Skip to main content
Log in

Arbitrary axisymmetric steady streaming: flow, force and propulsion

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Squires TM, Quake SR (2005) Microfluidics: fluid physics at the nanoliter scale. Rev Mod Phys 77:977–1026

    Article  ADS  Google Scholar 

  2. Sia SK, Whitesides GM (2003) Microfluidic devices fabricated in poly(dimethylsiloxane) for biological studies. Electrophoresis 24:3563–3576

    Article  Google Scholar 

  3. Stone HA, Kim S (2001) Microfluidics: basic issues, applications, and challenges. AIChE J 47:1250–1254

    Article  Google Scholar 

  4. Wensink HH, Dunkel J, Heidenreich S, Drescher K, Goldstein RE, Lowen H, Yeomans JM (2012) Meso-scale turbulence in living fluids. Proc Natl Acad Sci USA 109:14308–14313

    Article  ADS  MATH  Google Scholar 

  5. Lauga E, Powers TR (2009) The hydrodynamics of swimming micro-organisms. Rep Prog Phys 72:096601

    Article  ADS  Google Scholar 

  6. Drescher K, Goldstein RE, Michael N, Polin M, Tuval I (2010) Direct measurement of the flow field around swimming microorganisms. Phys Rev Lett 105(16):168101

    Article  ADS  Google Scholar 

  7. Pedley TJ, Kessler JO (1992) Hydrodynamic phenomena in suspension of swimming microorganisms. Annu Rev Fluid Mech 24:313–358

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Saintillan D, Shelley MJ (2008) Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys Rev Lett 100:178103

    Article  ADS  Google Scholar 

  9. Dunkel J, Heidenreich S, Drescher K, Wensink HH, Bar M, Goldstein RE (2013) Fluid dynamics of bacterial turbulence. Phys Rev Lett 110:228102

    Article  ADS  Google Scholar 

  10. Leptos KC, Guasto JS, Gollub JP, Pesci AI, Goldstein RE (2009) Dynamics of enhanced tracer diffusion in suspension of swimming eukaryotic microorganisms. Phys Rev Lett 103(19):198103

    Article  ADS  Google Scholar 

  11. Lin Z, Thiffeault J, Childress S (2011) Stirring by squirmers. J Fluid Mech 669:167–177

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Zaid IM, Dunkel J, Yeomans JM (2011) Levy fluctuations and mixing in dilute suspensions of algae and bacteria. J R Soc Interface 8:1314–1331

    Article  Google Scholar 

  13. Blake JR (1971) A spherical envelope approach to ciliary propulsion. J Fluid Mech 46:199–208

    Article  ADS  MATH  Google Scholar 

  14. Brennen C (1974) An oscillating-boundary-layer theory for cilliary propulsion. J Fluid Mech 65:799–824

    Article  ADS  MATH  Google Scholar 

  15. Lighthill MJ (1952) On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun Pur Appl Math 5:109–118

    Article  MathSciNet  MATH  Google Scholar 

  16. Pak O, Lauga E (2014) Generalized squirming motion of a sphere. J Eng Math 88:1–28

    Article  MathSciNet  MATH  Google Scholar 

  17. Li G, Ardekani AM (2014) Hydrodynamic interaction of microswimmers near a wall. Phys Rev E 90(1):013010

    Article  ADS  Google Scholar 

  18. Spagnolie SE, Lauga E (2012) Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J Fluid Mech 700:105–147

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Ishimoto K, Gaffney EA (2014) Swimming efficiency of spherical squirmer: beyond the Lighthill theory. Phys Rev E 90(1):012704

    Article  ADS  Google Scholar 

  20. Wang S, Ardekani AM (2012) Unsteady swimming of small organisms. J Fluid Mech 702:286–297

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Ishimoto K (2013) A spherical squirming swimmer in unsteady Stokes flow. J Fluid Mech 723:163–189

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Rao PM (1988) Mathematical model for unsteady ciliary propulsion. Math Comput Model 10:839–851

    Article  MATH  Google Scholar 

  23. Wang S, Ardekani A (2012) Inertial squirmer. Phys Fluids 24(10):101902

    Article  ADS  Google Scholar 

  24. Khair AS, Chisholm NG (2014) Expansions at small Reynolds numbers for the locomotion of a spherical squirmer. Phys Fluids 26:011902

