. Stronger shear thinning raises the critical gain by reducing ?_G, but it has little influence on the final decay rate once spin relaxation becomes limiting. These conclusions provide a rigorous basis for velocity-based reconstruction in microstructured non-Newtonian flows.

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Copyright

Copyright © 2026 Subhash Kumar, Dr. H. K. Pandey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.