Acceleration-Induced Shear Viscosity

• Acceleration-induced shear viscosity is a phenomenon where non-uniform or time-dependent accelerations generate shear stresses and couple to normal stresses.
• It is analyzed via molecular dynamics, relativistic hydrodynamics, and quantum field theory to capture memory effects and complex force profiles in diverse systems.
• Understanding these effects sharpens simulations of ultrafast flows and enhances predictions in cosmological, accretion, and viscoelastic contexts.

Acceleration-induced shear viscosity refers to viscous phenomena arising from non-uniform or time-dependent accelerations in fluids and solids, leading to both shear stress responses and unique couplings to normal stresses, entropy production, and energy dissipation. Unlike classical viscosity—often modeled via velocity gradients in the Navier–Stokes framework—acceleration-induced effects appear when the acceleration field $$\mathbf{a}(\mathbf{r},t) = \partial_t \mathbf{v}(\mathbf{r},t)$$ itself exhibits spatial inhomogeneity or temporal structure, or in contexts where non-inertial frames (such as Rindler observers, cosmological expansion, or strong curvature near black holes) bring fundamentally new kinematic sources of dissipation. The study of acceleration-induced shear viscosity traverses molecular simulations, continuum theory, relativistic hydrodynamics, horizon-thermodynamics, and rheology under finite deformations.

1. Acceleration-Induced Shear Viscosity in Simple Fluids

Recent advances have extended the conventional velocity-gradient-based viscous response to include acceleration-derived contributions. In molecular and statistical settings, the viscous force density is split into velocity and acceleration-driven bulk and shear parts, built from the divergence and curl of both the velocity and acceleration fields. The general constitutive relation is given by time-convolutions with exponentially decaying memory kernels specific to velocity or acceleration-driven forces: $$f_s(\mathbf{r},t) = -\frac{1}{\rho(\mathbf{r},t)} \int_0^t d\tau\, K_s^v(t-\tau) \nabla \times [\rho(\mathbf{r},t) \rho(\mathbf{r},\tau) \nabla \times \mathbf{v}(\mathbf{r},\tau)] -\frac{1}{\rho(\mathbf{r},t)} \int_0^t d\tau\, K_s^a(t-\tau) \nabla \times [\rho(\mathbf{r},t) \rho(\mathbf{r},\tau) \nabla \times \mathbf{a}(\mathbf{r},\tau)]$$ where $K_s^v$ and $K_s^a$ are shear memory kernels for velocity and acceleration, respectively. Empirical measurements via custom-flow molecular dynamics reveal that the shear acceleration viscosity kernel $K_s^a$ has a comparable memory time to $K_s^v$ but is diminished in amplitude; its omission leads to substantial inaccuracies in the predicted stress and force profiles for fast, inhomogeneous flows (Renner et al., 2022).

2. Relativistic and Cosmological Contexts: Backreaction and Shear Viscosity

Acceleration-induced shear viscosity acquires central significance in relativistic hydrodynamics and cosmology. In the framework of a perturbed Friedmann–Lemaître–Robertson–Walker metric, the dissipation from local velocity gradients and peculiar velocities (in dark-sector fluids or general baryonic matter) contributes a backreaction term $D$: $$D = \frac{1}{a^2} \langle \eta [(\partial_i v_j)(\partial_i v_j) + (\partial_i v_j)(\partial_j v_i) - \frac{2}{3} (\partial_i v_i)^2 ] \rangle + \cdots$$ where $\eta$ is the shear viscosity. When the magnitude of $D$ is sufficiently large, this term shifts the deceleration parameter $q$ into the accelerating regime ($q < 0$) without requiring a cosmological constant: $$\frac{4\pi G_N D}{3H^3} > \frac{1 + 3\hat w}{1 - 3\hat w}$$ The effect is directly linked to the power spectra of velocity perturbations and the microscopic fluid properties, with "shear acceleration viscosity" manifesting through enhanced energy dissipation during non-linear structure formation and inhomogeneous flows (Floerchinger et al., 2014).

