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

Abstract

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.

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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.

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Giusteri, G.G., Fried, E. Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization. Meccanica 49, 2153–2167 (2014). https://doi.org/10.1007/s11012-014-9890-4

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Keywords

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

Mathematics Subject Classification (2000)

  • 76D07
  • 76A05