    Article  ADS  Google Scholar 

  25. Chisholm N, Legendre D, Lauga E, Khair A (2016) A squirmer across Reynolds numbers. J Fluid Mech 796:233–256

    Article  ADS  Google Scholar 

  26. Hessel V, Lowe H, Schonfeld F (2005) Micromixers—a review on passive and active mixing principles. Chem Eng Sci 8–9:2479–2501

    Article  Google Scholar 

  27. Nguyen N-T, Wu Z (2004) Micromixers—a review. J Micromech Microeng 15(2):R1

    Article  Google Scholar 

  28. Riley N (2001) Steady streaming. Annu Rev Fluid Mech 33:43–65

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Chen Y, Lee S (2014) Manipulation of biological objects using acoustic bubbles: a review. Integr Comp Biol 54(6):959–968

    Article  Google Scholar 

  30. Elder SA (1959) Cavitation microstreaming. J Acoust Soc Am 31(1):54–64

    Article  ADS  Google Scholar 

  31. Marmottant P, Hilgenfeldt S (2004) A bubble-driven microfluidic transport element for bioengineering. Proc Natl Acad Sci USA 101:9523–9527

    Article  ADS  Google Scholar 

  32. Marmottant P, Hilgenfeldt S (2003) Controlled vesicle deformation and lysis by single oscillating bubbles. Nature 423(6936):153–156

    Article  ADS  Google Scholar 

  33. Feng J, Yuan J, Cho SK (2015) Micropropulsion by an acoustic bubble for navigating microfluidic spaces. Lab Chip 15:1554–1562

    Article  Google Scholar 

  34. Wang C, Rallabandi B, Hilgenfeldt S (2013) Frequency dependence and frequency control of microbubble streaming flows. Phys Fluids 25(2):022002

    Article  ADS  Google Scholar 

  35. Ahmed D, Mao X, Juluri BK, Huang T (2009) A fast microfluidic mixer based on acoustically driven sidewall-trapped microbubbles. Microfluid Nanofluid 7:727–731

    Article  Google Scholar 

  36. Wang SS, Jiao ZJ, Huang XY, Yang C, Nguyen NT (2009) Acoustically induced bubbles in a microfluidic channel for mixing enhancement. Microfluid Nanofluid 6:847–852

    Article  Google Scholar 

  37. Riley N (1966) On a sphere oscillating in a viscous fluid. Q J Mech Appl Math 19:461–472

    Article  Google Scholar 

  38. Davidson BJ, Riley N (1971) Cavitation microstreaming. J Sound Vib 15:217–233

    Article  ADS  MATH  Google Scholar 

  39. Longuet-Higgins MS (1998) Viscous streaming from an oscillating spherical bubble. Proc R Soc Lond A 454:725–742

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Maksimov AO (2007) Viscous streaming from surface waves on the wall of acoustically-driven gas bubbles. Eur J Mech B 26:28–42

    Article  MATH  Google Scholar 

  41. Ahmed D, Lu M, Nourhani A, Lammert PE, Stratton Z, Muddana HS, Crespi VH, Huang TJ (2015) Selectively manipulable acoustic-powered microswimmers. Sci Rep 5:9744

    Article  ADS  Google Scholar 

  42. Bertin N, Spelman T, Stephan O, Gredy L, Bouriau M, Lauga E, Marmottant P (2015) Propulsion of bubble-based acoustic microswimmers. Phys Rev Appl 4:064012

    Article  ADS  Google Scholar 

  43. Batchelor BK (1967) An introduction to fluid mechanics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  44. Ishida H, Shigenaka Y (1988) Cell model contraction in the ciliate Spirostomum. Cell Motil Cytoskelet 9:278–282

    Article  Google Scholar 

  45. Gaunt JA (1929) The triplets of helium. Phil Trans R Soc A 228:151–196

    Article  ADS  MATH  Google Scholar 

  46. Yu-Lin X (1996) Fast evaluation of the Gaunt coefficients. Math Comput Am Math Soc 65:1601–1612

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Grosswald E (1978) Bessel polynomials. Springer, Berlin

