Skip to main content

Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization


A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical slender-body theories. This approach has the potential to open up the way to more effective computational techniques, since geometrical complexities can be significantly reduced. This, however, is achieved at the expense of involving higher-order derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Barta E (2011) Motion of slender bodies in unsteady Stokes flow. J Fluid Mech 688:66–87

    ADS  Article  MATH  MathSciNet  Google Scholar 

  2. Barta E, Liron N (1988) Slender body interactions for low Reynolds numbers. II. Body-body interactions. SIAM J Appl Math 48(6):1262–1280

    Article  MATH  MathSciNet  Google Scholar 

  3. Batchelor GK (1970) Slender-body theory for particles of arbitrary cross-section in Stokes flow. J Fluid Mech 44:419–440

    ADS  Article  MATH  MathSciNet  Google Scholar 

  4. Burgers JM (1938) On the motion of small particles of elongated form suspended in a viscous liquid. Chap. III of Second Report of Viscosity and Plasticity. Kon Ned Akad Wet Verhand (Erste Sectie) 16:113

  5. Chwang AT, Wu TY-T (1975) Hydromechanics of low-Reynolds-number flow. II. Singularity method for Stokes flows. J Fluid Mech 67:787–815

    ADS  Article  MATH  MathSciNet  Google Scholar 

  6. Cortez R, Nicholas M (2012) Slender body theory for Stokes flows with regularized forces. Commun Appl Math Comput Sci 7(1):33–62

    Article  MATH  MathSciNet  Google Scholar 

  7. Cox RG (1970) The motion of long slender bodies in a viscous fluid. Part 1 General theory. J Fluid Mech 44(4):791–810

    ADS  Article  MATH  Google Scholar 

  8. de Mestre NJ (1973) Low-reynolds-number fall of slender cylinders near boundaries. J Fluid Mech 58(4):641–656

    ADS  Article  MATH  Google Scholar 

  9. Edwardes D (1892) Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis. Q J Pure Appl Math 26:70–78

    Google Scholar 

  10. Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher transcendental functions, vol III. McGraw-Hill Book Company Inc., New York

    MATH  Google Scholar 

  11. Fried E, Gurtin ME (2006) Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch Ration Mech Anal 182(3):513–554

    Article  MATH  MathSciNet  Google Scholar 

  12. Giusteri GG (2013) The multiple nature of concentrated interactions in second-gradient dissipative liquids. Z Angew Math Phys (ZAMP), 64(2):371–380

    Article  MATH  MathSciNet  Google Scholar 

  13. Giusteri GG, Marzocchi A, Musesti A (2010) Three-dimensional nonsimple viscous liquids dragged by one-dimensional immersed bodies. Mech Res Commun 37(7):642–646

    Article  MATH  Google Scholar 

  14. Giusteri GG, Marzocchi A, Musesti A (2011) Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures. Acta Mech 217:191–204

    Article  MATH  Google Scholar 

  15. Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Lond A 102:161–179

    ADS  Article  Google Scholar 

  16. Johnson RE (1980) An improved slender-body theory for Stokes flow. J Fluid Mech 99(2):411–431

    ADS  Article  MATH  MathSciNet  Google Scholar 

  17. Keller JB, Rubinow SI (1976) Slender-body theory for slow viscous flow. J Fluid Mech 75(4):705–714

    ADS  Article  MATH  Google Scholar 

  18. Kim S, Karrila SJ (2005) Microhydrodynamics: principles and selected applications. Dover Publications Inc., New York

    Google Scholar 

  19. Lighthill J (1975) Mathematical biofluiddynamics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  20. Lighthill J (1976) Flagellar hydrodynamics. SIAM Rev 18(2):161–230

    Article  MATH  MathSciNet  Google Scholar 

  21. Lions J-L (1969) Quelques methodes de résolution des problemès aux limites non linéaires. Dunod, Paris

  22. Musesti A (2009) Isotropic linear constitutive relations for nonsimple fluids. Acta Mech 204:81–88

    Article  MATH  Google Scholar 

  23. Oberbeck A (1876) Über stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung. Crelle 81:62–80

    Google Scholar 

  24. Oseen CW (1927) Neuere Methoden und Ergebnisse in der Hydrodynamik. Akademische Verlagsgesellschaft, Leipzig

    MATH  Google Scholar 

  25. Stokes GG (1851) On the effect of the internal friction of fluids on the motion of pendulums. Trans Camb Phil Soc 9:8–106

