Stokes law | PhysicsRank

A small bead falling through a viscous liquid can settle into a steady speed. Gravity has not switched off; the opposing forces have caught up. The interesting question is which drag model remains valid for that bead and fluid.

We will connect shear stress to velocity gradients, derive the terminal speed of a rigid sphere and distinguish creeping flow from the transition criteria used in pipes. Every prediction will include a check of the assumptions that made it possible.

What you will learn to do

Relate Newtonian shear stress to a velocity gradient.
Define Reynolds number with an explicit speed and length convention.
Derive a critical mean speed from a specified pipe criterion.
State the assumptions behind Stokes drag.
Include buoyancy when deriving a signed terminal velocity.
Recheck the flow regime after a parameter change or a viscosity measurement.

Useful before this chapterPressure, floating and fluid support →Continuity and energy in flowing fluids →From interactions to equations →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

If a falling object reaches constant speed, which is zero: its velocity, its weight or its net force?

Compare your reasoning

Its net force is zero. Velocity and weight can remain nonzero. Terminal motion is a force balance, not a disappearance of gravity.

Revisit the explanation →

01 / Viscosity, terminal motion and flow regimes

Viscosity responds to relative motion between neighbouring layers

A fluid can flow even when it resists deformation. Imagine liquid trapped between two broad parallel plates. The bottom plate is stationary and the top plate slides horizontally. With no slip at the solid boundaries, the fluid beside each plate shares that plate’s velocity. Between them, adjacent layers move at different speeds and transmit tangential stresses.

The relevant quantity is the velocity gradient, not the velocity alone. A fluid translating uniformly has no shear from this mechanism because neighbouring layers do not slide relative to one another. For a Newtonian fluid, the magnitude of shear stress is viscosity η times the magnitude of the transverse velocity gradient.

For steady flow between plates separated by a small distance d, neglecting edge effects and any imposed horizontal pressure gradient, the velocity profile is linear. If the top speed is U, the gradient is U/d. The external force needed to maintain its motion is ηAU/d. The viscous force on that plate opposes its motion; the forces transmitted at the two walls need not have the same direction.

|\tau|=\eta\left|\frac{dv_x}{dy}\right|,\qquad F=\eta A\frac{U}{d}

The second expression applies to the stated parallel-plate model. Viscosity η has units Pa·s = kg/(m·s), while shear stress τ has units Pa.

WORKED EXAMPLE

Infer viscosity from a controlled shear experiment

A plate of area 0.020 m² slides at 0.030 m/s over a stationary parallel plate with a 1.0 mm liquid gap. A steady tangential force of 0.12 N maintains the motion. Ignore edges and pressure-driven flow. Find the viscosity and the rate of mechanical energy transfer to the fluid.

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STOP & THINK

A Newtonian fluid moves everywhere with the same constant velocity. What is its shear stress from a velocity gradient?

Zero
Proportional to that velocity
Infinite

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: Viscosity responds to relative motion between neighbouring layers
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
|\tau|=\eta\left|\frac{dv_x}{dy}\right|,\qquad F=\eta A\frac{U}{d}
Meaning: The second expression applies to the stated parallel-plate model. Viscosity η has units Pa·s = kg/(m·s), while shear stress τ has units Pa.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Viscosity, terminal motion and flow regimes

A dimensionless comparison helps you choose a flow model

A fast flow is not necessarily turbulent, and a slow flow is not necessarily in the creeping-flow regime. Density, viscosity and geometry matter too. The Reynolds number compares inertial and viscous effects using a characteristic speed U and length L: Re = ρUL/η.

Always state what U and L mean. For flow through a circular pipe, a common convention uses cross-sectional mean speed and internal diameter. For a sphere moving relative to otherwise still fluid, a common convention uses the relative speed and sphere diameter. A result quoted with radius instead of diameter differs by a factor of two, so its threshold must use the same convention.