    Book  MATH  Google Scholar 

  48. Longuet-Higgins MS (1953) Mass transport in water waves. Philos Trans R Soc A 245:535–581

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Magar V, Pedley TJ (2005) Average nutrient uptake by a self-propelled unsteady squirmer. J Fluid Mech 539:93–112

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Ishikawa T, Simmonds MP, Pedley TJ (2006) Hydrodynamic interaction of two swimming model micro-organisms. J Fluid Mech 568:119–160

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Michelin S, Lauga E (2010) Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys Fluids 22(11):111901

    Article  ADS  Google Scholar 

  52. Keller SR, Wu TY (1977) A porous prolate-spheroidal model for ciliated micro-organisms. J Fluid Mech 80:259–278

    Article  ADS  MATH  Google Scholar 

  53. Aranson IS (2013) Active colloids. Physics-Uspekhi 56(1):79

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was funded in part by the EPSRC (TS) and the European Union through a Marie Curie CIG Grant (EL).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eric Lauga.

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,

$$\begin{aligned} \varvec{n}\cdot \varvec{\sigma }\cdot \varvec{t}=0\quad \text {at}\quad r=R,\,\,\theta =\varTheta , \end{aligned}$$
(127)

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

$$\begin{aligned} \varvec{t}= & {} \mathbf{e}_{\theta }+\frac{\partial R}{\partial \theta }{} \mathbf{e}_{r}+O(\epsilon ^{3}),\end{aligned}$$
(128)
$$\begin{aligned} \varvec{n}= & {} \mathbf{e}_{r}-\frac{\partial R}{\partial \theta }\mathbf{e}_{\theta }+O(\epsilon ^{3}). \end{aligned}$$
(129)

Using these equations, the no-stress condition can be expanded in terms of \(\epsilon \) giving the first two terms as

$$\begin{aligned} \sigma _{\theta r}+\frac{\partial R}{\partial \theta _{0}}\left( \sigma _{rr}-\sigma _{\theta \theta }\right) +O(\epsilon ^{3})=0\quad \text {at}\quad r=R,\,\,\theta =\varTheta , \end{aligned}$$
(130)

becoming

$$\begin{aligned} \sigma _{\theta r}+(R-1)\frac{\partial \sigma _{\theta r}}{\partial r}+(\varTheta -\theta _{0})\frac{\partial \sigma _{\theta r}}{\partial \theta }+\left( \sigma _{rr}-\sigma _{\theta \theta }\right) \frac{\partial R}{\partial \theta }+O(\epsilon ^{3})=0 \quad \text {at}\quad r=1,\,\,\theta =\theta _{0} . \end{aligned}$$
(131)

1.2 (b) Leading-order solution

At \(O(\epsilon )\), Eq. 131 reduces to the non-dimensional equation

$$\begin{aligned} \left( \frac{1}{r}\frac{\partial u_{r}}{\partial \theta }+\frac{\partial u_{\theta }}{\partial r}-\frac{u_{\theta }}{r}\right) =0\quad \text {at}\quad r=1,\,\,\theta =\theta _{0}. \end{aligned}$$
(132)

By substituting in the known first-order solution \(\psi _{1}\), (26), this equation can be used to determine \(W_{n}\) giving

$$\begin{aligned} W_{n}=-nV_{n}+\delta \frac{2n(n+2)}{(1+\mathrm{i})}V_{n}+\delta ^{2} \mathrm{i}n(n+2)^{2}V_{n}+O(\delta ^{3}). \end{aligned}$$
(133)

1.3 (c) Second-order solution

At \(O(\epsilon ^{2})\), after Taylor expansion of \(\sigma _{\theta r}\),  (131) reduces to the non-dimensional equation:

$$\begin{aligned} \sigma _{\theta r}^{(\epsilon ^{2})}=-(R-1)\frac{\partial \sigma _{\theta r}^{(\epsilon )}}{\partial R}-(\varTheta -\theta _{0})\frac{\partial \sigma _{\theta r}^{(\epsilon )}}{\partial \theta }-\frac{\partial R}{\partial \theta _{0}}\left( \sigma _{rr}^{(\epsilon )}-\sigma _{\theta \theta }^{(\epsilon )}\right) \quad \text {at}\quad r=R,\,\,{\theta =\varTheta }, \end{aligned}$$
(134)

