Fluid intro
Fluids are materials that flow
when you push or drag on them. Unlike solids, they cannot hold a shear force
without moving – even a tiny tangential force causes continuous deformation[1][2]. This means fluids (liquids and
gases) have no fixed shape of their own and will keep deforming as long
as the force is applied[1][3]. In technical terms, a fluid has zero
shear modulus[3]. In a fluid, stress is proportional
to strain rate (how fast layers slide past each other) rather than to
strain itself[4].
In this blog, we explain fluids from
the ground up: what defines a fluid, how we model it as a continuous material,
and how shear forces cause flow. We introduce shear stress and shear
strain rate, and show how Newton’s law of viscosity (τ = μ·dγ/dt = μ·du/dy)
links them in a Newtonian fluid[5]. We contrast Newtonian fluids
(constant viscosity) with non-Newtonian fluids (viscosity changes with flow)[5][6], giving familiar examples like
water, air, honey, and blood. We describe typical flows (Couette and
Poiseuille) and their boundary conditions, and we discuss real-world
applications and misconceptions. The discussion is structured with clear
definitions, diagrams, tables, and even a timeline of key fluid-mechanics
milestones. By the end, you should understand why fluids continuously deform
under shear and how scientists use this idea in engineering and nature.
What Is a Fluid?
A fluid is any liquid or gas (or similar substance) that flows
under shear. That is, if you apply a tangential force, even a small one, the
fluid will start to deform and keep deforming[1][3]. In everyday words, fluids can’t
resist shear – they don’t “spring back” like a solid. For example, water in
a cup will start flowing or sloshing if you tilt the cup (because gravity
applies shear inside it), whereas a block of rubber would just twist a little
and stay bent[1][7].
Mathematically, this means fluids have zero shear modulus[3]. If you twist a fluid a
little, it will not hold that shape; it flows indefinitely. In fact, the
Encyclopedia Britannica states: “A fluid… is any liquid or gas… that cannot
sustain a tangential, or shearing, force when at rest and that undergoes a
continuous change in shape when subjected to such a stress.”[1]. In simple terms: solids
resist shear (stress ∝ strain) and hold their shape, but fluids deform
continuously under shear (stress ∝ strain rate)[7][4].
|
Property |
Solid |
Fluid |
|
Shear response |
Resists with elastic force |
Flows (no static shear resistance)[1] |
|
Deformation under constant shear |
Stops (reversible) |
Keeps going (irreversible flow)[1] |
|
Stress–strain |
Stress ∝ strain |
Stress ∝ strain rate[4] |
|
Shear modulus |
Nonzero (stiffness) |
Zero (cannot hold shape)[3] |
|
Shape (at rest) |
Fixed shape |
Takes container shape; no fixed shape |
<span style="font-size:0.8em;">*Table: Comparing solids
and fluids under shear.</span>
Fluids include liquids and gases. A simple example: water and
air both flow under shear, so they are fluids[8]. Even very viscous liquids
like honey eventually flow under their own weight, so honey is still a fluid
(specifically, a Newtonian fluid with high viscosity). Blood is also a
fluid, but its flow behavior changes with stress (it is non-Newtonian,
we’ll explain later)[6][9].
The Continuum
Assumption
Although fluids are made of molecules, engineers and scientists usually
ignore the discrete nature of matter and treat fluids as continuous. In the continuum
assumption, we imagine the fluid as a continuous material with smoothly
varying properties (density, velocity, pressure, etc.)[10]. We imagine a
“tiny” volume that contains many molecules, so small fluctuations average out,
but still large enough to be much smaller than the flow’s length scales[10].
Under this view, properties like velocity and pressure are well-defined
at every point and change gradually. For example, in air flow in a duct,
individual molecules zip around at ~1000 m/s, but the air’s average velocity
might be only a few m/s. We smooth over the molecular chaos by averaging in a
small volume[10]. The sampled
volume must be large enough (thousands of molecules) that random molecular
motion averages out, yet small enough to capture variations in flow[10]. This lets us use
calculus: we can talk about velocity gradients (du/dy), pressure fields, and so
on, as if the fluid were a smooth continuum[10].