Re is dimensionless: ρUL has units kg/(m·s), the same as dynamic viscosity η. Large viscosity favours viscous effects; large speed or size favours inertia. The same fluid may therefore support very different regimes around a tiny settling particle and in a large fast pipe.

The ratio η/ρ is called kinematic viscosity and has units m²/s. Then Re = UL/ν. Do not substitute kinematic viscosity into a formula that already contains density unless you also rearrange the formula correctly.

\mathrm{Re}=\frac{\rho UL}{\eta}=\frac{UL}{\nu},\qquad \nu=\frac{\eta}{\rho}

Specify the geometry and length convention before calculating. A Reynolds number alone does not describe every instability or boundary condition.

WORKED EXAMPLE

Check a flow regime with the correct length

A circular pipe has internal diameter 8.0 mm. A liquid of density 1000 kg/m³ and viscosity 0.0020 Pa·s flows with mean speed 0.10 m/s. Calculate the diameter-based Reynolds number.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: A dimensionless comparison helps you choose a flow model
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
\mathrm{Re}=\frac{\rho UL}{\eta}=\frac{UL}{\nu},\qquad \nu=\frac{\eta}{\rho}
Meaning: Specify the geometry and length convention before calculating. A Reynolds number alone does not describe every instability or boundary condition.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Viscosity, terminal motion and flow regimes

A critical speed requires a specified transition criterion

For ordinary straight circular pipes, introductory models often use a Reynolds number near 2000 as a conservative laminar-flow boundary, with a transition range above it. The exact onset depends on disturbances, inlet conditions, pipe roughness and the definition of transition. There is no universal critical speed shared by all fluids and geometries.

If a problem supplies a threshold Re_c under a stated pipe model, solve U_c = Re_c η/(ρD). At fixed fluid properties and threshold, the critical mean speed is inversely proportional to pipe diameter. A larger pipe reaches the same Reynolds number at a lower speed.

Volume flow rate is speed times cross-sectional area. At that prescribed threshold, Q_c = (πD²/4)U_c, so Q_c is proportional to D, even though U_c is proportional to 1/D. This distinction matters: a larger pipe can carry more volume per second while having a lower allowed mean speed.

Do not transfer a pipe-transition threshold to a small sphere. The Stokes drag approximation introduced next needs Re much less than one, a far more restrictive condition than laminar pipe flow. Laminar and creeping are different descriptions.

U_c=\frac{\mathrm{Re}_c\eta}{\rho D},\qquad Q_c=\frac{\pi\,\mathrm{Re}_c\eta D}{4\rho}

These equations use the given diameter-based criterion. They do not establish a universal turbulence threshold.

WORKED EXAMPLE

Separate the limiting speed from the limiting flow rate

For a specified straight-pipe model, use Re_c = 2000. A liquid has η = 0.0010 Pa·s and ρ = 1000 kg/m³. Find the critical mean speed for D = 10 mm. Compare the speed and volume-flow limits when D doubles.

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STOP & THINK

For the same pipe-transition criterion, fluid density and diameter, viscosity becomes three times as large. What is the ratio of new to old critical mean speed?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: A critical speed requires a specified transition criterion
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
U_c=\frac{\mathrm{Re}_c\eta}{\rho D},\qquad Q_c=\frac{\pi\,\mathrm{Re}_c\eta D}{4\rho}
Meaning: These equations use the given diameter-based criterion. They do not establish a universal turbulence threshold.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Viscosity, terminal motion and flow regimes

Stokes drag is a carefully limited model

A small rigid sphere moving slowly through a Newtonian fluid experiences a drag force of magnitude 6πηrv in the creeping-flow approximation. Here r is the sphere radius and v is its speed relative to the surrounding fluid far away. The drag acts opposite that relative velocity.

The assumptions include Re ≪ 1, a rigid sphere with a no-slip surface, and boundaries sufficiently far away. The fluid is treated as a continuum and the steady-flow drag relation is used. A narrow container, a deforming drop or a clean gas bubble can require corrections because the boundary conditions differ.