where superscripts indicate the order at which each term is to be taken. Upon substitution of R , \(\varTheta \) and \(\sigma \), the (134) becomes

$$\begin{aligned}&\bigg ((1-\mu ^{2})\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial \mu ^{2}}+2\frac{\partial \psi _{2}^\mathrm{i}}{\partial r}-\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial r^{2}}\bigg ) \nonumber \\&\qquad =\mathfrak {R}\left[ \mathrm{i}\sum _{n=0}^{\infty }V_{n}P_{n}(\mu )\mathrm{e}^{\mathrm{i}t}\right] \mathfrak {R}\left[ (1-\mu ^{2})\frac{\partial ^{3}\psi _{1}}{\partial \mu ^{2}\partial r}+3\frac{\partial ^{2}\psi _{1}}{\partial r^{2}}-\frac{\partial ^{3}\psi _{1}}{\partial r^{3}}-3(1-\mu ^{2})\frac{\partial ^{2}\psi _{1}}{\partial \mu ^{2}}-4\frac{\partial \psi _{1}}{\partial r}\right] \nonumber \\&\qquad \;+\;\mathfrak {R}\left[ \epsilon \mathrm{i}\sum _{n=1}^{\infty }W_{n}\left( \int _{\mu }^{1}P_{n}(x)\mathrm{d}x\right) \mathrm{e}^{\mathrm{i}t}\right] \mathfrak {R}\left[ \frac{\mu }{(1-\mu ^{2})}\left( (1-\mu ^{2})\frac{\partial ^{2}\psi _{1}}{\partial \mu ^{2}}+2\frac{\partial \psi _{1}}{\partial r}-\frac{\partial ^{2}\psi _{1}}{\partial r^{2}}\right) \right. \nonumber \\&\qquad \;\left. +\;{\partial \over \partial \mu }\left( (1-\mu ^{2})\frac{\partial ^{2}\psi _{1}}{\partial \mu ^{2}}\right) +2\frac{\partial ^{2}\psi _{1}}{\partial r\partial \mu }-\frac{\partial ^{3}\psi _{1}}{\partial r^{2}\partial \mu }\right] -\mathfrak {R}\left[ \epsilon \mathrm{i} \sum _{n=0}^{\infty }n(n+1)V_{n}\left( \int _{\mu }^{1}P_{n}(x)\mathrm{d}x\right) \mathrm{e}^{\mathrm{i}t}\right] \nonumber \\&\qquad \;\times \;\mathfrak {R}\left[ 6\frac{\partial \psi _{1}}{\partial \mu }-4\frac{\partial \psi _{1}}{\partial r\partial \mu }-2\frac{\mu }{(1-\mu ^{2})}\frac{\partial \psi _{1}}{\partial r}\right] \text {at}\quad r=1,\,\,\theta =\theta _{0}. \end{aligned}$$
(135)

The forcing on the right-hand side of (135) can be calculated explicitly from the derivatives of \(\psi _{1}\), (26), simplifying the equation to

$$\begin{aligned}&\bigg \langle (1-\mu ^{2})\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial \mu ^{2}}+2\frac{\partial \psi _{2}^\mathrm{i}}{\partial r}-\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial r^{2}} \bigg \rangle \\ \nonumber&\qquad =\; \frac{\mathrm{i}}{2}\sum _{k=0}^{\infty }\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }a_{knm}\bigg \{-\bar{V}_{n}V_{m}m(m^{2}+2)+\bar{V}_{n}W_{m}(m+2)^{2}-4\bar{W}_{n}V_{m}m(m+1) \\ \nonumber&\qquad \quad +\; \bar{V}_{n}(W_{m}+mV_{m})\bigg [2\mathrm{i}\frac{1}{\delta ^{2}}+\frac{1}{\delta }(m+3)(1+\mathrm{i})+ \frac{1}{2}m(2m+3)\bigg ]\\\nonumber&\qquad \quad -\;\frac{1}{\delta }(\bar{W}_{n}+n\bar{V}_{n})W_{m}(1-\mathrm{i})+\bar{V}_{n}W_{m}n(n+2)-\bar{W}_{n}W_{m}(n+2) \\ \nonumber&\qquad \quad +\;\sum _{j=1}^{\infty }\frac{C_{nj}}{j(j+1)}\bigg [\frac{1}{\delta }W_{m} (\bar{W}_{j}+j\bar{V}_{j})\frac{(1-\mathrm{i})}{j(j+1)}+W_{m} (\bar{W}_{j}-j\bar{V}_{j})\frac{(j+2)}{j(j+1)} \\ \nonumber&\qquad \quad +\;2V_{m}\bar{W}_{j}m(m+1)\bigg ]\bigg \} \int _{\mu }^{1}P_{k}(x)\mathrm{d}x. \end{aligned}$$
(136)