The continuum model is incredibly useful. For example, aircraft and car
designers assume air acts as a continuous medium described by the Navier–Stokes
equations. Only at extreme scales (very high altitudes, or nanoscales) do
molecular effects (kinetic theory) become important. In everyday conditions,
continuum fluid mechanics works extremely well.
Shear
Stress, Strain Rate, and Viscosity
Shear Stress
Shear stress (τ) is the tangential force per unit area between layers of fluid.
Imagine stacking many thin layers (lamina) of fluid sliding over each other. If
the top layer is pulled faster than the bottom layer, the layers resist
relative motion by an internal force – that’s the shear stress. Formally, τ has
units of force/area (Pa, same as pressure) and acts tangentially to a surface
within the fluid.
Shear Strain and Strain Rate
When layers
slide, the fluid is deforming. If you mark two originally perpendicular lines
in the fluid, shear causes the angle between them to change. The shear
strain (γ) measures that angle change. If the top layer moves relative to
the bottom, the fluid’s shape distorts; shear strain quantifies that.
The shear
strain rate (dγ/dt or sometimes denoted 
) measures how fast this distortion happens. For simple shear
between parallel plates, the strain rate equals the velocity difference divided
by the distance: if the top plate moves at speed 
relative to a stationary bottom
plate separated by 
, then 
. More generally, in a 3D flow, the local shear strain rate in one
direction is 
, the gradient of velocity in the perpendicular direction.
Newton’s Law of Viscosity
Sir Isaac Newton
first noted that for many fluids, shear stress is proportional to shear strain
rate. This is called Newton’s law of viscosity:
where μ (mu) is the
dynamic viscosity of the fluid[5]. Viscosity μ is
the constant of proportionality: it measures a fluid’s “thickness” or
resistance to flow. For Newtonian fluids, μ is constant (independent of
shear rate). The Anton Paar Rheology Wiki explains: “According to Newton’s
Law, shear stress is viscosity times shear rate. Thus the viscosity (η) is
shear stress divided by shear rate… Only Newtonian liquids can be described by
this simple relation.”[5]. The equation can
be rearranged to define viscosity:
In SI units,
viscosity is Pa·s (equivalently N·s/m²). For example, water at room temperature
has μ ≈ 0.001 Pa·s, air about 0.000018 Pa·s, and honey roughly 2–10 Pa·s (much
thicker)[5]. A useful
mnemonic: 1 Pa·s = 1000 mPa·s (or centipoise).
Behavior of Solids vs. Fluids
Recall
solids and fluids differ fundamentally in their stress–strain relations. For a solid
(like rubber), apply a shear stress and it deforms until an equilibrium strain,
then holds that shape (elastic or plastic deformation). In contrast, a fluid
will keep deforming under constant shear stress, never reaching a steady
strain. In fact, as the fluid mechanics texts say, “in a solid, shear stress is
a function of strain, but in a fluid, shear stress is a function of strain
rate”[4].
This is why
dropping a fluid from a tap forms a continuous stream – the stress from gravity
causes ever-growing deformation (flow). No shear stress is “sustained” without
motion; at rest the maximum shear stress in a fluid is zero[2].
Figure: Couette flow between parallel plates. The top plate
moves at speed U, the bottom is fixed. Fluid layers slide past each other with
a constant velocity gradient 
. According to Newton’s law of
viscosity, the shear stress τ = μ·(U/h) is the same across all layers[11].
Newtonian vs.
Non-Newtonian Fluids
Not all fluids follow a simple linear law (τ = μ·γ̇). Newtonian
fluids have a constant μ. In these fluids, if you double the shear rate,
the shear stress doubles. A plot of τ vs γ̇ is a straight line (viscosity =
slope)[5][12]. Examples
of Newtonian fluids (in everyday conditions) include water, air, alcohol,
glycerol, and many oils[8][9]. Even
honey and corn syrup, though viscous, behave Newtonian (their viscosity stays
roughly constant with shear) under normal stirring.