The linear dependence on v is characteristic of this regime. It is not the general drag law for every falling object. A large fast sphere may experience substantial inertial drag and cannot be analysed by extending the same formula without checking Re.

The formula gives a force, not a speed or acceleration by itself. To determine motion, draw the other forces and use Newton’s law. In a settling experiment the important additional forces are weight and buoyancy, and both depend on the sphere volume.

\mathbf F_{\mathrm d}=-6\pi\eta r\,\mathbf v_{\mathrm{relative}}

The vector sign means drag opposes motion relative to the far-field fluid. Radius is used here, while the Reynolds check below uses diameter.

STOP & THINK

A sphere moves downward through liquid that itself moves downward faster. In which direction is its creeping-flow drag?

Upward because the sphere moves down
Downward because the sphere moves upward relative to the liquid
Zero because both move down

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: Stokes drag is a carefully limited model
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
\mathbf F_{\mathrm d}=-6\pi\eta r\,\mathbf v_{\mathrm{relative}}
Meaning: The vector sign means drag opposes motion relative to the far-field fluid. Radius is used here, while the Reynolds check below uses diameter.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Viscosity, terminal motion and flow regimes

Terminal speed comes from balancing three forces

Choose downward as positive for a sphere settling in still liquid. Its weight is ρ_sVg downward, buoyancy is ρ_fVg upward, and Stokes drag is upward when its velocity is downward. At terminal motion acceleration is zero, so the effective weight (ρ_s − ρ_f)Vg is balanced by drag.

Insert the sphere volume 4πr³/3 and divide by 6πηr. This produces a terminal velocity proportional to r² and the density difference. Buoyancy must not be discarded merely because the object is sinking; it can still be a substantial fraction of the weight.

The signed expression also describes a light rigid sphere rising through liquid: when ρ_s < ρ_f, the calculated downward terminal velocity is negative. For equal densities the gravity-driven terminal velocity is zero in this model. These sign checks are useful before any numerical substitution.

Terminal does not mean stationary. The object can move at a nonzero constant velocity while the net force is zero. A sphere released from rest initially accelerates, but the steady drag formula alone should not be presented as an exact description of every transient fluid effect.

v_t=\frac{2r^2(\rho_s-\rho_f)g}{9\eta}

This is a signed downward terminal velocity under the rigid-sphere creeping-flow assumptions. Re must be checked using its magnitude.

WORKED EXAMPLE

Calculate a speed and then test the approximation

A rigid sphere of radius 1.0 mm and density 2000 kg/m³ settles in a wide container of still liquid with density 1000 kg/m³ and viscosity 1.0 Pa·s. Take g = 10 m/s². Estimate its terminal speed and diameter-based Reynolds number.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: Terminal speed comes from balancing three forces
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
v_t=\frac{2r^2(\rho_s-\rho_f)g}{9\eta}
Meaning: This is a signed downward terminal velocity under the rigid-sphere creeping-flow assumptions. Re must be checked using its magnitude.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Viscosity, terminal motion and flow regimes

A changed radius can invalidate an otherwise correct scaling

Within the Stokes terminal-speed model, doubling radius multiplies terminal speed by four. But the Reynolds number contains both diameter and speed, so it grows by a factor of eight. Repeating the radius scaling many times can quickly carry the system outside the assumptions used to derive it.

This is a general habit worth developing: use the model to make a prediction, then test whether the predicted state still belongs to the model’s domain. If it does not, the algebra has identified a failure of the approximation rather than a reliable final answer.

Viscosity enters both the drag and Reynolds number. At fixed sphere and densities, increasing η reduces terminal speed as 1/η, and consequently reduces the terminal Reynolds number as 1/η². A denser or larger object does not automatically settle in the same regime as a smaller calibration sphere.

