# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 2 Swimming at LRN by Shape Changes-the Exterior Problem  
The Navier-Stokes equations for an incompressible fluid of density \(\rho\), viscosity \(\mu\), and velocity \(\mathbf{u}\) are  
\[\rho\frac{\partial\mathbf{u}}{\partial t}+\rho\big{(}\mathbf{u} \cdot\
abla\big{)}\mathbf{u} = -\
abla p+\mu\Delta\mathbf{u}+\mathbf{f}_{\text{ext}}\] \[\
abla\cdot\mathbf{u} = 0\]  
where \(\mathbf{f}_{\text{ext}}\) is the external force field. Herein we assume that the swimmer is self-propelled and does not rely on any exterior force, and therefore we require that \(\mathbf{f}_{\text{ext}}=\mathbf{0}\). The Reynolds number based on a characteristic length scale \(L\) and speed scale \(U\) is \(\text{Re}=\rho LU/\mu\), and when converted to dimensionless form and the symbols re-defined, the equations read  
\[\text{Re}\Big{[}\text{Sl}\frac{\partial\mathbf{u}}{\partial t}+ \big{(}\mathbf{u}\cdot\
abla\big{)}\mathbf{u}\Big{]} = -\
abla p+\Delta\mathbf{u} \tag{1}\] \[\
abla\cdot\mathbf{u} = 0\]  
Here \(\text{Sl}=\omega L/U\) is the Strouhal number and \(\omega\) is a characteristic frequency of the shape changes. When \(\text{Re}\ll 1\) the convective momentum term in Equation (1) can be neglected, but the time variation requires that \(\text{Re}\text{Sl}=\omega L^{2}/\
u\ll 1\). When both terms are neglected, which we assume throughout, the flow is governed by the Stokes equations:  
\[\mu\Delta\mathbf{u}-\
abla p=\mathbf{0},\qquad\
abla\cdot\mathbf{u}=0 \tag{2}\]  
We also consider the propulsion problem in an infinite domain and impose the condition \(\mathbf{u}|_{\mathbf{x}\to\infty}=\mathbf{0}\) on the velocity field. In the LRN regime time does not appear explicitly, momentum is assumed to equilibrate instantaneously, and bodies move by exploiting the viscous resistance of the fluid. As a result, time-reversible deformations produce no motion, which is the content of the _scallop theorem_[1]. For a self-propelled swimmer, there is no net force or torque, and therefore movement is a purely geometric process: the net displacement of a swimmer during a stroke is independent of the rate at which the stroke is executed, as long as the Reynolds and Strouhal numbers remain small enough. Suppose that a swimmer occupies the closed compact domain \(\Omega(t)\subset\mathbb{R}^{3}\), at time \(t\), and let \(\partial\Omega(t)\) denote its prescribed time-dependent boundary. A _swimming stroke_\(\gamma\) is specified by a time-dependent sequence of the boundary \(\partial\Omega(t)\), and it is _cyclic_ if the initial and final shapes are identical, i.e., \(\partial\Omega(0)=\partial\Omega(T)\) where \(T\) is the period [35]. The swimmer's boundary velocity \(\mathbf{V}\) relative to fixed coordinates can be written as a part \(\mathbf{v}\) that defines the intrinsic shape deformations, and a rigid motion \(\mathbf{U}\). If \(\mathbf{u}\) denotes the velocity field in the fluid exterior to \(\Omega\), then a standard LRN self-propulsion problem is:  
Given a cyclic shape deformation \(\gamma(t)\) that is specified by \(\mathbf{v}\), solve the Stokes equations subject to:  
\[\text{Boundary condition:}\quad\mathbf{u}\big{|}_{\partial\Omega(t)} =\mathbf{V}=\mathbf{v}+\mathbf{U},\quad\mathbf{u}|_{\mathbf{x}\to \infty}=\mathbf{0}\] \[\text{Force-free condition:}\quad\int_{\partial\Omega(t)}\mathbf{f}= \mathbf{0}\] \[\text{Torque-free condition:}\quad\int_{\partial\Omega(t)}\mathbf{r} \wedge\mathbf{f}=\mathbf{0}\]In order to treat general shape changes of a swimmer defined by \(\Omega(t)\subset\mathbb{R}^{3}\) with boundary \(\partial\Omega(t)\), one must solve the exterior Stokes Equations (2) for \(\mathbf{u}\), with a prescribed velocity \(\mathbf{v}(t)\) on \(\partial\Omega(t)\) and subject to the decay conditions \(\mathbf{u}\sim 1/r\) and \(p\sim 1/r^{2}\) as \(r\rightarrow\infty\).