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Viscous Force Formula

Force and laws of motion formulasFluidsClass 11Updated 1 Nov 2026

F=ηAdvdxF = \eta A \frac{dv}{dx}
SI unit
newton (N)
Dimensions
[M¹ L¹ T⁻²]
where
  • F viscous (fluid friction) force N
  • η coefficient of viscosity Pa s
  • A area of the layer in contact m²
  • dv difference in speed between layers m/s
  • dx distance between the layers m

The viscous force formula is F=ηAdvdxF = \eta A \frac{dv}{dx}, which says the drag between two layers of a flowing liquid equals the viscosity η times the area of contact times the velocity gradient. Drag a 0.25 m² glass sheet at 4.0 cm/s across oil 1.0 mm deep, with η = 0.50 Pa s, and the oil resists with 5.0 N.

This is Newton's law of viscosity. A minus sign is sometimes written in front, as a reminder that the drag acts against the sliding of the layers; this page works with sizes. For a small sphere moving slowly through a fluid the same physics gives Stokes' law, F=6πηrvF = 6\pi\eta r v. Viscous force is fluid friction, a close cousin of the solid friction met in the laws of motion.

Quick facts

  • Layers of fluid

    F=ηAdvdxF = \eta A \frac{dv}{dx}

  • Sphere, slow flow

    F=6πηrvF = 6\pi\eta r v (Stokes' law)

  • SI unit

    newton (N)

  • Unit of η

    Pa s, equal to N s/m² or kg/(m s)

  • Dimensions of η

    [M¹ L⁻¹ T⁻¹]

  • Direction

    Opposes the relative motion of the layers

Formula of viscous force, symbol by symbol

Symbols in F = ηA dv/dx
SymbolMeaningSI unit
FFTangential force between layersN
η\etaViscosity: how thick or sticky the fluid isPa s
AAArea of the layerm²
dvdx\frac{dv}{dx}Velocity gradient: change of speed per metre across the flows⁻¹

Picture honey on a spoon as a stack of thin sheets. The sheet touching the spoon moves with it, the far sheet barely moves, and each sheet drags on its neighbours. The steeper the change in speed across the stack, the larger the drag. The velocity gradient and coefficient of viscosity pages treat each factor on its own.

Stokes' law for a falling sphere

For a small ball (radius r) drifting slowly at speed v in a liquid or gas, with smooth flow around it, the viscous force is

F=6πηrvF = 6\pi\eta r v

It grows in step with speed. A ball bearing dropped in oil speeds up until this drag plus the buoyant force balances its weight. From then on it falls at a steady speed given by vt=29⋅r2(ρ−σ)gηv_t = \frac{2}{9}\cdot\frac{r^2 (\rho - \sigma) g}{\eta}, where ρ is the density of the sphere and σ that of the fluid. That speed is the terminal velocity.

Worked examples on the viscous force formula

Example 1: a sliding glass sheet

A glass sheet of area 0.25 m² is drawn at 4.0 cm/s across a 1.0 mm layer of oil lying on a fixed bench. For this oil η = 0.50 Pa s. Find the viscous force.

In SI units dv = 0.040 m/s and dx = 0.0010 m, so the velocity gradient is 0.040 ÷ 0.0010 = 40 s⁻¹.

F=0.50×0.25×40=5.0 NF = 0.50 \times 0.25 \times 40 = 5.0\ \text{N}

Example 2: Stokes' law, with mm and cm/s

A steel bead 1.5 mm in radius sinks at 2.0 cm/s through a liquid whose viscosity is 1.2 Pa s. Find the viscous drag.

In metres and m/s: r = 1.5 × 10⁻³ m, v = 0.020 m/s.

F=6π×1.2×1.5×10−3×0.020=6.79×10−4 N≈6.8×10−4 NF = 6\pi \times 1.2 \times 1.5 \times 10^{-3} \times 0.020 = 6.79 \times 10^{-4}\ \text{N} \approx 6.8 \times 10^{-4}\ \text{N}

Example 3: finding η

A force of 0.60 N keeps a 0.050 m² plate moving at 0.30 m/s over a fluid layer 1.0 mm thick. Find the viscosity.

η=F dxA dv=0.60×0.00100.050×0.30=6.0×10−40.015=0.040 Pa s\eta = \frac{F\,dx}{A\,dv} = \frac{0.60 \times 0.0010}{0.050 \times 0.30} = \frac{6.0 \times 10^{-4}}{0.015} = 0.040\ \text{Pa s}

Units and dimensions

Being a force, the viscous drag is measured in N, dimensionally [M¹ L¹ T⁻²]. Rearranging, η=FA (dv/dx)\eta = \frac{F}{A\,(dv/dx)} has unit N ÷ (m² × s⁻¹) = N s/m², which is the pascal second. Its dimensions are [M L T⁻²] ÷ ([L²][T⁻¹]) = [M¹ L⁻¹ T⁻¹], with the dimensions of η worked out step by step on a separate page. In CGS work you will meet the poise; ten poise make one pascal second.

Viscous force compared with solid friction

Viscous force against sliding friction
FeatureViscous forceSliding friction
Depends on speedYes, grows with relative speedNearly independent of speed
Depends on areaYes, proportional to ANo, for dry surfaces
Depends on normal forceNoYes, f = μN
Effect of heatingLiquids get less viscousLittle change

At higher speeds the flow becomes turbulent and drag grows roughly with the square of speed instead; that regime is handled by the drag force formula.

Common errors with viscous force

  • Leaving the layer thickness in mm while the speed is in m/s.
  • Using diameter instead of radius in Stokes' law.
  • Applying Stokes' law to fast or large objects, such as a falling person, where the flow is turbulent.
  • Writing the unit of η as Pa/s; it is Pa s, pascal multiplied by second.

Frequently asked questions

What is the formula of viscous force?

Between fluid layers, F = ηA(dv/dx). For a small sphere moving slowly, Stokes' law gives F = 6πηrv.

In which unit is viscous force measured?

The newton, as for any force. The coefficient of viscosity η is measured in pascal second, Pa s.

Why does viscous force depend on speed?

A faster layer sets up a steeper velocity gradient across the fluid, and the drag between layers is proportional to that gradient.

What happens to viscosity when a liquid is heated?

For liquids it falls, so the same flow produces less viscous force. For gases it rises slightly with temperature.