    ADS  Google Scholar 

  26. Tillett JPK (1970) Axial and transverse Stokes flow past slender axisymmetric bodies. J Fluid Mech 44:401–417

    ADS  Article  MATH  MathSciNet  Google Scholar 

  27. Tuck EO (1964) Some methods for flows past blunt slender bodies. J Fluid Mech 18:619–635

    ADS  Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Giulio G. Giusteri.

Appendix A: Uniform flow past a straight rod

Appendix A: Uniform flow past a straight rod

To compute the low-Reynolds-number flow past a uniformly translating rigid rod, we follow the method of \(\S\)4.1. Specifically, we start from the solution \(\mathbf{u}^{ps},\) given by Chwang and Wu [5], for the disturbance field generated by a prolate spheroid with axis along \(\mathbf{e}_1\) and foci at x 1 =  ± a, uniformly translating with velocity \(\mathbf{U}=U_1\mathbf{e}_1+U_2\mathbf{e}_2.\) In view of the symmetry of the spheroid, there is no loss of generality in taking U 3 = 0. A particular solution of (12) is \(\mathbf{u}^{*}=\mathbf{u}^{ps}+L^2\Updelta\mathbf{u}^{ps}.\) Setting

$$ r:=\sqrt{x_2^2+x_3^2} ,\quad R_1:=\sqrt{(x_1+a)^2+r^2} ,\quad R_2:=\sqrt{(x_1-a)^2+r^2}, $$
$$ D_0:=\frac{R_2-(x_1-a)}{R_1-(x_1+a)} ,\quad D_1:=\frac{1}{R_2}-\frac{1}{R_1} ,\quad D_2:=\frac{x_1+a}{R_1}-\frac{x_1-a}{R_2}, $$
$$ D_3:=\frac{1}{R_2^3}-\frac{1}{R_1^3} ,\quad D_4:=\frac{x_1-R_2}{R_2^2-R_2(x_1-a)}-\frac{x_1-R_1}{R_1^2-R_1(x_1+a)}, $$
$$ D_5:=\frac{1}{R_2^2-R_2(x_1-a)}-\frac{1}{R_1^2-R_1(x_1+a)}, $$

we can express its components as

$$u^*_1({\mathbf{x}})={}-2(\alpha_1+\beta_1)\log D_0-2\beta_1 x_1D_1 +2L^2\alpha_1 x_1D_3+(\beta_2x_2-2\beta_1x_1)D_4 +\alpha_1D_2-\alpha_2 x_2D_1+\beta_2\frac{x_2}{r^2}(R_1-R_2)+2L^2\alpha_2\frac{x_2}{r^2}D_1+\frac{x_2x_1(x_1+a)}{r^2}\left(\frac{2L^2\alpha_2}{R_1^3}-\frac{\beta_2}{R_2}\right)+\frac{x_2x_1(x_1-a)}{r^2}\left(\frac{\beta_2}{R_1}-\frac{2L^2\alpha_2}{R_2^3}\right),$$
$$ u^*_2({\mathbf{x}})=(\beta_2-\alpha_2)\log D_0-(\alpha_1x_2+2\beta_1 x_2)D_1+2L^2\alpha_1 x_2D_3+\frac{x_1-a}{r^2}\left(\beta_2R_1+\frac{2L^2\alpha_2}{R_2}\right)-\frac{x_1+a}{r^2}\left(\beta_2R_2+\frac{2L^2\alpha_2}{R_1}\right)+\frac{\alpha_2x_2^2D_2}{r^2}\left(1+\frac{4L^2}{r^2}\right) +\frac{2L^2\alpha_2x_2^2}{r^2}\left(\frac{x_1+a}{R_1^3}-\frac{x_1-a}{R_2^3}\right)+\frac{\beta_2x_2^2}{r^2}\left[(x_1-a)\left(\frac{1}{R_1}-\frac{2R_1}{r^2}\right)-(x_1+a)\left(\frac{1}{R_2}-\frac{2R_2}{r^2}\right)\right]+(\beta_2x_2^2-2\beta_1x_1x_2)D_5,$$
$$u^*_3({\mathbf{x}})=-(\alpha_1x_3+2\beta_1 x_3)D_1+2L^2\alpha_1 x_3D_3 +\frac{\alpha_2x_2x_3D_2}{r^2}+\frac{\beta_2x_2x_3}{r^2}\left[(x_1-a)\left(\frac{1}{R_1}-\frac{2R_1}{r^2}\right) -(x_1+a)\left(\frac{1}{R_2}-\frac{2R_2}{r^2}\right)\right]+\frac{2L^2\alpha_2x_2x_3}{r^2}\left[(x_1+a)\left(\frac{1}{R_1^3}+\frac{2}{R_1r^2}\right) -(x_1-a)\left(\frac{1}{R_2^3}+\frac{2}{R_2r^2}\right)\right]+(\beta_2x_2x_3-2\beta_1x_1x_3)D_5,$$