WORKED EXAMPLE

Audit a proposed extrapolation

A sphere has a Stokes-model terminal Reynolds number of 0.020. Someone uses the same model for a sphere five times its radius in the same liquid. What Reynolds number does that prediction imply, and is the extrapolation self-consistent?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

If radius doubles while all other settling parameters stay fixed, the Stokes-model terminal Reynolds number changes by what factor?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: A changed radius can invalidate an otherwise correct scaling
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Viscosity, terminal motion and flow regimes

Design a viscosity measurement around the assumptions

A falling-sphere viscosity experiment can infer η from a measured terminal speed, known radius and density difference. Mark a timing interval far enough below release for the observed speed to become steady, and far enough from the bottom and walls to limit boundary effects. Agreement of speeds over successive equal distances is evidence of approximate steady motion.

Radius deserves careful measurement because it enters squared. Repeated timing can reduce random scatter, but it cannot remove a systematic radius error, an incorrect density or a persistent wall effect. Keep temperature controlled or recorded because viscosity can change substantially with temperature.

Estimate the Reynolds number using the inferred viscosity and measured speed. If it is not small, fitting Stokes’ formula does not turn the experiment into a valid Stokes measurement. Compare several small spheres or container widths to detect size and wall dependencies rather than treating one fitted value as definitive.

WORKED EXAMPLE

Infer viscosity and identify the sensitive input

A small rigid sphere has radius 0.50 mm and density 3000 kg/m³. It settles steadily at 1.0 mm/s in a liquid of density 1000 kg/m³. Take g = 10 m/s² and assume the Stokes conditions are independently satisfied. Infer viscosity.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Viscosity, terminal motion and flow regimes
Focus: Design a viscosity measurement around the assumptions
Learning goals:
- Relate Newtonian shear stress to a velocity gradient.
- Define Reynolds number with an explicit speed and length convention.
- Derive a critical mean speed from a specified pipe criterion.
- State the assumptions behind Stokes drag.
- Include buoyancy when deriving a signed terminal velocity.
- Recheck the flow regime after a parameter change or a viscosity measurement.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Viscous shear depends on a velocity gradient, not on uniform translation.
Reynolds number compares inertia and viscosity and depends on a declared length convention.
A pipe-transition criterion is not the creeping-flow criterion for a sphere.
Stokes terminal velocity follows from weight, buoyancy and linear drag.
Increasing sphere radius can invalidate the model through its strong effect on Reynolds number.
Repeated timing does not remove systematic radius, temperature or wall errors.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A Stokes-model sphere has terminal speed 2 mm/s. Its radius is halved with all fluid and density parameters unchanged. What terminal speed does the model predict?

Answer in mm/s.

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SELF CHECK 2

For a specified diameter-based threshold Re_c = 2000, use η = 0.002 Pa·s, ρ = 1000 kg/m³ and D = 0.020 m. Find the critical mean speed.

Answer in m/s.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

A sphere settles at terminal speed in still liquid. Which balance is correct?

Weight equals drag only
Weight equals buoyancy plus drag
The sphere has no forces

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SELF CHECK 4

May the rigid-sphere Stokes formula be transferred automatically to a clean gas bubble?

Yes, shape alone is sufficient
No; interfacial boundary conditions can change the drag
Only by replacing radius with diameter

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

State the flow criterion firstIntermediate · Hints and guided reasoning A larger sphere falls through syrupIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

In the Stokes regime, why does a sphere’s terminal speed scale as radius squared rather than radius cubed?

Compare your reasoning

Effective weight scales as radius cubed, while drag at a fixed speed scales as radius. Balancing them leaves terminal speed proportional to radius squared for fixed material and fluid properties.

Revisit the explanation →

A few days later: change the situation

You greatly increase the size of a falling sphere. Why should you check the Reynolds number before trusting the same terminal-speed formula?

Compare your reasoning

The flow regime may change as size and speed increase. Stokes drag assumes creeping flow at small Reynolds number. A neat radius-squared prediction does not prove that its own assumptions still hold.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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