Stokes flow (low Reynolds number)
Stokes flow (low Reynolds number): Theoretical Foundations
Overview
Professor, how is Stokes flow different from the regular Navier-Stokes equations?
In flows with a very small Reynolds number ($Re \ll 1$), inertial forces are negligible compared to viscous forces. The Stokes equation is the linear equation obtained by dropping the advection term (nonlinear term) from the NS equations. It is important in areas such as sedimentation of fine particles, microfluidic devices, and biofluid mechanics.
Stokes Equation
The steady, incompressible Stokes equations are as follows.
Momentum Equation:
This is the form obtained by removing $\rho(\partial\mathbf{u}/\partial t + \mathbf{u}\cdot\nabla\mathbf{u})$ from the NS equations. The pressure gradient and viscous forces are perfectly balanced.
Under what conditions can the advection term be ignored?
It is determined by the Reynolds number.
| Subject | Typical Re | Stokes Approximation |
|---|---|---|
| Bacterial swimming | $10^{-5}$~$10^{-4}$ | Very good |
| Sperm swimming | $10^{-2}$ | Good |
| Particle sedimentation ($d=10\mu$m, in water) | $10^{-3}$ | Good |
| High-viscosity polymer flow | $10^{-2}$~$10^{-1}$ | Approximately valid |
| Microchannel ($L=100\mu$m) | $0.1$~$10$ | Case dependent |
Stokes' Drag Law
The drag force on a sphere (radius $R$) moving with uniform velocity $U$ in a viscous fluid, derived by Stokes in 1851.
This is Stokes' drag law. In terms of drag coefficient,
where $Re = \rho U (2R)/\mu$. This relationship holds for $Re < 1$.
Characteristics of Stokes Flow
Stokes flow has unique mathematical properties.
- Reversibility: The flow field remains the same under time reversal (indistinguishable even if a video is played backwards)
- Instantaneous Response: No inertia, so it responds immediately to changes in boundary conditions
- Linearity: Superposition of solutions holds. Interactions of multiple particles can be treated analytically
- Uniqueness: The solution is unique given the boundary conditions
Reversibility is interesting. That's impossible in normal flows.
A famous demonstration is in a Taylor-Couette device: a band of ink in a viscous fluid is "stirred" and then rotated in reverse, returning to its original state. This is an intuitive demonstration of the reversibility of Stokes flow.
Bacteria Live in a World Where "Swimming Backwards Still Moves You Forward"
The mechanism by which bacteria (like E. coli) swim by rotating their flagella only works in the world of Stokes flow (Reโช1). A famous result called the "Scallop Theorem" proves that "reversible motion (reciprocal motion) yields zero propulsion in flows with Reโช1." This is why bacteria cannot swim like humans. Bacteria achieve irreversible motion by rotating their flagella in a helical manner, avoiding this paradox. This principle is also an important constraint in the design of microrobots and medical nanomachines.
Computational Methods for Stokes flow (low Reynolds number)
Numerical Methods for the Stokes Equations
Please teach me how to solve the Stokes equations numerically.
The Stokes equations are linear, so they are easier to solve than the NS equations. However, the velocity-pressure coupling (saddle-point problem) remains.
Formulation as a Saddle-Point Problem
Discretizing the Stokes equations with FEM yields the following saddle-point type system of equations.
Here, $A$ is the stiffness matrix for the viscous term, and $B$ is the discrete version of the divergence operator.
The zero diagonal block looks difficult to solve.
Exactly. A combination of elements satisfying the inf-sup condition (LBB condition) is required.
| Velocity Element | Pressure Element | LBB Stable | Name |
|---|---|---|---|
| P2 (quadratic triangle) | P1 (linear triangle) | Stable | Taylor-Hood element |
| P1+bubble | P1 | Stable | MINI element |
| Q2 (quadratic quadrilateral) | Q1 (linear quadrilateral) | Stable | Standard |
| P1 | P1 | Unstable | Stabilization needed |
| P1 | P0 | Unstable | Not usable |
Solution with FVM
The SIMPLE algorithm can be used directly in the finite volume method, but special considerations are needed due to the extremely low Re number.
