Is viscosity a vector?

NCERT Exemplar · Q11

Q. Is viscosity a vector?

Andhra Pradesh BieapShort· 2mImportance★★★★★est

77% · 41/53 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Fluid Properties

Fluid Properties: From Intuition to Precision

Imagine you're holding a glass of water. Now imagine holding a glass of honey. You know instantly they behave differently — honey pours slowly, water splashes easily. That difference is what fluid properties capture. A fluid is anything that flows: liquids, gases, even some granular materials like sand (though we'll stick to liquids and gases here).

The key idea: fluids deform continuously under any shear stress, no matter how small. A solid resists deformation; a fluid gives way. But not all fluids give way the same way — that's where properties come in.

Density ( $\rho$ )

Intuition: A kilogram of feathers takes up much more space than a kilogram of lead. Density tells you how much mass is packed into a given volume.

Precise statement: Density is mass per unit volume.

\rho = \frac{m}{V}

Units: $\text{kg/m}^3$ in SI. Water at $4^\circ\text{C}$ has $\rho \approx 1000\ \text{kg/m}^3$ — a useful benchmark. Air at sea level is about $1.2\ \text{kg/m}^3$ .

Note

Density changes with temperature and pressure, especially for gases. For liquids, it's nearly constant — that's why we often call them "incompressible."

Specific Weight ( $\gamma$ )

Intuition: How heavy is that fluid? Not just mass — weight. A bucket of water feels heavier than the same bucket of air because gravity pulls harder on the denser fluid.

Precise statement: Specific weight is weight per unit volume.

\gamma = \rho g

where $g \approx 9.81\ \text{m/s}^2$ . Units: $\text{N/m}^3$ . For water, $\gamma \approx 9810\ \text{N/m}^3$ .

Specific Gravity (SG)

Intuition: "How many times heavier than water is this fluid?" A number without units — pure comparison.

Precise statement: The ratio of a fluid's density to the density of water at a reference temperature (usually $4^\circ\text{C}$ ).

\text{SG} = \frac{\rho_{\text{fluid}}}{\rho_{\text{water}}}

Mercury has SG $\approx 13.6$ — it's 13.6 times denser than water. That's why a small column of mercury can balance a tall column of water in a barometer.

Viscosity ( $\mu$ )

Intuition: Honey is "thick," water is "thin." Viscosity measures a fluid's resistance to flow — its internal friction. Imagine sliding a thin layer of fluid between two plates: the more viscous the fluid, the harder you must pull.

Precise statement: Viscosity (dynamic viscosity) is the proportionality constant between shear stress $\tau$ and the velocity gradient (rate of shear strain) in the fluid.

For a fluid between two parallel plates separated by distance $dy$ , with top plate moving at speed $dV$ :

\tau = \mu \frac{dV}{dy}

This is Newton's law of viscosity. Units: $\text{Pa·s}$ (or $\text{N·s/m}^2$ ). Water at $20^\circ\text{C}$ has $\mu \approx 1.0 \times 10^{-3}\ \text{Pa·s}$ ; honey is about $2\ \text{Pa·s}$ — two thousand times more viscous.

Watch out

Viscosity is not density. Mercury is dense but flows easily (low viscosity). Honey is less dense but flows slowly (high viscosity). Don't confuse them.

Kinematic viscosity ( $\nu$ ) is dynamic viscosity divided by density:

\nu = \frac{\mu}{\rho}

Units: $\text{m}^2/\text{s}$ . It appears naturally in problems where both inertial and viscous forces matter.

Surface Tension ( $\sigma$ )

Intuition: A water strider walks on water. A needle floats even though steel is denser than water. The surface of a liquid acts like a stretched elastic membrane.

Precise statement: Surface tension is the force per unit length acting along the surface of a liquid, tending to minimize the surface area.

\sigma = \frac{F}{L}

Units: $\text{N/m}$ . For water-air at $20^\circ\text{C}$ , $\sigma \approx 0.073\ \text{N/m}$ . It arises because molecules at the surface experience a net inward pull (fewer neighbors above), creating tension.

Tip

Surface tension explains why small droplets are spherical — a sphere has the smallest surface area for a given volume.

