Abstract
A well-developed method to induce mixing on microscopic scales is to exploit flows generated by steady streaming. Steady streaming is a classical fluid dynamics phenomenon whereby a time-periodic forcing in the bulk or along a boundary is enhanced by inertia to induce a non-zero net flow. Building on classical work for simple geometrical forcing and motivated by the complex-shaped oscillations of elastic capsules and bubbles, we develop the mathematical framework to quantify the steady streaming of a spherical body with arbitrary axisymmetric time-periodic boundary conditions. We compute the flow asymptotically for small-amplitude oscillations of the boundary in the limit where the viscous penetration length scale is much smaller than the body. In that case, the flow has a boundary layer structure, and the fluid motion is solved by asymptotic matching. Our results, presented in the case of no-slip boundary conditions and extended to include the motion of vibrating free surfaces, recover classical work as particular cases. We illustrate the flow structure given by our solution and propose one application of our results for small-scale force generation and synthetic locomotion.
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Acknowledgements
This work was funded in part by the EPSRC (TS) and the European Union through a Marie Curie CIG Grant (EL).
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Appendices
Appendix 1: Out-of-phase streaming around a bubble
In order to apply the general steady streaming model (72) specifically to a bubble, the boundary condition of no tangential stress on the boundary of the spherical body,
needs to be applied. This will determine the angular motion on the surface of the bubble \(W_{n}\) and \(F_{n}\) in terms of a prescribed radial motion \(V_{n}\).
1.1 (a) Boundary condition
We denote the unit tangent vector to the body’s surface in the plane through the axis of axisymmetry \(\varvec{t}\) and the normal vector \(\varvec{n}\). Both can be calculated in terms of \(\mathbf{e}_{r}\) and \(\mathbf{e}_{\theta }\) measured from the centre of the rest position of the body giving
Using these equations, the no-stress condition can be expanded in terms of \(\epsilon \) giving the first two terms as
becoming
1.2 (b) Leading-order solution
At \(O(\epsilon )\), Eq. 131 reduces to the non-dimensional equation
By substituting in the known first-order solution \(\psi _{1}\), (26), this equation can be used to determine \(W_{n}\) giving
1.3 (c) Second-order solution
At \(O(\epsilon ^{2})\), after Taylor expansion of \(\sigma _{\theta r}\), (131) reduces to the non-dimensional equation:
where superscripts indicate the order at which each term is to be taken. Upon substitution of R , \(\varTheta \) and \(\sigma \), the (134) becomes
The forcing on the right-hand side of (135) can be calculated explicitly from the derivatives of \(\psi _{1}\), (26), simplifying the equation to
The left-hand side of (135) can be evaluated using \(\psi _{2}^\mathrm{i}\) from (56), and the value of the three constants of integration, \(L_{k}\), \(M_{k}\) and \(N_{k}\), are needed. \(L_{k}\) and \(M_{k}\) were calculated in (57) and (58). From the asymptotic matching, \(L_{k}\) and \(M_{k}\) are known in terms of \(T_{k}\) and \(S_{k}\), (65) and (66). Matching at one higher order determines \(N_{k}\) in terms of \(T_{k}\) and \(S_{k}\), then using (65) and (66), \(N_{k}\) can be written in terms of \(L_{k}\) and \(M_{k}\) giving
Equation (135) leads to an equality for \(\sum _{k=1}^{\infty }F_{k} \left( \int _{\mu }^{1}P_{k}(x)\mathrm{d}x\right) \) at leading order, O(1). As this must hold for all \(\mu \in [-1,1]\), the coefficients of \(\left( \int _{\mu }^{1}P_{k}(x)\mathrm{d}x\right) \) must equate for each k, and thus this will give an equation for every \(F_{k}\) separately. Upon substitution of (133), the equation for \(F_{k}\) reduces to
Then substituting \(F_{k}^{(\delta )}\) into (74) and (75) finally gives
and
which with (73) gives the value of all the constants in the streaming solution (72).
Appendix 2: In-phase streaming around a bubble
Notice that, from the solution of the steady streaming around a bubble given in Appendix 1, all terms in (140), (139) and (73) are multiplies of \(\mathrm{i}V_{m}\bar{V}_{k}\) for some integers \(m,k\ge 0\). As such, Appendix 1 only gives the solution for out-of-phase motion of the bubble, as otherwise that solution is identically zero. Therefore, for these cases, the steady streaming needs to be calculated to the next order, namely \(O(\delta )\).
Since here the analysis is to determine one higher order in \(\delta \), this could require more stringent conditions than \(\epsilon \ll \delta \ll 1\) relationship. However, the order change is due to terms being identically zero which we expect to continue at higher orders in \(\epsilon \) so the same relationship should hold.