3. Accelerated Frames, Horizon Thermodynamics, and Quantum Viscosity

Quantum field theory in non-inertial (accelerated) frames reveals that the Minkowski vacuum behaves effectively as a quantum fluid with finite temperature (Unruh effect) and a universal shear viscosity, even in absence of physical particle collisions. The kinematically induced viscosity, derived via Kubo formulas from stress-tensor correlators in Rindler space, yields

$$\eta = \frac{1}{240\pi^2 l_c^2}, \quad s = \frac{1}{60\pi l_c^2}$$

for massless Dirac fields, and a matching Kovtun–Son–Starinets ratio $\eta/s = 1/4\pi$ for both spin-½ and spin-1 fields, saturating the universal bound. Locally, this ratio can dip below $1/4\pi$ near the stretched horizon, reaching $1/8\pi$ on the membrane surface. The quantum viscosity is a consequence of entanglement across the horizon and is independent of microscopic coupling strengths (Lapygin et al., 25 Feb 2025).

4. Relativistic Accretion Flows: Shear Viscosity from Radial Drift Acceleration

In accretion disks around Kerr black holes, any nonzero radial acceleration directly sources components of the shear tensor: $$\sigma^{\mu\nu} = \frac{1}{2} [u^\mu_{;r} h^{r\nu} + u^\nu_{;r} h^{r\mu}] - \frac{1}{3}\Theta h^{\mu\nu}$$ and the associated shear stress is given by

$\pi^{\mu\nu} = -2\eta\sigma^{\mu\nu}$

where $\eta$ is the shear viscosity, $u^\mu$ the four-velocity, and $h^{\mu\nu}$ the projection tensor. Radial profiles of the four-velocity form the acceleration field, and regions with strong acceleration (near ISCO) manifest maximal shear stresses; for nearly geodesic flow, these stresses vanish. Thus, acceleration in the drift velocity modulates shear viscosity and the angular momentum transport in relativistic disks (Moghaddas, 2017).

5. Continuum Simulations and Numerical Implementation

In particle-based methods such as Godunov-SPH, the viscous acceleration is discretized to include second derivatives of the velocity field, critical for capturing acceleration-induced viscous effects: $$\mathbf{a}_{\rm visc} = \dot{\mathbf{v}}_\nu = \frac{1}{\rho} \nabla\cdot(\rho\nu\boldsymbol{\sigma})$$ where $\boldsymbol{\sigma}$ encodes the symmetrized velocity gradients. The double-summation approach for evaluating second derivatives resolves the noise associated with naive Laplacian approximations. Simulation studies confirm analytic and experimental predictions across laminar and turbulent regimes, including dispersion and interface stability in variable viscosity layers (Cha et al., 2016).

6. Finite Deformation, Viscoelasticity, and Acceleration Coupling

In incompressible viscoelastic solids subject to finite acceleration, acceleration-induced shear viscosity is central to nontrivial rheological phenomena. The Gordon–Schowalter derivative formulates the evolution of the deviatoric stress tensor under combined acceleration and velocity gradients, yielding

$\eta_{\rm eff}(a,t) = \frac{S_{12}(t)}{u t}$

where $S_{12}(t)$ is the time-dependent shear stress and $u$ the imposed acceleration. Notably, under acceleration the shear rate couples into normal stresses—manifesting as the Poynting effect—and the variance of stress tensor components is tunable via the choice of objective derivative parameter $a$. Randomized shear rates further reveal the shifting of stress dispersions among principal axes, illustrating the direct impact acceleration has on macroscopic viscosity and stress profiles (Kozhukhov, 14 Dec 2025).

7. Significance, Cross-Disciplinary Connections, and Open Questions

Acceleration-induced shear viscosity bridges condensed matter physics, cosmology, quantum field theory, and nonlinear rheology. It is pivotal to modeling fast, inhomogeneous flows; interpreting the quantum nature of spacetime horizons; and understanding dissipative mechanisms in accelerating cosmological backgrounds. Its quantification via power spectra and memory kernels, and its saturation of universal entropy bounds, continue to stimulate research spanning simulation, analytic theory, and experiment.

A plausible implication is that nonlocal and acceleration-coupled viscosities will become standard across simulations of ultra-fast flows and multi-component systems, and may reveal deeper connections between quantum entanglement and emergent hydrodynamics in both gravitational and condensed matter contexts.