Capillarity

Intuition: Water climbs up a narrow glass tube; mercury is pushed down. That's capillarity — the combined effect of surface tension and adhesion (attraction to the tube walls) versus cohesion (attraction within the liquid).

Precise statement: The rise (or fall) of a liquid in a narrow tube due to surface tension is given by:

h = \frac{2\sigma \cos\theta}{\rho g r}

where $\theta$ is the contact angle (wetting angle), $r$ is the tube radius. For water in clean glass, $\theta \approx 0^\circ$ (rises); for mercury, $\theta \approx 130^\circ$ (falls).

Bulk Modulus ( $K$ )

Intuition: How hard is it to squeeze a fluid? Gases compress easily; liquids barely compress at all. Bulk modulus measures resistance to uniform compression.

Precise statement: The ratio of pressure increase to the resulting volumetric strain (fractional change in volume):

K = -V \frac{dP}{dV}

Units: $\text{Pa}$ . For water, $K \approx 2.2 \times 10^9\ \text{Pa}$ — enormous. For air at atmospheric pressure, $K \approx 1.4 \times 10^5\ \text{Pa}$ — about 15,000 times smaller.

Important

In most engineering problems, liquids are treated as incompressible ( $K \to \infty$ ). Gases are compressible unless the pressure changes are very small.

Vapor Pressure ( $P_v$ ) …

Start your 14-day free trial to unlock the full solution →

Viscosity is a measure of a fluid's resistance to flow. It is a scalar quantity for most common fluids (Newtonian and isotropic) and a tensor quantity for more complex fluids, but it is never a vector quantity.

To understand whether viscosity is a vector, we first need to clarify what a vector is and what viscosity represents.

Understanding Physical Quantities

Physical quantities are broadly classified based on whether they have direction in addition to magnitude:

Scalar quantities possess only magnitude. Examples include mass, temperature, density, and energy.
Vector quantities possess both magnitude and direction. Examples include force, velocity, acceleration, and momentum.
Tensor quantities are more complex mathematical objects that generalize scalars and vectors. A scalar is a 0th-order tensor, and a vector is a 1st-order tensor. Higher-order tensors describe relationships between vectors or other tensors, often representing properties that vary with direction in a more intricate way than a simple vector. Stress and strain are examples of 2nd-order tensors.

What is Viscosity?

Viscosity is a fundamental property of fluids that quantifies their resistance to flow or deformation under shear stress. Imagine stirring honey versus water; honey is much more viscous, meaning it resists flow more strongly. This resistance arises from the internal friction between layers of the fluid moving at different velocities.

\tau

Now, let's determine if viscosity fits the definition of a vector.

Viscosity as a Scalar (for Newtonian, Isotropic Fluids):

For the vast majority of fluids encountered in introductory physics and engineering (like water, air, oils under normal conditions), viscosity is considered a scalar quantity. This is because:
- The resistance to flow (viscosity) is an intrinsic property of the fluid itself.
- It does not inherently point in a specific direction. If you shear a fluid from left to right, its resistance to that shear is the same as if you shear it from top to bottom (assuming the fluid is isotropic, meaning its properties are uniform in all directions).
- In the formula $\tau = \mu \frac{du}{dy}$ , $\mu$ acts as a proportionality constant. While shear stress ( $\tau$ ) and velocity gradient ( $\frac{du}{dy}$ ) are related to directional quantities (force, area, velocity, distance), the viscosity $\mu$ itself is a single value that describes the fluid's internal resistance, irrespective of the orientation of the applied shear.
Viscosity as a Tensor (for Complex Fluids): …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

AP EAPCET 2026Set eng-2026-05-18-FN1 markMCQ
$Q. Match the law/ principle with concerned applications: a) Hydraulic Lift — i) Bernoulli's Principle; b) Speed of Efflux — ii) Torricelli's Law; c) Dynamic lift — iii) Stoke's Law; d) Viscous drag force — iv) Pascal's Law (A) a - i, b - ii, c - iii, d - iv (B) a - i, b - iv, c - ii, d - iii (C) a - iv, b - i, c - iii, d - ii (D) a - iv, b - ii, c - i, d - iii$
›Reveal solutionSolution
This tests matching fluid-mechanics laws to their standard named applications. Answer: a-iv, b-ii, c-i, d-iii.

