1
$\begingroup$

The formula is from the first paragraph in the paper "Second Kind Integral Equation Formulation of Stokes' Flows Past a Particle of Arbitrary Shape" by Power and Miranda:

... the governing equations for the auxiliary perturbed fluid velocity $\vec{v}=(v_1,v_2,v_3)$ and pressure $p$ can be approximated by the creeping motion and continuity equations $ \begin{align} &\frac{\partial^2 v_i}{\partial x_j\partial x_j}(x)=\frac{\partial p}{\partial x_i}(x)\\ &\frac{\partial v_i}{\partial x_i}(x)=0 \end{align} $

Here are my questions:

  • How should I understand the first formula? Is the Einstein summation convention applied here, i.e., $ \sum_j\frac{\partial^2 v_i}{\partial x_j\partial x_j}(x)=\frac{\partial p}{\partial x_i}(x)? $

  • Is there any reference for the derivation of the first formula? (I don't find any in the paper.)

  • 0
    The second formula in the question, I think, means $\nabla\cdot v=0$, instead of $\nabla v=0$.2011-11-20

1 Answers 1

1

See the Wikipedia article Creeping Motion.

Einstein summation convention is indeed implied. In other words, the equation can be written as $\nabla ^2\mathbf{v}=\mathbf{\nabla}p$.

As for the derivation, according to the article, it follows directly from the Navier-Stokes equations, but as I don't know what they mean by ". . . the inertial forces are assumed to be negligible . . .", so I'm afraid I can't explain this, although perhaps this will help.

Note: The original reason I did not want to post this as an answer was because I don't know how to completely answer your second point. In any case, if you are happy with this answer, then I am certainly happy to post it as one :).

  • 0
    The "inertial forces" neglected in the Stokes equation are simply the time derivative terms. Navier Stokes has the form "Forces = mass* acceleration" as in classical mechanics, but formulated for a fluid element and "mass * acceleration" is what they call inertial forces. The reason this is neglected is that the limit of Reynolds number $ \to \infty$ is taken. Another way of deriving this equation is by minimizing the dissipation $\int_V (\partial_i v_j) (\partial_i v_j) \mathrm{d}V$ for an incompressible flow.2018-07-26