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Deformation of ambient chemical gradients by sinking spheres
- Bryce G. Inman, Christopher J. Davies, Carlos R. Torres, Peter J. S. Franks
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- Journal:
- Journal of Fluid Mechanics / Volume 892 / 10 June 2020
- Published online by Cambridge University Press:
- 08 April 2020, A33
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- Article
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A sphere sinking through a chemical gradient drags fluid with it, deforming the gradient. The sphere leaves a trail of gradient enhancement that persists longer than the velocity disturbance in the Reynolds $10^{-2}\leqslant Re\leqslant 10^{2}$, Froude $10^{-1}\leqslant Fr\leqslant 10^{3}$ and Péclet $10^{2}<Pe\leqslant 10^{6}$ regime considered here. We quantify the enhancement of the gradient and the diffusive flux in the trail of disturbed chemical left by the passing sphere using a combination of numerical simulations and scaling analyses. When $Fr$ is large and buoyancy forces are negligible, dragged isosurfaces of chemical form a boundary layer of thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}$ around the sphere with diameter $l$. We derive the scaling $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}/l\sim \mathit{Pe}^{-1/3}$ from the balance of advection and diffusion in the chemical boundary layer. The sphere displaces a single isosurface of chemical a maximum distance $\mathit{L}_{Def}$ that increases as $\mathit{L}_{Def}/l\sim l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim \mathit{Pe}^{1/3}$. Increased flux through the chemical boundary layer moving with the sphere is described by a Sherwood number, $Sh\sim l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim \mathit{Pe}^{1/3}$. The gradient enhancement trail extends much farther than $\mathit{L}_{Def}$ as displaced isosurfaces slowly return to their original positions through diffusion. In the reference frame of a chemical isosurface moving past the sphere, a new quantity describing the Lagrangian flux is found to scale as $\mathit{M}\sim (\mathit{L}_{Def}/l)^{2}\sim \mathit{Pe}^{2/3}$. The greater $\mathit{Pe}$ dependence of $\mathit{M}$ versus $Sh$ demonstrates the importance of the deformation trail for determining the total flux of chemical in the system. For $\mathit{Fr}\geqslant 10$, buoyancy forces are weak compared to the motion of the sphere and the preceding results are retained. Below $\mathit{Fr}=10$, an additional Froude dependence is found and $l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim Sh\sim Re^{1/6}Fr^{-1/6}Pe^{1/3}$. Buoyancy forces suppress gradient deformation downstream, resulting in $\mathit{L}_{Def}/l\sim Re^{-1/3}Fr^{1/3}Pe^{1/3}$ and $\mathit{M}\sim Re^{-1/3}Fr^{1/3}Pe^{2/3}$. The productivity of marine plankton – and therefore global carbon and oxygen cycles – depends on the availability of microscale gradients of chemicals. Because most plankton exist in the fluids regime under consideration, this work describes a new mechanism by which sinking particles and plankton can stir weak ambient chemical gradients a distance $\mathit{L}_{Def}$ and increase chemical flux in the trail by a factor of $\mathit{M}$.