In contrast, non-Newtonian fluids do not have a single constant
μ. Their viscosity changes with shear rate or time. Some thicken under shear
(shear-thickening), like a cornstarch-water “oobleck” (feels solid when you
punch it). Others thin under shear (shear-thinning), like ketchup or blood
(they flow more easily at higher strain rates). There are also viscoelastic
fluids (e.g. Silly Putty, silly glue) that behave partly like solids and partly
like fluids. For instance, Silly Putty stretches slowly (fluid-like) but can
bounce like a rubber ball if thrown (solid-like under sudden force)[13].
|
|
Newtonian Fluids |
Non-Newtonian Fluids |
|
Definition |
τ ∝ γ̇ (linear) (constant μ)[5] |
τ not ∝ γ̇ (μ varies with γ̇ or time) |
|
Viscosity (η) |
Constant (depends only on temperature)[5] |
Variable (depends on shear rate, history, etc.) |
|
Examples |
Ketchup, blood, paints, polymer solutions |
|
|
Behavior |
τ–γ̇ plot is straight line |
τ–γ̇ plot is curved or has yield stress |
<span style="font-size:0.8em;">Table: Newtonian vs.
non-Newtonian fluids.</span>
One key example: blood is a shear-thinning non-Newtonian fluid.
As shear rate increases (e.g. faster flow), blood’s viscosity drops[6].
Medical fluid dynamics often assumes blood Newtonian for simplicity, but
acknowledges real blood (with cells) has a higher apparent viscosity at low
shear and thins out at higher shear[6]. In
summary, “blood is a non-Newtonian fluid with a shear-thinning nature”[6].
Viscosity units: We mentioned viscosity μ has
units Pa·s. Sometimes a convenient unit is the Poise (P): 1 P = 0.1 Pa·s. So
water (≈0.001 Pa·s) is 0.01 P, air (~0.000018 Pa·s) is 0.00018 P. In fluid
mechanics, SI units (Pa·s) are most common[5].
Simple
Shear Flows: Couette and Poiseuille
To build intuition, consider two classic flows:
- Couette flow: Fluid between two parallel
plates, one moving sideways at velocity U while the other is fixed.
This creates a simple shear flow with velocity varying linearly
from 0 at the fixed plate to U at the moving plate (no pressure
gradient). The shear rate is
(with h = plate
separation). Newton’s law then gives a constant shear stress 
everywhere in the fluid[11]. Couette flow is used to measure viscosity and illustrates
shear-driven flow. Figure [41] shows the layers: velocity increases
linearly, and arrows indicate the shear stress constant throughout[11]. - Poiseuille flow: Fluid driven through a
pipe or between stationary plates by a pressure difference. For flow in a
circular pipe, the steady laminar solution is a parabolic velocity
profile: fastest at the center, zero at the walls. Mathematically
(Hagen–Poiseuille law)[14]. The shear rate is zero at the center and highest at the wall
(due to no-slip). The resulting shear stress in a Newtonian fluid is
proportional to du/dr, and the pressure drop is 
where Q is flow rate[14]. Plane Poiseuille (between parallel plates) is similar: a
parabolic profile across the gap. In both cases, flows obey τ = μ·du/dy
(or du/dr) locally.
In practical terms, Couette flow helps define viscosity, while
Poiseuille flow explains how much pressure is needed to pump a fluid through
pipes.
Real-World
Examples and Applications
Fluids are all around us. Water and air are common
Newtonian fluids[8]; our
weather, rivers, and drinking water involve fluid dynamics. Oils
(lubricants, engine oil) are viscous Newtonian fluids[9]. Honey
and syrup are also Newtonian but very viscous (pours slowly)[5].
Non-Newtonian fluids are everywhere too. Blood
(flowing through your arteries) is shear-thinning[6].