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

$$\begin{aligned} N_{k}=\left[ \frac{(2-k)k}{2}L_{k}+\frac{(1-2k)}{2}\frac{M_{k}}{\delta }\right] \delta ^{2}. \end{aligned}$$
(137)

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

$$\begin{aligned} \frac{1}{\delta }F_{k}^{(\delta )}= & {} -\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }a_{knm}\left[ \frac{1}{2} \bar{V}_{n}V_{m} m(m+4+n)\mathrm{i}+\sum _{j=1}^{\infty }\left( \frac{C_{nj}}{j+1}\right) \frac{1}{2}\bar{V}_{m}V_{j}m\mathrm{i}\right] \nonumber \\&-\;V_{0}\bar{V}_{k}\frac{3k\mathrm{i}}{(2k+1)}-\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }g_{knm}\bar{V}_{n}V_{m}\frac{9k\mathrm{i}}{2(2k+1)}+\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }f_{knm}\bar{V}_{n}V_{m}\frac{3k(n+2)\mathrm{i}}{2(2k+1)(n+1)(m+1)}\nonumber \\&+\;\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{(2k+1)}\left[ \bar{V}_{n}V_{m}\left( n^{2}-4nm-4n+m^{2}-m-3\right) m\mathrm{i} \right. \nonumber \\&\left. +\;\sum _{j=1}^{\infty }\left( \frac{C_{nj}}{j+1}\right) V_{m}\bar{V}_{j}m(3m+5)\mathrm{i}\right] . \end{aligned}$$
(138)

Then substituting \(F_{k}^{(\delta )}\) into (74) and (75) finally gives

$$\begin{aligned} T_{k}= & {} \mathfrak {R}\left\{ V_{0}\bar{V}_{k}\frac{(1-k^{2})\mathrm{i}}{(2k+1)}+\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }f_{knm}\bar{V}_{n}V_{m}\frac{\left( k^{2}-k-1\right) (n+2)\mathrm{i}}{2(2k+1)(n+1)(m+1)}\right. \nonumber \\&+\;\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }g_{knm}\bar{V}_{n}V_{m}\frac{3(1-k^{2})\mathrm{i}}{2(2k+1)}+\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{2(2k+1)}\Bigg [\bar{V}_{n}V_{m}(n^{2}-4nm-4n\nonumber \\&\left. +\;m^{2}-m-3)\mathrm{i}m+\sum _{j=1}^{\infty }\left( \frac{C_{nj}}{j+1}\right) V_{m}\bar{V}_{j}m(3m+5)\mathrm{i}\Bigg ]\right\} +O(\delta ) \end{aligned}$$
(139)

and

$$\begin{aligned} S_{k}= & {} \mathfrak {R}\bigg \{ V_{0}\bar{V}_{k}\frac{k(k+2)\mathrm{i}}{(2k+1)}-\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }f_{knm}\bar{V}_{n}V_{m}\frac{(2k+4)k(n+2)\mathrm{i}}{4(2k+1)(n+1)(m+1)} \nonumber \\&+\;\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }g_{knm}\bar{V}_{n}V_{m}\frac{3k(k+2)\mathrm{i}}{2(2k+1)}-\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{2(2k+1)}\bigg [\bar{V}_{n}V_{m}m\mathrm{i} (n^{2}-4nm-4n \nonumber \\&+\;m^{2}-m-3)+\sum _{j=1}^{\infty }\bigg (\frac{C_{nj}}{j+1}\bigg )V_{m}\bar{V}_{j}m(3m+5)\mathrm{i}\bigg ]\bigg \} +O(\delta ) \end{aligned}$$
(140)

