Stokes flow (low Reynolds number)
Theory and Physics
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 (Re≪1). A famous result called the "Scallop Theorem" proves that "reversible motion (reciprocal motion) yields zero propulsion in flows with Re≪1." 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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, splashing manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to the heartbeat, and the flow fluctuations each time an engine valve opens and closes are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost is significantly reduced, solving first for steady state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the air, as a "carrier," transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity makes the diffusion term stronger, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re flows, convection overwhelms, and diffusion becomes a minor player.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to confusion between absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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 Flow—Why the Boundary Element Method (BEM) is More Advantageous than the Finite Volume Method
The Stokes equations (the low Re limit where Re→0) 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 N²) requires storage capacity O(N²) and solution cost O(N³) 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.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is judged when residuals for Continuity, momentum, and energy decrease by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2〜0.3, Velocity: 0.5〜0.7. Reduce the factor if diverging. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each timestep until a steady solution converges. Guideline for internal iteration count: 5〜20 times. If residuals fluctuate between timesteps, review the timestep size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised with the corrected pressure—this catchball is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
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 Flow—Why the Boundary Element Method (BEM) is More Advantageous than the Finite Volume Method
The Stokes equations (the low Re limit where Re→0) 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 N²) requires storage capacity O(N²) and solution cost O(N³) 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.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is judged when residuals for Continuity, momentum, and energy decrease by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2〜0.3, Velocity: 0.5〜0.7. Reduce the factor if diverging. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each timestep until a steady solution converges. Guideline for internal iteration count: 5〜20 times. If residuals fluctuate between timesteps, review the timestep size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised with the corrected pressure—this catchball is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
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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