where α 1,  α 2,  β 1, and β 2 are constants to be determined.

We introduce prolate spheroidal coordinates (uvϕ), defined by

$$\begin{array}{l} x_1=a\cosh{u}\cos v,\\ x_2=a\sinh{u}\sin v\cos\phi,\\ x_3=a\sinh{u} \sin v\sin\phi, \end{array} $$

with \(u\in[0,+\infty),\,v\in[0,\pi],\) and \(\phi\in[0,2\pi).\) The surfaces given by u = c, with c > 0, are confocal prolate spheroids with foci at x 1 =  ± a, and u = 0 describes the degenerate spheroid \(\Uplambda.\) Using these coordinates, we have

$$ \begin{array}{ll} r=a\sinh u\sin v ,\\ R_1=a(\cosh u+\cos v),\\ R_2=a(\cosh u-\cos v),\\ R_1-(x_1+a)=a(\cosh u-1)(1-\cos v),\\ R_2-(x_1-a)=a(\cosh u+1)(1-\cos v) , \end{array}$$

and we can rewrite the components of \(\mathbf{u}^*\) as

$$ u^*_1(u,v,\phi)=-2(\alpha_1+\beta_1)\log\left(\frac{\cosh u+1}{\cosh u-1}\right) -4\beta_1 \frac{\cosh u\cos^2 v}{\cosh^2 u-\cos^2 v}+2\alpha_1\frac{\cosh u\sin^2 v}{\cosh^2 u-\cos^2 v}+\frac{4L^2\alpha_1}{a^2}\frac{\cosh u\cos^2 v(3\cosh^2 u+\cos^2 v)}{(\cosh^2 u-\cos^2 v)^3}-2\alpha_2 \frac{\cos\phi\sinh u\sin v\cos v}{\cosh^2 u-\cos^2 v}+2\beta_2\frac{\cos\phi\cos v}{\sinh u\sin v}+\frac{4L^2\alpha_2}{a^2}\frac{\cos\phi\cos v}{\sinh u\sin v\left(\cosh^2 u-\cos^2 v\right)}+\frac{\cos\phi \cosh u\cos v(\cosh u\cos v+1)}{\sinh u\sin v}\left(\frac{2aL^2\alpha_2}{R_1^3}-\frac{a\beta_2}{R_2}\right)+\frac{\cos\phi \cosh u\cos v(\cosh u\cos v-1)}{\sinh u\sin v}\left(\frac{a\beta_2}{R_1}-\frac{2aL^2\alpha_2}{R_2^3}\right)+a(\beta_2 \sinh u \sin v\cos\phi-2\beta_1\cosh u\cos v)D_4 ,$$
$$ u^*_2(u,v,\phi)=(\beta_2-\alpha_2)\log\left(\frac{\cosh u+1}{\cosh u-1}\right)-(2\alpha_1+4\beta_1)\frac{\cos\phi\sinh u\sin v\cos v}{\cosh^2 u-\cos^2 v}+\frac{4L^2\alpha_1}{a^2}\frac{\cos\phi\sinh u\sin v\cos v(3\cosh^2 u+\cos^2 v)}{(\cosh^2 u-\cos^2 v)^3} +\frac{\cosh u\cos v-1}{\sinh^2 u\sin^2 v}\left(\frac{\beta_2R_1}{a}+\frac{2L^2\alpha_2}{aR_2}\right)-\frac{\cosh