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:

$$\begin{aligned} \frac{\delta ^{2}}{2} \langle D^{4} \psi _{2}^\mathrm{i} \rangle =\frac{1}{r^{2}} \left\langle \frac{\partial (\psi _{1},D^{2}\psi _{1})}{\partial (r,\mu )}+2L\psi _{1}D\psi _{1} \right\rangle . \end{aligned}$$
(141)

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

$$\begin{aligned} \frac{\delta ^{2}}{2} \langle D^{4} \psi _{2}^\mathrm{i} \rangle =\frac{1}{2\delta ^{2}}\frac{\partial ^{4} \langle \psi _{2}^\mathrm{i} \rangle }{\partial \eta ^{4}}+(1-\mu ^{2})\frac{\partial ^{4} \langle \psi _{2}^\mathrm{i} \rangle }{\partial \eta ^{2}\partial \mu ^{2}}+O(\delta ). \end{aligned}$$
(142)

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

$$\begin{aligned}&\frac{\partial ^{4} \langle \psi _{2}^\mathrm{i} \rangle }{\partial \eta ^{4}}=\frac{2\delta ^{2}}{r^{2}} \left\langle \frac{\partial (\psi _{1},D^{2}\psi _{1})}{\partial (r,\mu )}+2L\psi _{1}D\psi _{1}\right\rangle \nonumber \\&\qquad \qquad \quad \;+\;\delta ^{2}\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }a_{knm}\left[ 2\bar{V}_{n}(W_{m}+mV_{m})k(k+1)\mathrm{e}^{-(1+\mathrm{i})\eta }\right] \left( \int _{\mu }^{1}P_{k}(x)\mathrm{d}x\right) +O(\delta ^{3}). \end{aligned}$$
(143)

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

$$\begin{aligned}&\left\langle \frac{\partial ^{2} \psi _{2}^\mathrm{i}}{\partial \eta ^{2}}\right\rangle ^{(\delta ^{2})}=\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }a_{knm}\Bigg \{ -(\bar{W}_{n}+n\bar{V}_{n})V_{m}m\mathrm{i}\left( \frac{n}{2}-m-3\right) \nonumber \\&\qquad \qquad \qquad \qquad +\;(\bar{W}_{n}+n\bar{V}_{n})(W_{m}+mV_{m})\left( \frac{\mathrm{i}}{2}\left( 1-m+n\right) -2-\frac{n}{2}\right) \nonumber \\&\qquad \qquad \qquad \qquad +\;\bar{V}_{n}\left( W_{m}+mV_{m}\right) \mathrm{i}\left( -\frac{5n^{2}}{4}-\frac{21n}{4}+nm+m+\frac{m^{2}}{4}-3-t^2-t\right) \nonumber \\&\qquad \qquad \qquad \qquad +\;\sum _{j=1}^{\infty }\left( \frac{C_{nj}}{j(j+1)}\right) \Bigg [(W_{m}+mV_{m})(\bar{W}_{j}+j\bar{V}_{j})\left( 1+\mathrm{i}-\frac{m}{2}-\frac{\mathrm{i}j}{2}+j\right) \nonumber \\&\qquad \qquad \qquad \qquad +\;(W_{m}+mV_{m})\bar{V}_{j}\mathrm{i}j(2j+6-m) \Bigg ]\Bigg \} \int _{\mu }^{1}P_{k}(x)\mathrm{d}x . \end{aligned}$$
(144)

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

$$\begin{aligned} \langle \psi _{2}^\mathrm{i} \rangle =\sum _{k=1}^{\infty }\left[ f_{k}(\eta )+\delta g_{k}(\eta )+\delta ^{2}h_{k}(\eta )+L_{k}+M_{k}\eta +N_{k}\eta ^{2}+O(\delta ^{3})\right] \left( \int _{\mu }^{1}P_{k}(x)\mathrm{d}x\right) , \end{aligned}$$
(145)

where f and g are known from (56) and h could be found by integrating (144). The no-tangential stress boundary condition then gives

