On steady linear diffusion-driven flow

Page, Michael A.; Johnson, E. R.

doi:10.1017/s002211200800178x

Cited by 6 publications

(34 citation statements)

References 14 publications

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“…As in Page & Johnson (2008), there are three key flow regions for steady 'diffusiondriven flow' in a closed container: on vertical or sloping surfaces there are 'buoyancy layers'; on some horizontal lines there can be thin 'R 1/3 layers'; while the remainder is the 'outer flow'. Modifications to each of these regions in the nonlinear case are described below.…”

Section: Flow Regionsmentioning

confidence: 99%

“…(In contrast, Page & Johnson (2008) used the constant background temperature gradient dT * 0 /dz * to determine the temperature scale, so their scaled linearized temperature, written asT here, is equivalent to T = 4z + 2 √ σT .) The velocity components of the flow are non-dimensionalized with L * and the buoyancy frequency N * = (g * α * T * /L * ) 1/2 , so (u, w) = (u * , w * )/N * L * , without the factor of used in Page & Johnson (2008). The pressure in the fluid is predominantly hydrostatic, and variations are quantified by a scaled pressure p that is nondimensionalized with ρ * 00 (N * L * ) 2 .…”

Section: Configuration and Governing Equationsmentioning

confidence: 99%

“…The governing equations for steady flow can then be written as Taking into account the different scaling of the pressure and temperature here, without the factors of √ σ used in Page & Johnson (2008) and Wunsch (1970), the nonlinear equations (2.2)-(2.4) are equivalent to (1)-(3) in Wunsch (1970) when his = 1. It is assumed that R 1, with σ taken to be O(1) with respect to R, and under those conditions it will be seen that p and T are dominantly functions of z over most of the container.…”

Section: Configuration and Governing Equationsmentioning

confidence: 99%

“…It is assumed that R 1, with σ taken to be O(1) with respect to R, and under those conditions it will be seen that p and T are dominantly functions of z over most of the container. The only significant term on the left-hand sides of (2.2)-(2.4) turns out to be wT z , and the z-variation of the coefficient of w in that term provides the key difference from the linear analysis in Page & Johnson (2008).…”

Section: Configuration and Governing Equationsmentioning

confidence: 99%

“…More recently, Page & Johnson (2008) extended that analysis to the case of a contained fluid when the imposed temperature gradient at the boundary is small relative to the background temperature gradient. Their governing equations are linear, which assists in the analysis of the flow structure, but they are not applicable to many practical situations.…”

Section: Introductionmentioning

confidence: 99%

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Steady nonlinear diffusion-driven flow

Page

Johnson

2009

J. Fluid Mech.

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An imposed normal temperature gradient on a sloping surface in a viscous stratified fluid can generate a slow steady flow along a thin 'buoyancy layer' against that surface, and in a contained fluid the associated mass flux leads to a broader-scale 'outer flow'. Previous analysis for small values of the Wunsch-Phillips parameter R is extended to the nonlinear case in a contained fluid, when the imposed temperature gradient is comparable with the background temperature gradient. As for the linear case, a compatibility condition relates the buoyancy-layer mass flux along each sloping boundary to the outer-flow temperature gradient. This condition allows the leadingorder flow to be determined throughout the container for a variety of configurations.

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Section: Flow Regionsmentioning

confidence: 99%

Section: Configuration and Governing Equationsmentioning

confidence: 99%

Section: Configuration and Governing Equationsmentioning

confidence: 99%

Section: Configuration and Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Steady nonlinear diffusion-driven flow

Page

Johnson

2009

J. Fluid Mech.

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Low-Reynolds-number diffusion-driven flow around a horizontal cylinder

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Page

2017

J. Fluid Mech.

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Diffusion-driven flow occurs when any insulated sloping surface is in contact with a quiescent stratified and viscous fluid. This startling fluid motion is generally very slow, and is caused by a hydrostatic pressure imbalance due to bending of isotherms near the surface. In contrast to previous studies of the phenomenon, the low-Reynolds-number case is considered here, for which the induced steady motion is influenced by viscous and density diffusion over a much larger length scale than the size of the insulated object. The relevant linear equations of steady two-dimensional motion in a linearly stratified fluid are solved for a circular cylinder using a matched two-region approach that yields analytical solutions for the streamfunction and the density variations both close to and far from the object. The exact analytical expressions for the solutions in the ‘outer-flow region’ are new, and after matching enable accurate solutions to be evaluated easily at any point. Similar qualitative behaviour is expected under similar conditions near isolated objects of other shapes, including for a sphere. Implications for multiple objects are also discussed.

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The rotation of a sedimenting spheroidal particle in a linearly stratified fluid

2021

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We derive analytically the angular velocity of a spheroid, of an arbitrary aspect ratio $\kappa$ , sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit $Re, Ri_v \ll 1$ , where $Re = \rho _0UL/\mu$ and $Ri_v =\gamma L^3\,g/\mu U$ , the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here, $L$ is the spheroid semi-major axis, $U$ an appropriate settling velocity scale, $\mu$ the fluid viscosity and $\gamma \ (>0)$ the (constant) density gradient characterizing the stably stratified ambient, with the fluid density $\rho_0$ taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an $O(Re)$ inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an $O(Ri_v)$ hydrostatic contribution due to the ambient stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification whose nature depends on $Pe$ ; $Pe = UL/D$ being the Péclet number with $D$ the diffusivity of the stratifying agent. For $Pe \ll 1$ , this contribution is $O(Ri_v)$ and orients prolate spheroids edgewise for all $\kappa \ (>1)$ . For oblate spheroids, it changes sign across a critical aspect ratio $\kappa _c \approx 0.41$ , orienting oblate spheroids with $\kappa _c < \kappa < 1$ edgewise and those with $\kappa < \kappa _c$ broadside-on. For $Pe \ll 1$ , the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For $Pe \gg 1$ , the hydrodynamic contribution is dominant, being $O(Ri_v^{{2}/{3}}$ ) in the Stokes stratification regime characterized by $Re \ll Ri_v^{{1}/{3}}$ , and orients the spheroid edgewise regardless of $\kappa$ . Consideration of the inertial and large- $Pe$ stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the $Ri_v/Re^{{3}/{2}}$ – $\kappa$ plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large $Pe$ are broadly consistent with observations.

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On steady linear diffusion-driven flow

Cited by 6 publications

References 14 publications

Steady nonlinear diffusion-driven flow

Steady nonlinear diffusion-driven flow

Low-Reynolds-number diffusion-driven flow around a horizontal cylinder

The rotation of a sedimenting spheroidal particle in a linearly stratified fluid

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