5 results
Isotropically active colloids under uniform force fields: from forced to spontaneous motion
- Saikat Saha, Ehud Yariv, Ory Schnitzer
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- Journal:
- Journal of Fluid Mechanics / Volume 916 / 10 June 2021
- Published online by Cambridge University Press:
- 14 April 2021, A47
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We consider the inertia-free motion of an isotropic chemically active particle which is exposed to a weak uniform force field. This problem is characterised by two velocity scales, a ‘chemical’ scale associated with diffusio-osmosis and a ‘mechanical’ scale associated with the external force. The motion animated by the force deforms the originally spherically symmetric solute cloud surrounding the particle, thus resulting in a concomitant diffusio-osmotic flow which, in turn, modifies the particle speed. A weak-force linearisation furnishes a closed-form expression for the particle velocity as a function of the intrinsic Péclet number $\alpha$ associated with the chemical velocity scale. We find that the predicted velocity may become singular at $\alpha =4$, and that this happens under the same conditions on the surface parameters for which the associated unforced problem is known to exhibit, for $\alpha >4$, a symmetry-breaking instability giving rise to steady spontaneous motion (Michelin, Lauga & Bartolo, Phys. Fluids, vol. 25, 2013, 061701). Here, a local analysis in a distinguished region near $\alpha =4$, wherein the velocity scaling is amplified, yields a closed-form description of the imperfect bifurcation which bridges between a perturbed stationary state and a perturbed spontaneous motion. Remarkably, while the direction of spontaneous motion in the absence of an external force is random, in the perturbed case that motion is rendered steady solely in the directions parallel or antiparallel to the external force.
Burnett-order constitutive relations, second moment anisotropy and co-existing states in sheared dense gas–solid suspensions
- Saikat Saha, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 887 / 25 March 2020
- Published online by Cambridge University Press:
- 21 January 2020, A9
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The Burnett- and super-Burnett-order constitutive relations are derived for homogeneously sheared gas–solid suspensions by considering the co-existence of ignited and quenched states and the anisotropy of the second moment of velocity fluctuations ($\unicode[STIX]{x1D648}=\langle \boldsymbol{C}\boldsymbol{C}\rangle ,C$ is the fluctuation or peculiar velocity) – this analytical work extends our previous works on dilute (Saha & Alam, J. Fluid Mech., vol. 833, 2017, pp. 206–246) and dense (Alam et al., J. Fluid Mech., vol. 870, 2019, pp. 1175–1193) gas–solid suspensions. For the combined ignited–quenched theory at finite densities, the second-moment balance equation, truncated at the Burnett order, is solved analytically, yielding expressions for four invariants of $\unicode[STIX]{x1D648}$ as functions of the particle volume fraction ($\unicode[STIX]{x1D708}$), the restitution coefficient ($e$) and the Stokes number ($St$). The phase boundaries, demarcating the regions of (i) ignited, (ii) quenched and (iii) co-existing ignited–quenched states, are identified via an ordering analysis, and it is shown that the incorporation of excluded-volume effects significantly improves the predictions of critical parameters for the ‘quenched-to-ignited’ transition. The Burnett-order expressions for the particle-phase shear viscosity, pressure and two normal-stress differences are provided, with their Stokes-number dependence being implicit via the anisotropy parameters. The roles of ($St,\unicode[STIX]{x1D708},e$) on the granular temperature, the second-moment anisotropy and the nonlinear transport coefficients are analysed using the present theory, yielding quantitative agreements with particle-level simulations over a wide range of ($St,\unicode[STIX]{x1D708}$) including the bistable regime that occurs at $St\sim O(5)$. For highly dissipative particles ($e\ll 1$) that become increasingly important at large Stokes numbers, it is shown that the Burnett-order solution is not adequate and further higher-order solutions are required for a quantitative agreement of transport coefficients over the whole range of control parameters. The latter is accomplished by developing an approximate super-super-Burnett-order theory for the ignited state ($St\gg 1$) of sheared dense gas–solid suspensions in the second part of this paper. An extremum principle based on viscous dissipation and dynamic friction is discussed to identify ignited–quenched transition.