u\cos v+1}{\sinh^2 u\sin^2 v}\left(\frac{\beta_2R_2}{a}+\frac{2L^2\alpha_2}{aR_1}\right)+2\alpha_2\frac{\cos^2 \phi\cosh u\sin^2 v}{\cosh^2 u-\cos^2 v}\left(1+\frac{4L^2}{a^2\sinh^2 u\sin^2 v}\right)+2L^2\alpha_2\cos^2 \phi\left(\frac{x_1+a}{R_1^3}-\frac{x_1-a}{R_2^3}\right)+\beta_2\cos^2 \phi\left[(x_1-a)\left(\frac{1}{R_1}-\frac{2R_1}{a^2\sinh^2 u\sin^2 v}\right)-(x_1+a)\left(\frac{1}{R_2}-\frac{2R_2}{a^2\sinh^2 u\sin^2 v}\right)\right] +(\beta_2a^2\cos^2 \phi\sinh^2 u\sin^2 v-2\beta_1 a^2\cos\phi\cosh u\cos v \sinh u\sin v)D_5, $$
$$ u^*_3(u,v,\phi)=-(2\alpha_1+4\beta_1)\times \frac{\sin\phi\sinh u\sin v\cos v}{\cosh^2 u-\cos^2 v} +\frac{4L^2\alpha_1}{a^2}\frac{\sin\phi\sinh u\sin v\cos v(3\cosh^2 u+\cos^2 v)}{(\cosh^2 u-\cos^2 v)^3} +2\alpha_2\frac{\cos\phi\sin\phi\cosh u\sin^2 v}{\cosh^2 u-\cos^2 v} +\beta_2\cos\phi\sin\phi\left[(x_1-a)\left(\frac{1}{R_1}-\frac{2R_1}{a^2\sinh^2 u\sin^2 v}\right) -(x_1+a)\left(\frac{1}{R_2}-\frac{2R_2}{a^2\sinh^2 u\sin^2 v}\right)\right] +2L^2\alpha_2\cos\phi\sin\phi(x_1+a)\left(\frac{1}{R_1^3}+\frac{2}{R_1a^2\sinh^2 u\sin^2 v}\right) -2L^2\alpha_2\cos\phi\sin\phi(x_1-a)\left(\frac{1}{R_2^3}+\frac{2}{R_2a^2\sinh^2 u\sin^2 v}\right)+D_5\beta_2a^2\cos\phi\sin\phi\sinh^2 u\sin^2 v -2D_5\beta_1 a^2\sin\phi\cosh u\cos v \sinh u\sin v .$$

Notice that, in the limit \(u\to0,\,\mathbf{u}^*\) is divergent. Since we wish to impose the velocity of the fluid precisely on the set \(\Uplambda\) defined by u = 0, we must add a suitable solution of the homogeneous equation associated with (12), thereby canceling the divergent terms.

We introduce external prolate spheroidal wave functions (using the notation of Erdélyi et al. [10]) defined for \({n\in\mathbb N}\) and \(l=0,\ldots,n:\)

$$ W_n^{\pm l}(u,v,\phi)=S_n^{l(3)}(\cosh u,\kappa^2a^2/4)\times \mathrm{Ps}_n^l(\cos v,\kappa^2a^2/4)e^{\pm il\phi}. $$