Ketchup and mustard are classic shear-thinning fluids: they stick on the bottle
until you shake them (applying stress), then they flow out more easily. Paints
and some cosmetics are shear-thinning to help application. Toothpaste,
mayonnaise, and cream are examples of yield-stress fluids (a type of
non-Newtonian) – they behave like a solid until a certain stress makes them
flow. Some slurries (wet cement, mud) are shear-thickening or thixotropic.
Applications: Engineers use fluid properties
in countless ways. Water flow principles design pipes and dams. Aerodynamics of
air uses fluid equations. Blood flow research uses non-Newtonian models in
medical devices. Microfluidics and inkjet printers rely on fluid rheology.
Rheometers measure viscosity as described by τ = μ·γ̇[5].
Hydraulics exploits fluids’ incompressibility to transmit forces. Even cooking
(mixing batter, ketchup flow) involves fluid ideas.
Common Misconceptions
·
“Fluids can’t resist any
force.” Not true: fluids resist normal (perpendicular)
stresses via pressure. They do push back equally from all sides
(Pascal’s law). What they cannot resist is shear stress: a fluid at rest
has zero internal shear stress[2].
·
“Fluids aren’t solids so they
have no strength.” Fluids don’t have shear rigidity,
but they do have viscosity (internal friction) and thus resist motion to some
extent. Think honey: it flows, but very slowly if undisturbed because of
high viscosity.
·
“Only liquids are fluids.” Gases are fluids too. Air, oxygen, helium – all behave as fluids under
shear, especially at everyday pressures (air’s viscosity ~1.8×10^-5 Pa·s). Both
liquids and gases share the property of flowing under shear[1][3].
·
“All non-Newtonian substances
are solids.” Not so – some non-Newtonian fluids appear
solid-like only under certain conditions. For example, Silly Putty is a
non-Newtonian polymer: if you slowly stretch it, it flows (fluid-like), but if
you yank it quickly, it snaps like a solid[13]. Substances like
molten polymers or gelatin desserts change behavior with temperature or force.
Mermaid diagrams often show how we classify fluids. Below is a
historical timeline of key discoveries in fluid mechanics, illustrating how our
understanding evolved:
timeline
title
History of Fluid Mechanics
250 BC :
Archimedes discovers buoyancy (fluids)[15]
1687 :
Newton formulates laws of motion and introduces idea of viscosity[16]
1757 : Euler
derives equations for inviscid (frictionless) fluid flow
1785 :
Navier adds viscosity to fluid equations (Navier–Stokes)
1845 :
Stokes publishes law of viscous friction for spheres
1883 :
Osborne Reynolds discovers laminar vs turbulent flow (Reynolds number)
1904 :
Prandtl develops boundary layer theory
1960s: Bird,
Stewart & Lightfoot formalize fluid rheology (transport phenomena)[9]
Classroom
Demonstrations and Experiments
·
Viscosity race: Tilt two identical channels, one with water and one with honey.
Observe how honey flows much slower. Use a stopwatch to compare times – this
shows how higher viscosity means slower deformation.
·
Paint brushes or shampoo: These often show non-Newtonian behavior (shear-thinning). Stir yogurt
or ketchup gently and observe they resist motion at first, then flow faster
when stirred harder.
·
Couette flow demo: Attach two smooth plates with a small gap and sandwich a fluid (e.g.
glycerin). Move the top plate slowly by hand; insert a dye or small dot on the
fluid layers to watch linear flow profile (velocity vs depth). This illustrates
shear layers moving relative to each other[11]
.
·
Oobleck (cornstarch+water): Mix corn starch and water to get a runny paste that turns solid-like
under sudden impact. Jump or hit it and see it resist (shear-thickening), then
stand still and watch it settle (fluid-like).
·
Ball drop in fluids: Drop identical balls into air and into oil. The ball slows quickly in
oil due to viscous drag. Use high-viscosity fluids (glycerin) to exaggerate the
effect (demonstrates viscosity).
·
Viscometer using gravity: Fill a graduated cylinder with fluid and drop a steel ball; time how
long to fall a certain distance. Use Stokes’ law (for low speeds) to calculate
viscosity μ = (force/gravity)/rate-of-strain.
