Velocity Gradient Formula: dv/dx, SI Unit and Examples

The velocity gradient formula is $\frac{dv}{dx} = \frac{\Delta v}{\Delta x}$ : take the difference in speed between two layers of a flowing fluid and divide by the distance between them, measured at right angles to the flow. The result, in units of per second, tells you how sharply the speed changes as you move across the stream.

Engineers often call the same quantity the shear rate. It matters because viscous drag depends on it, not on the speed itself. Two layers both racing along at 5 m/s feel no friction between them; layers at 5 m/s and 4 m/s, a millimetre apart, rub hard. That is why the gradient appears in Newton's law of viscosity (see the formula of viscosity) and throughout the fluids formula sheet.

Quick facts

Formula
$\frac{dv}{dx} = \frac{\Delta v}{\Delta x}$
SI unit
s⁻¹, from (m/s) ÷ m
Dimensional formula
[M⁰ L⁰ T⁻¹]
Also called
shear rate or rate of shear strain
Used in
$F = \eta A \frac{dv}{dx}$
Direction of x
Across the flow, perpendicular to the velocity

Meaning of each term

Terms in dv/dx = Δv / Δx
Symbol	Meaning	SI unit
$\frac{dv}{dx}$	Velocity gradient at a point (the limit of the ratio)	s⁻¹
$\Delta v$	Faster layer speed minus slower layer speed	m/s
$\Delta x$	Separation of the two layers, measured across the flow	m

The letter x here is not the direction of flow. If the fluid moves left to right, x runs from the still wall out into the stream. Students who mix these up get a meaningless answer. When the speed rises evenly from the wall, as in a thin film under a sliding plate, the gradient is the same everywhere and the simple ratio gives the exact value.

Using the velocity gradient formula step by step

Find two layers whose speeds you know, often a still surface (speed zero) and a moving one.
Subtract the speeds to get Δv in m/s.
Measure the gap between those layers at right angles to the flow and convert it to metres.
Divide Δv by Δx. The answer is in s⁻¹.

Put the result into $F = \eta A \frac{dv}{dx}$ to get the viscous force, or rearrange it with the coefficient of viscosity formula to find η. The calculator above handles the gradient step.

Worked examples

Example 1: two layers of water

Two layers of water 0.50 mm apart move at speeds that differ by 0.20 m/s. Find the velocity gradient.

Convert: Δx = 0.50 mm = 5.0 × 10⁻⁴ m.

$\frac{dv}{dx} = \frac{0.20}{5.0 \times 10^{-4}} = 400\ \text{s}^{-1}$

Example 2: gradient and then the drag

A plate of area 0.050 m² slides at 3.0 cm/s over a 1.5 mm film of oil (η = 0.80 Pa s) resting on a fixed table. Find the velocity gradient and the force needed.

Convert: Δv = 0.030 m/s, Δx = 1.5 × 10⁻³ m.

$\frac{dv}{dx} = \frac{0.030}{1.5 \times 10^{-3}} = 20\ \text{s}^{-1}$

$F = \eta A \frac{dv}{dx} = 0.80 \times 0.050 \times 20 = 0.80\ \text{N}$

Example 3: an average gradient in a river

A stream flows at 1.2 m/s at its surface and is almost still at the bed, 3.0 m below. Estimate the average velocity gradient.

$\frac{dv}{dx} \approx \frac{1.2 - 0}{3.0} = 0.40\ \text{s}^{-1}$

The true profile is curved, steepest near the bed, so this is only an average.

Keep both lengths in metres. A gap in millimetres left unconverted makes the gradient 1000 times too small.

Units, dimensions and the speed profile

Dividing metres per second by metres leaves per second:

$\left[\frac{dv}{dx}\right] = \frac{[\text{L T}^{-1}]}{[\text{L}]} = [\text{M}^0\,\text{L}^0\,\text{T}^{-1}]$

So the velocity gradient has the same dimensions as frequency and angular velocity, though it means something quite different. On a graph of speed against distance from the wall, dv/dx is the slope, and a straight line tells you the gradient is uniform. In a round pipe with smooth flow the speed profile is a parabola, fastest on the axis, so the gradient is zero at the centre and largest at the wall. That is where the viscous stress on the pipe is greatest.

Mistakes to watch for

Dividing by a distance along the flow. Δx must run across the layers.
Using one speed instead of a difference. The gradient needs two speeds.
Giving the unit as m/s². The correct unit is s⁻¹; there is no length left.
Assuming a uniform gradient in a wide channel. That is fine for thin films but only an estimate for rivers and pipes.

Frequently asked questions

What is the formula of velocity gradient?

Velocity gradient = Δv / Δx, the difference in speed between two fluid layers divided by their separation measured perpendicular to the flow. In calculus form it is dv/dx.

In which unit is a velocity gradient measured?

Per second, s⁻¹, which comes from (m/s) divided by m. Its dimensions work out to [M⁰ L⁰ T⁻¹].

Is velocity gradient the same as acceleration?

No. Acceleration is the change in velocity per unit time, in m/s². Velocity gradient is the change in velocity per unit distance across a flow, in s⁻¹.

Why is velocity gradient important in viscosity?

The viscous force between layers is proportional to it: F = ηA(dv/dx). A steeper gradient means stronger internal friction for the same fluid and area.