Those are solutions of \(\Updelta W+\kappa^2 W=0,\) regular at infinity but divergent on \(\Uplambda.\) In our problem κ = i/L, and we are interested in the cases n = 0, 1, 2, 3. We have

$$ W_0^{0}(u,v,\phi)=S_0^{0(3)}(\cosh u,-a^2/4l^2)\times {\mathrm{Ps}}_0^0(\cos v,-a^2/4l^2) =s_0^0\sum_{2r\geq 0}\sum_{2p\geq 0}(-1)^p a_{0,r}^0 a_{0,p}^0\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi r}K_{2r+\frac{1}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p}^{0}(\cos v) ,$$
$$ W_1^{0}(u,v,\phi)=S_1^{0(3)}(\cosh u,-a^2/4l^2)\mathrm{Ps}_1^0(\cos v,-a^2/4l^2) =s_1^0\sum_{2r\geq -1}\sum_{2p\geq -1}(-1)^p a_{1,r}^0a_{1,p}^0\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+1/2)}K_{2r+\frac{3}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+1}^{0}(\cos v) ,$$
$$ W_1^{\pm 1}(u,v,\phi)=S_1^{1(3)}(\cosh u,-a^2/4l^2)\times \mathrm{Ps}_1^1(\cos v,-a^2/4l^2)e^{\pm i\phi} =e^{\pm i\phi}\left(1-\frac{1}{\cosh^{2} u}\right)^{\frac{1}{2}}s_1^{-1}\times \sum_{2r\geq 0}\sum_{2p\geq 0}(-1)^p a_{1,r}^{-1}a_{1,p}^1\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+1/2)}_{2r+\frac{3}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+1}^{1}(\cos v) , $$
$$ W_2^{0}(u,v,\phi)=S_2^{0(3)}(\cosh u,-a^2/4l^2)\mathrm{Ps}_2^0(\cos v,-a^2/4l^2) =s_2^0\sum_{2r\geq -2}\sum_{2p\geq -2}(-1)^p a_{2,r}^0a_{2,p}^0\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+1)}K_{2r+\frac{5}{2}}\left(\frac{a\cosh u}{L} \right)P_{2p+2}^{0}(\cos v) ,$$
$$ W_2^{\pm 1}(u,v,\phi)=S_2^{1(3)}(\cosh u,-a^2/4l^2)\times \mathrm{Ps}_2^1(\cos v,-a^2/4l^2)e^{\pm i\phi} =e^{\pm i\phi}\left(1-\frac{1}{\cosh^{2} u}\right)^{\frac{1}{2}}s_2^{-1}\times\sum_{2r\geq -1}\sum_{2p\geq -1}(-1)^p a_{2,r}^{-1}a_{2,p}^1\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+1)}K_{2r+\frac{5}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+2}^{1}(\cos v) , $$
$$ W_2^{\pm 2}(u,v,\phi)=S_2^{2(3)}(\cosh u,-a^2/4l^2)\times\mathrm{Ps}_2^2(\cos v,-a^2/4l^2)e^{\pm i2\phi} =e^{\pm i2\phi}\left(1-\frac{1}{\cosh^{2} u}\right)s_2^{-2}\times \sum_{2r\geq 0}\sum_{2p\geq 0}(-1)^p a_{2,r}^{-2}a_{2,p}^2\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+1)}K_{2r+\frac{5}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+2}^{2}(\cos v) , $$
$$ W_3^{0}(u,v,\phi)=S_3^{0(3)}(\cosh u,-a^2/4l^2)\mathrm{Ps}_3^0(\cos v,-a^2/4l^2) =s_3^0\sum_{2r\geq -3}\sum_{2p\geq -3}(-1)^p a_{3,r}^0a_{3,p}^0\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+3/2)}K_{2r+\frac{7}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+3}^{0}(\cos v) , $$
$$ W_3^{\pm 1}(u,v,\phi)=S_3^{1(3)}(\cosh u,-a^2/4l^2)\times \mathrm{Ps}_3^1(\cos v,-a^2/4l^2)e^{\pm i\phi} =e^{\pm i\phi}\left(1-\frac{1}{\cosh^{2} u}\right)^{\frac{1}{2}}s_3^{-1}\times\sum_{2r\geq -2}\sum_{2p\geq -2}(-1)^p a_{3,r}^{-1}a_{3,p}^1\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+3/2)}K_{2r+\frac{7}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+3}^{1}(\cos v) , $$
$$ W_3^{\pm 2}(u,v,\phi)=S_3^{2(3)}(\cosh u,-a^2/4l^2)\times \mathrm{Ps}_3^2(\cos v,-a^2/4l^2)e^{\pm i2\phi} =e^{\pm i2\phi}\left(1-\frac{1}{\cosh^{2} u}\right)s_3^{-2}\times\sum_{2r\geq -1}\sum_{2p\geq -1}(-1)^p a_{3,r}^{-2}a_{3,p}^2\sqrt{\frac{2L}{a\pi\cosh u}} \times e^{-i\pi(r+3/2)}K_{2r+\frac{7}{2}}\left(\frac{a\cosh u}{L}\right)P_{2p+3}^{2}(\cos v) , $$