These simple demos bring out the idea that some substances flow easily,
others resist, and that fluids always flow under shear (given enough
force).
Summary
Fluids
are materials (liquid or gas) defined by their inability to resist static
shear: they flow under shear[1][3]. In technical terms, fluids have zero shear modulus and
their stress is tied to the rate of deformation not the deformation
itself[1][4]. Viscosity μ quantifies internal friction: Newton’s law
of viscosity (τ = μ·dγ/dt) links shear stress and shear rate[5].
We
classify fluids into Newtonian (constant viscosity) and non-Newtonian (variable
viscosity) based on how τ relates to γ̇. Common fluids like water, air, oils,
and honey are Newtonian[8][9], while blood, ketchup, polymers and others show
non-Newtonian behavior[6]. Typical flows like Couette flow (between moving
plates) and Poiseuille flow (pressure-driven pipe flow) illustrate these ideas
with linear or parabolic velocity profiles.
In
real life, fluid mechanics underpins hydraulics, weather, blood circulation,
and countless technologies. A key take-home: Even though fluids have no
rigidity, they still have resistance (viscosity) and continuously deform under
shear. There’s no “at rest” shear stress in a true fluid[2], so fluids will always move if any shear force is
applied. Understanding this helps engineers design engines, pipes, medical
devices, and lets scientists explain natural flows from river currents to air
in your lungs.
Quick Check – MCQs:
1. Which statement is true? (A) A solid resists shear indefinitely; a
fluid keeps deforming under shear. (B) A fluid resists shear like a spring; a
solid flows under shear.
2. Newton’s law of viscosity is τ = μ·(du/dy). What are the units of μ
in SI? (A) Pa·s (B) Pa/m (C) N/m (D) Pa·s/m.
3. Which is a Newtonian fluid at room conditions? (A) Water (B) Ketchup
(C) Toothpaste (D) Cornstarch suspension.
Answers: 1) (A). 2) (A) Pa·s (Pascal-second)[5]. 3) (A) Water – its viscosity stays constant with
shear; ketchup and cornstarch fluid are non-Newtonian (viscosity changes with
shear), and toothpaste has yield-stress[5][6].
[1] Fluid | Definition, Models, Newtonian Fluids, Non-Newtonian Fluids,
& Facts | Britannica
https://www.britannica.com/science/fluid-physics
[2] Full text of "Introduction To The Mechanics Of A Continuous
Medium"
[3] [4] [13] Fluid - Wikipedia
https://en.wikipedia.org/wiki/Fluid
[5] Basics of viscometry | Anton Paar Wiki
https://wiki.anton-paar.com/us-en/basic-of-viscometry/
[6] Frontiers | Comparison of Newtonian and Non-newtonian Fluid Models in
Blood Flow Simulation in Patients With Intracranial Arterial Stenosis
https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2021.718540/full
[7] Fluid mechanics_inroductionIt is about materials enginerring course
similar to william callister subject code is MT30001.It is our sirs slides iam
from iit kgp and applications_Classes 1 to 3.pptx
[8] Newtonian fluid - Wikipedia
https://en.wikipedia.org/wiki/Newtonian_fluid
[9] Newton’s Law | Anton Paar Wiki
https://wiki.anton-paar.com/us-en/basics-of-rheology/newtons-law/
[10] Continuum Assumption - an overview | ScienceDirect Topics
https://www.sciencedirect.com/topics/engineering/continuum-assumption
[11] Couette flow - Wikipedia
https://en.wikipedia.org/wiki/Couette_flow
[12] Viscosity of Newtonian and Non-Newtonian Fluids
https://www.rheosense.com/applications/viscosity/newtonian-non-newtonian
[14] Hagen–Poiseuille equation - Wikipedia
https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation
[15] [16] Fluid Mechanics - an overview | ScienceDirect Topics
https://www.sciencedirect.com/topics/engineering/fluid-mechanics
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