- Pressure Relaxation Factor: Can be set larger than usual (0.5~0.8). More stable because there is no advection term
- Advection Scheme: Not needed (advection term is zero)
- Convergence: Linear problem, so typically converges in tens of iterations
Boundary Integral Method (BEM)
The boundary integral method is a powerful technique that leverages the linearity of Stokes flow. Using Green's functions (Stokeslets), unknowns are placed only on boundary surfaces, not throughout the entire domain.
$$ u_j(\mathbf{x}) = -\frac{1}{8\pi\mu}\oint_S G_{ij}(\mathbf{x}, \mathbf{y})f_i(\mathbf{y})\,dS(\mathbf{y}) + \frac{1}{8\pi}\oint_S T_{ijk}(\mathbf{x}, \mathbf{y})u_i(\mathbf{y})n_k(\mathbf{y})\,dS(\mathbf{y}) $$
Here, $G_{ij}$ is the Oseen-Burgers tensor (free-space Green's function). For 3D problems, volume meshing becomes unnecessary, significantly reducing computational cost.
So for Stokes flow, it can be solved just on the boundary. This is impossible for the NS equations, right?
Correct. The nonlinearity of the NS equations prevents the existence of a Green's function. This is a privileged method made possible by the linearity of Stokes flow.
Coffee Break Trivia
Numerical Methods for Stokes FlowโWhy the Boundary Element Method (BEM) is More Advantageous than the Finite Volume Method
The Stokes equations (the low Re limit where Reโ0) are linear partial differential equations, making the Boundary Element Method (BEM) particularly effective. BEM does not mesh the interior of the fluid domain, only discretizing the boundary surfaces, so even for 3D problems, only 2D meshes are needed. Especially for external flows (open domains), it naturally handles boundaries at infinity, eliminating the need for the "far-field boundary" that the finite volume method must artificially set. However, the BEM matrix (a dense matrix of Nยฒ) requires storage capacity O(Nยฒ) and solution cost O(Nยณ) relative to the mesh count N. For large-scale problems, the Fast Multipole Method (FMM) is combined to compress this to O(N log N). It is also an effective method for many-body problems involving particle/bubble groups in fluids.
The boundary integral method is a powerful technique that leverages the linearity of Stokes flow. Using Green's functions (Stokeslets), unknowns are placed only on boundary surfaces, not throughout the entire domain.
Here, $G_{ij}$ is the Oseen-Burgers tensor (free-space Green's function). For 3D problems, volume meshing becomes unnecessary, significantly reducing computational cost.
So for Stokes flow, it can be solved just on the boundary. This is impossible for the NS equations, right?
Correct. The nonlinearity of the NS equations prevents the existence of a Green's function. This is a privileged method made possible by the linearity of Stokes flow.
Numerical Methods for Stokes FlowโWhy the Boundary Element Method (BEM) is More Advantageous than the Finite Volume Method
The Stokes equations (the low Re limit where Reโ0) are linear partial differential equations, making the Boundary Element Method (BEM) particularly effective. BEM does not mesh the interior of the fluid domain, only discretizing the boundary surfaces, so even for 3D problems, only 2D meshes are needed. Especially for external flows (open domains), it naturally handles boundaries at infinity, eliminating the need for the "far-field boundary" that the finite volume method must artificially set. However, the BEM matrix (a dense matrix of Nยฒ) requires storage capacity O(Nยฒ) and solution cost O(Nยณ) relative to the mesh count N. For large-scale problems, the Fast Multipole Method (FMM) is combined to compress this to O(N log N). It is also an effective method for many-body problems involving particle/bubble groups in fluids.
Stokes flow (low Reynolds number) in Practice
Practical Guide
Please tell me about cases where Stokes flow analysis is performed in practical work.
Let me list the main application areas.
Calculation of Particle Sedimentation Velocity
The terminal sedimentation velocity of a spherical particle can be obtained directly from Stokes' drag law. Under the condition where gravity and drag force balance,
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