$$\begin{aligned}&\left\langle (1-\mu ^{2})\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial \mu ^{2}}+2\frac{\partial \psi _{2}^\mathrm{i}}{\partial r}-\frac{\partial ^{2}\psi _{2}^\mathrm{i}}{\partial r^{2}}\right\rangle =\sum _{k=1}^{\infty }\left\{ -\frac{1}{\delta ^{2}}\frac{\partial ^{2}f_{k}}{\partial \eta ^{2}}+\frac{1}{\delta }\left( 2\frac{\partial f_{k}}{\partial \eta }-\frac{\partial ^{2}g_{k}}{\partial \eta ^{2}}\right) \right. \nonumber \\&\quad \left. +\;\left[ -k(k+1)f_{k}(\eta )+2\frac{\partial g_{k}}{\partial \eta }-\frac{\partial ^{2}h_{k}}{\partial \eta ^{2}}+2\frac{1}{\delta }M_{k}-2\frac{1}{\delta ^{2}}N_{k}-k(k+1)L_{k}\right] \right\} \left( \int _{\mu }^{1}P_{k}(x)\mathrm{d}x\right) . \end{aligned}$$
(146)

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

$$\begin{aligned} -\frac{1}{\delta }M_{k}(2k+1)= & {} -3L_{k}k+\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }a_{knm}\bigg (\frac{1}{\delta }\bigg \{ \frac{1}{2}(\bar{W}_{n}+n\bar{V}_{n})(W_{m}+mV_{m})(\mathrm{i}-1) \nonumber \\&+\;\frac{1}{2}\bar{V}_{n}(W_{m}+mV_{m})(1-\mathrm{i})(2n+3) +\; \sum _{j=1}^{\infty }\frac{C_{nj}}{j(j+1)}\bigg [\frac{1}{2}(W_{m}+mV_{m})(\bar{W}_{j}+j\bar{V}_{j})(1+\mathrm{i}) \nonumber \\&-\; (W_{m}+mV_{m})\bar{V}_{j}j(1-\mathrm{i})-\frac{1}{2}W_{m}\bigg (\bar{W}_{j}+j\bar{V}_{j}\bigg )\frac{(1+\mathrm{i})}{j(j+1)}\bigg ]\bigg \} \nonumber \\&+\; \bigg \{ (\bar{W}_{n}+n\bar{V}_{n})V_{m}\bigg (\frac{n}{2}-m-2\bigg )m\mathrm{i} -\;\frac{1}{2}(\bar{W}_{n}+n\bar{V}_{n})(W_{m}+mV_{m})(\mathrm{i}n-\mathrm{i}m+n-1) \nonumber \\&-\bar{V}_{n}(W_{m}+mV_{m})\mathrm{i}\bigg [-\frac{5n^{2}}{4}-\frac{9n}{4}+nm +\;\frac{m^{2}}{4}+1-\frac{3}{2}k(k+1)+\frac{1}{4}(2m^{2}+3m)\bigg ] \nonumber \\&-\frac{1}{2}\bar{V}_{n}W_{m}n(n+2)\mathrm{i}+\frac{1}{2}\bar{W}_{n}W_{m}(n+2)\mathrm{i} +\;\frac{1}{2}\bar{V}_{n}V_{m}m(m^{2}+2)\mathrm{i}-\frac{1}{2}\bar{V}_{n}W_{m}(m+2)^{2}\mathrm{i} \nonumber \\&+2\bar{W}_{n}V_{m}m(m+1)\mathrm{i} +\;\sum _{j=1}^{\infty }\frac{C_{nj}}{j(j+1)}\bigg [-(W_{m}+mV_{m})(\bar{W}_{j}+j\bar{V}_{j}) \nonumber \\&\times \; \bigg (\frac{1}{2}(1+\mathrm{i})-\frac{1}{2}(m+\mathrm{i}j)+j\bigg ) -\;(W_{m}+mV_{m})\bar{V}_{j}\mathrm{i}j(2j+4-m)-V_{m}\bar{W}_{j}m(m+1)\mathrm{i} \nonumber \\&-\;\frac{1}{2}W_{m}(\bar{W}_{j}-j\bar{V}_{j})\frac{(j+2)\mathrm{i}}{j(j+1)}\bigg ]\bigg \} \bigg )+O(\delta ). \end{aligned}$$
(147)