where s l n and a l n,m are coefficients dependent on κ2 a 2/4 =  − a 2/4l 2,  K ν is a modified Bessel function of the second kind, and P μν is an associated Legendre function of the first kind.

The divergence of the logarithmic terms in \(u^*_1\) and \(u^*_2\) can be cancelled by adding multiples of W 00 . If \(U_2=\alpha_2=\beta_2=0,\) which corresponds to a uniform translation of the rod along its axis, the dependence on \(\phi\) and the radial decay of \(\mathbf{u}^*\) suggest that a solution \(\mathbf{u}^{\parallel}\) of (12) can be written in the form

$$\begin{array}{l} u^{\parallel}_1=u^*_1+c_1W_0^0+d_1W_1^0+k_1W_2^0+e_1W_3^0,\\ u^{\parallel}_2=u^*_2+c_2(W_1^1+W_1^{-1})+d_2(W_2^1+W_2^{-1})+k_2(W_3^1+W_3^{-1}),\\ u^{\parallel}_3=u^*_3+ic_3(W_1^1-W_1^{-1})+id_3(W_2^1-W_2^{-1})+ik_3(W_3^1-W_3^{-1}).\end{array}$$

The no-slip condition on \(\Uplambda\) requires that \(\mathbf{u}(0,v,\phi)=-\mathbf{U},\) which fixes the value of α1, once the boundedness of \(\mathbf{u}^{\parallel}\) is ensured by a suitable choice of the the coefficients \(c_i,\,d_i,\) \(k_i,\) and \(e_1\) as functions of \(\alpha_1.\)

On the other hand, in the case U 1 = α1 = β1 = 0, corresponding to a translation of the rod with direction in the plane orthogonal to \(\mathbf{e}_1,\) we may seek a solution \(\mathbf{u}^{\perp}\) of (12) of the form

$$\begin{array}{ll} u^{\perp}_1=u^*_1+c_1(W_1^1+W_1^{-1})+d_1(W_2^1+W_2^{-1})+k_1(W_3^1+W_3^{-1}),\\ u^{\perp}_2=u^*_2+c_2W_0^0+d_2(W_2^2+W_2^{-2}+A_2W_2^0+B_2W_0^0) +k_2(W_3^2+W_3^{-2}+A_3W_3^0+B_3W_0^0),\\ u^{\perp}_3=u^*_3+id_3(W_2^2-W_2^{-2})+ik_3(W_3^2-W_3^{-2}).\\ \end{array}$$

As above, the no-slip condition on \(\Uplambda\) fixes the value of α2, once the boundedness of \(\mathbf{u}^{\perp}\) is ensured by a suitable choice of the coefficients A i ,   B i ,  c i ,  d i , and k i . Finally, the disturbance field \(\mathbf{u}\) for a uniform translation with generic velocity \(\mathbf{U}=U_1\mathbf{e}_1+U_2\mathbf{e}_2\) is given by \(\mathbf{u}=\mathbf{u}^{\parallel}+\mathbf{u}^{\perp}.\)

Notice that, since we can impose the corresponding boundary conditions on the surface described by u = c, for any value c ≥ 0, the previous expressions provide the general solution for the flow past any translating spheroid of the family with foci at x 1 =  ± a.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Giusteri, G.G., Fried, E. Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization. Meccanica 49, 2153–2167 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Slender-body theory
  • Hyperviscosity
  • Fluid-structure interaction
  • Dimensional reduction

Mathematics Subject Classification (2000)

  • 76D07
  • 76A05