Revisiting ignited–quenched transition and the non-Newtonian rheology of a sheared dilute gas–solid suspension
- Saikat Saha, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 833 / 25 December 2017
- Published online by Cambridge University Press:
- 03 November 2017, pp. 206-246
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The hydrodynamics and rheology of a sheared dilute gas–solid suspension, consisting of inelastic hard spheres suspended in a gas, are analysed using an anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the closed-form solutions for granular temperature and three invariants of the second-moment tensor are obtained as functions of the Stokes number ($St$), the mean density ($\unicode[STIX]{x1D708}$) and the restitution coefficient ($e$). Multiple states of high and low temperatures are found when the Stokes number is small, thus recovering the ‘ignited’ and ‘quenched’ states, respectively, of Tsao & Koch (J. Fluid Mech., vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional ($\unicode[STIX]{x1D708},St,e$)-space that delineates the regions of ignited and quenched states and their coexistence. The particle-phase shear viscosity and the normal-stress differences are analysed, along with related scaling relations on the quenched and ignited states. At any $e$, the shear viscosity undergoes a discontinuous jump with increasing shear rate at the ‘quenched–ignited’ transition. The first (${\mathcal{N}}_{1}$) and second (${\mathcal{N}}_{2}$) normal-stress differences also undergo similar first-order transitions: (i) ${\mathcal{N}}_{1}$ jumps from large to small positive values and (ii) ${\mathcal{N}}_{2}$ from positive to negative values with increasing $St$, with the sign change of ${\mathcal{N}}_{2}$ identified with the system making a transition from the quenched to ignited states. The superior prediction of the present theory over the standard Grad’s method and the Burnett-order Chapman–Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.
Normal stress differences, their origin and constitutive relations for a sheared granular fluid
- Saikat Saha, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 795 / 25 May 2016
- Published online by Cambridge University Press:
- 19 April 2016, pp. 549-580
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The rheology of the steady uniform shear flow of smooth inelastic spheres is analysed by choosing the anisotropic/triaxial Gaussian as the single-particle distribution function. An exact solution of the balance equation for the second-moment tensor of velocity fluctuations, truncated at the ‘Burnett order’ (second order in the shear rate), is derived, leading to analytical expressions for the first and second ($\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$) normal stress differences and other transport coefficients as functions of density (i.e. the volume fraction of particles), restitution coefficient and other control parameters. Moreover, the perturbation solution at fourth order in the shear rate is obtained which helped to assess the range of validity of Burnett-order constitutive relations. Theoretical expressions for both $\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$ and those for pressure and shear viscosity agree well with particle simulation data for the uniform shear flow of inelastic hard spheres for a large range of volume fractions spanning from the dilute regime to close to the freezing-point density (${\it\nu}\sim 0.5$). While the first normal stress difference $\unicode[STIX]{x1D615}_{1}$ is found to be positive in the dilute limit and decreases monotonically to zero in the dense limit, the second normal stress difference $\unicode[STIX]{x1D615}_{2}$ is negative and positive in the dilute and dense limits, respectively, and undergoes a sign change at a finite density due to the sign change of its kinetic component. It is shown that the origin of $\unicode[STIX]{x1D615}_{1}$ is tied to the non-coaxiality (${\it\phi}\neq 0$) between the eigendirections of the second-moment tensor $\unicode[STIX]{x1D648}$ and those of the shear tensor $\unicode[STIX]{x1D63F}$. In contrast, the origin of $\unicode[STIX]{x1D615}_{2}$ in the dilute limit is tied to the ‘excess’ temperature ($T_{z}^{ex}=T-T_{z}$, where $T_{z}$ and $T$ are the $z$-component and the average of the granular temperature, respectively) along the mean vorticity ($z$) direction, whereas its origin in the dense limit is tied to the imposed shear field.
Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method
- Saikat Saha, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 757 / 25 October 2014
- Published online by Cambridge University Press:
- 19 September 2014, pp. 251-296
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The non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $, pressure $p$, first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $, (ii) restitution coefficient $e$, (iii) dimensionless shear rate $R$, (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle $\phi $ between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier–Stokes order, recovering the known exact transport coefficients from the present analysis in the limit $\eta ,\phi \to 0$, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of $R$, $\eta $ and $\phi $ with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ($e \ll 1$). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$, $\mu $ and ${\mathcal{N}}_1$ over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ($e\sim 0.9$). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.