General properties of creeping flow
Contents
General properties of creeping flow¶
Before continuing we wish to break and describe two general (and important!) properties of the creeping flow equations: first, that for fixed boundary conditions, a given solution is unique; second, that the solutions are reversible (also known as the “scallop theorem”): if we flip the sign of the boundary conditions, we flip the sign of the whole flow, which presents a chalenge for low-\(Re\) swimming.
Uniqueness¶
The first thing we wish to show is that for fixed Dirichlet boundary conditions \({\vec u}({\vec x})\Big|_B = {\vec u}_B({\vec x})\), the solutions to the creeping flow equations are unique. Note that the equations
have no time derivatives. This means that the solution at any time is governed by the boundary conditions, and time-dependent flow will only occur throug time-dependent boundary conditions. This is very much in contrast to the full Navier-Stokes equations, in which the time derivative matters, and solutions depend on initial conditions.
To see this we consider the possibility that there are two solutions, \({\vec u}_{i=1,2}\) and \(p_{i=1,2}\), satisfying the same boundary conditions \({\vec u}_i({\vec x})\Big|_B = {\vec u}_B({\vec x})\). Because the creeping flow equations are linear, the difference \({\vec V} = {\vec u}_2 - {\vec u}_1\), \(P = p_2 - p_1\) also solves the creeping flow equations with \({\vec D}\Big|_B = 0\).
Now let us take the dot product of \({\vec V}\) with (creep) acting on \(P,{\vec V}\), and integrate the result over the domain D with boundary \(\p D = B\):
Here I have expressed the dot product in terms of sums over indices, using Einstein summation notation, to make sure we get the next step right, and kept the dimension general (could be 2 or 3, or 10, or whatever). We now wish to integrate the result by parts. Since the boundary terms are proportional to \(V\) (with no derivative acting on it), they will vanish. Furthermore, upon integrating by parts, the bulk part of the first term is proportional to \(\nabla^i V_i \equiv {\vec\nabla}\cdot{\vec V}\), which vanishes due to incompressibility of both \({\vec u}_i\). Thus, we are left with
But this is a sum (over \(i,j\) of squares at every point in \(D\), which means that \(\p_i V_j = 0\) for every \(i,j\) and every point in \(D\). Because it is true for very \(i\), \(V_j\) must be constant in every direction. Since it vanishes at the boundary, it must vanish everywhere, and \({\vec v}_1 = {\vec v}_2\), thus proving uniqueness.
Reversibility¶
The next statement follows from the above statement, together with the fact that the creeping flow equations have no time derivatives. Note that our argument about uniqueness was completely independent of time. Thus, the flow at any time is completely detemined by the boundary conditions at a given time. This is of course not exactly physical – knowledge that the boundary has moved should propagate with some speed less than that of light! – but is a result of our approximation. Nonetheless, or slow, low-\(Re\) flow it is basically correct.
A particular consequence is that the equations are time-reversible. Time reversal at a given time flips the sign of \({\vec u}\) – imagine a movie of a agged fluid parcel suddenly run backwards in time. The solution with \({\vec u} \to - {\vec u}\) is a solution with the reversed boundary conditions \({\vec u}_B \to - {\vec u}_B\), and is in fact the unique solution. A particular consequence, beautifully described in [Purcell, 1977], is that attempts to swim by a motion followed by its reverse will produce no net motion of the body. This is called the “Scallop theorem” by Purcell, as you can imagine a scallop trying to move by opening and closing its hinge. At high Reynolds number, it could close quickly, jet forward, and then open slowly. At low Reynolds number this will not work and you need to swim by some more complex protocol involving a nontrivial cycle in the space of shapes of your body.