1.1 (a) Inner solution at the third order in \(\delta \)
In order to find the steady streaming at \(O(\delta )\), the asymptotic matching must be carried out at one higher order. As such, the inner second-order solution must be calculated to an extra order in \(\delta \). Therefore, more terms will be required in the \(\delta \) expansions, so we first return to the second-order, inner governing equation:
When expanding the left-hand side in \(\delta \), the O(1) terms in \(\delta \) now contribute to the streaming as well as the \(O(\delta ^{-2})\) term, so we have
From (56), the O(1) (leading-order) solution of \(\langle \psi _{2}^\mathrm{i} \rangle \) is known. The second term in (142) will only make an O(1) or lower contribution from this term so the governing equation for the first three orders of \( \langle \psi _{2}^\mathrm{i} \rangle \) can be simplified to
Expanding the term \(\left\langle {\partial (\psi _{1},D^{2}\psi _{1})}/{\partial (r,\mu )}+2L\psi _{1}D\psi _{1}\right\rangle \) to \(O(\delta ^{2})\) and substituting into (143) then gives an equation for \(\left\langle {\partial ^{4} \psi _{2}^\mathrm{i} }/ {\partial \eta ^{4}} \right\rangle \) which can be integrated twice to find that the \(O(\delta ^{2})\) (only) term in \( \left\langle {\partial ^{2} \psi _{2}^\mathrm{i}} /{\partial \eta ^{2}} \right\rangle \) is
The \(O(\delta ^{2})\) contribution to \(\langle \psi _{2}^\mathrm{i} \rangle \) can be calculated by integrating twice more but in order to satisfy the no tangential stress boundary condition only \(\left\langle {\partial ^{2}\psi _{2}^\mathrm{i}}/{\partial \eta ^{2}}\right\rangle \) is required at \(O(\delta ^{2})\).
Notice that every term in \(\langle \psi _{2}^\mathrm{i} \rangle \) up to and including O(1) is a multiple of \(W_{n}+nV_{n}\) for some n, so every term will drop an order when the first-order stress condition \(W_{n}=-nV_{n}+O(\delta )\) is applied. Similarly, higher-order terms in \(\langle \psi _{2}^\mathrm{i} \rangle \) will also be multiples of \(W_{n}+nV_{n}\) since in \(\left\langle {\partial (\psi _{1},D^{2}\psi _{1})}/{\partial (r,\mu )}+2L\psi _{1}D\psi _{1}\right\rangle \) every term is a multiple of \(B_{n}\propto (W_{n}+nV_{n})\) and in \( \langle D^{4} \psi _{2}^\mathrm{i} \rangle \) extra terms in its delta expansion will be in terms of lower orders of \(\langle \psi _{2}^\mathrm{i} \rangle \) which are also proportional to \(W_{n}+nV_{n}\). Therefore, even when calculating the streaming to \(O(\delta )\) for a bubble, only the first three terms up to O(1) are needed in the equation for the inner streaming.
1.2 (b) Constants of integration
The second-order inner solution \(\langle \psi _{2}^\mathrm{i} \rangle \) is of the form
where f and g are known from (56) and h could be found by integrating (144). The no-tangential stress boundary condition then gives
The \(O(\delta ^{-2})\) terms will cancel with the \(O(\delta ^{-2})\) quantity in (136). The \(O(\delta ^{-1})\) terms do not cancel exactly but when taking \(W_{n}=-nV_{n}+O(\delta )\) (the first-order bubble condition) they do. Therefore, the O(1) terms will give the leading-order behaviour for which the value of the constants \(L_{k}\), \(M_{k}\) and \(N_{k}\) are needed. The constants \(L_{k}\) and \(M_{k}\) were calculated in (57) and (58) and \(N_{k}\) can be written in terms of \(L_{k}\) and \(M_{k}\), (137).
Equation (146) can then be equated with (136) to give the algebraic condition for no tangential stress. This will give a condition on \(M_{k}\) and \(L_{k}\) but \(L_{k}\) is uniquely determined by the boundary condition: the radial velocity of the bubble equals the radial velocity of the fluid adjacent to the bubble. \(M_{k}\) was also determined but is a function of the unknown \(F_{k}\) which this no tangential stress condition will determine. However, \(F_{k}\) uniquely determines \(M_{k}\) so this equation can be considered as just determining \(M_{k}\) giving
When applying the first-order stress boundary condition (133) all terms drop by one order in \(\delta \). Then assuming the O(1) terms cancel (which is required for the result in Appendix 1 not to give the solution) this gives \(M_{k}\) at \(O(\delta )\). Then \(L_{k}\) can be calculated at \(O(\delta )\) by applying (32) at \(O(\delta )\) to \(\langle \psi _{2}^\mathrm{i} \rangle \) (56). This gives
and the simplified \(M_{k}\) expression
1.3 (c) Outer streaming constants
Using the matching conditions (68) and (69) we finally obtain
and
which, with \(Y_{knm}=0\), give the value of all the constants in the Lagrangian streaming solution (72).
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Spelman, T.A., Lauga, E. Arbitrary axisymmetric steady streaming: flow, force and propulsion. J Eng Math 105, 31–65 (2017). https://doi.org/10.1007/s10665-016-9880-8
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DOI: https://doi.org/10.1007/s10665-016-9880-8