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

$$\begin{aligned} L_{k}= & {} \delta \left\{ \sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{1}{2}a_{knm}\bar{V}_{n}V_{m}m(m+2)(1+\mathrm{i})-\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\frac{1}{2}g_{knm}\bar{V}_{n}V_{m}m(m+2)(1+\mathrm{i})\right. \nonumber \\&\left. +\;\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{1}{2}f_{knm}\bar{V}_{n}V_{m}\frac{(n+2)(m+2)(1+\mathrm{i})}{(n+1)(m+1)}-\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\frac{1}{2}f_{knm}\bar{V}_{n}V_{m}\frac{(n+2)(1-\mathrm{i})}{(n+1)(m+1)}\right\} , \end{aligned}$$
(148)

and the simplified \(M_{k}\) expression

$$\begin{aligned} \frac{1}{\delta }M_{k}= & {} L_{k} \bigg ( \frac{3k}{2k+1} \bigg )-\delta \sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{(2k+1)}\bigg \{ -\bar{V}_{n}V_{m}nm(n+2)(2m+1)(1-\mathrm{i}) \nonumber \\&+\;\frac{1}{4}\bar{V}_{n}V_{m}(n^{2}+9n-5m^{2}+6t(t+1)-5m)(1+\mathrm{i})m(m+2)\nonumber \\&+\;\sum _{j=1}^{\infty }\bigg (\frac{C_{nj}}{j(j+1)}\bigg )V_{m}\bar{V}_{j}mj\bigg [(m+1)(j+2)(1-\mathrm{i})-(m+2)(j+4)(1+\mathrm{i})\bigg ]\bigg \}. \end{aligned}$$
(149)

1.3 (c) Outer streaming constants

Using the matching conditions (68) and (69) we finally obtain

$$\begin{aligned} T_{k}= & {} \delta \mathfrak {R}\bigg (\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }f_{knm}\bar{V_{n}}V_{m}\frac{(1-k^{2})(n+2)[(m+2)(1+\mathrm{i})-(1-\mathrm{i})]}{2(2k+1)(n+1)(m+1)} \nonumber \\&-\;\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }g_{knm}\bar{V}_{n}V_{m}\frac{(1-k^{2})m(m+2)(1+\mathrm{i})}{2(2k+1)}+\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{2(2k+1)}\bigg \{\frac{1}{4}\bar{V}_{n}V_{m} (n^{2}+9n \nonumber \\&-\;5m^{2}-5m+4+2k^{2}+6k)(1+\mathrm{i})m(m+2)-\bar{V}_{n}V_{m}n(n+2)m(2m+1)(1-\mathrm{i})\nonumber \\&+\;\sum _{j=1}^{\infty }\bigg (\frac{C_{nj}}{j+1}\bigg )V_{m}\bar{V}_{j} [m(m+1)(j+2)(1-\mathrm{i})-m(m+2)(j+4)(1+\mathrm{i})]\bigg \} \bigg ), \end{aligned}$$
(150)

and

$$\begin{aligned} S_{k}= & {} \delta \mathfrak {R}\bigg (\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }f_{knm}\bar{V}_{n}V_{m}\frac{k(k+2)(n+2)[(m+2)(1+\mathrm{i})-(1-\mathrm{i})]}{2(n+1)(m+1)(2k+1)} \nonumber \\&-\;\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }g_{knm}\bar{V}_{n}V_{m}\frac{m(m+2)k(k+2)(1+\mathrm{i})}{2(2k+1)}+\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }\frac{a_{knm}}{2(2k+1)}\bigg \{ -\frac{1}{4}\bar{V}_{n}V_{m}m(m+2) [n^{2}+9n \nonumber \\&-\;5m^{2}-5m+2k(k-1)](1+\mathrm{i})+\bar{V}_{n}V_{m}n(n+2)m (2m+1)(1-\mathrm{i})\nonumber \\&-\;\sum _{j=1}^{\infty }\bigg (\frac{C_{nj}}{j+1}\bigg )V_{m}\bar{V}_{j} [m(m+1)(j+2)(1-\mathrm{i})-m(m+2)(j+4)(1+\mathrm{i})]\bigg \}\bigg ), \end{aligned}$$
(151)

which, with \(Y_{knm}=0\), give the value of all the constants in the Lagrangian streaming solution (72).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-016-9880-8

Keywords

Mathematics Subject Classification

Navigation