Concept and Intuition

Each of these is a classic textbook pairing:

Pascal's Law: pressure applied to an enclosed fluid is transmitted undiminished in all directions — the working principle of a hydraulic lift/press.

Torricelli's Law: the speed of efflux of a liquid from an orifice under gravity, $v=\sqrt{2gh}$ — a specific named result (itself derivable from Bernoulli's equation, but conventionally cited by its own name for 'speed of efflux').

Bernoulli's Principle: relates pressure and velocity along a streamline; explains dynamic lift (faster flow over a curved/airfoil surface → lower pressure → net lift).

Stoke's Law: gives the viscous drag force on a small sphere moving through a fluid, $F=6\pi\eta rv$ .

Step-by-Step Solution

a) Hydraulic Lift → Pascal's Law → (iv).

b) Speed of Efflux → Torricelli's Law → (ii).

c) Dynamic lift → Bernoulli's Principle → (i). …
Start your 14-day free trial to see the full solution →
AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
$r_1$
›Reveal solutionSolution
Tests combining the excess-pressure-due-to-surface-tension formula for a soap bubble (which has two surfaces) with the ideal gas law to compare masses of trapped air. Answer: $r_1^2:r_2^2$ .

Concept and Intuition

A soap bubble has two liquid surfaces (inner and outer film surfaces), so its excess internal pressure is $\dfrac{4T}{r}$ (twice the single-surface value $2T/r$ for a droplet). Since the bubbles are in vacuum, there is no external atmospheric pressure to add — the entire internal pressure IS this surface-tension term. Using $PV=nRT$ (constant temperature, so mass $\propto PV$ since $n\propto$ mass) lets us find how mass scales with radius.

Step-by-Step Solution

Excess pressure inside a soap bubble (two surfaces): $P = \dfrac{4T}{r}$ . In vacuum, this is the total pressure of the trapped air (no atmosphere adds to it).

Volume of the bubble: $V = \dfrac{4}{3}\pi r^3$ .

At constant temperature, mass of enclosed air $m \propto PV$ (from $PV = \dfrac{m}{M}RT$ , with $M$ , $R$ , $T$ same for both): $m \propto \frac{4T}{r}\times\frac{4}{3}\pi r^3 = \frac{16\pi T}{3}r^2 \ \Rightarrow\ m \propto r^2$ …
Start your 14-day free trial to see the full solution →
AP EAPCET 2025Set eng-2025-05-26-AN1 markMCQ
$Q. When the temperature increases, the viscosity of (A) gases decreases but liquids increases (B) gases increases but liquids decreases (C) both gases and liquids increases (D) both gases and liquids decreases$
›Reveal solutionSolution
Gas viscosity and liquid viscosity respond oppositely to a temperature rise, because they arise from different microscopic mechanisms; the gas gets more viscous while the liquid gets less viscous.

Concept and Intuition

Viscosity is the internal friction between adjacent layers of a fluid moving at different velocities. In a liquid, molecules sit close together, and viscosity is dominated by the cohesive (attractive) intermolecular forces holding neighbouring layers together. Heating a liquid increases the average kinetic energy of its molecules, which overcomes these cohesive bonds more easily — so the resistance to flow (viscosity) falls.

In a gas, molecules are far apart and rarely interact via cohesive forces; instead, viscosity comes from the transfer of momentum as molecules randomly cross between adjacent layers moving at different bulk speeds — faster molecules diffusing into a slower layer speed it up, and vice versa. Kinetic theory gives gas viscosity $\eta \propto \sqrt{T}$ (through the mean molecular speed), so heating a gas makes molecules move faster, transfer momentum more effectively, and the gas viscosity rises.

Step-by-Step Solution

Identify the mechanism of viscosity in liquids: cohesive intermolecular forces.

Heating a liquid $\Rightarrow$ weaker effective cohesion $\Rightarrow$ liquid viscosity decreases. …
Start your 14-day free trial to see the full solution →
AP EAPCET 2024Set eng-2024-05-23-FN1 markMCQ
$7.5\times10^{-5}\ \mathrm{m^3s^{-1}}$
›Reveal solutionSolution
Computing the Reynolds number from the flow rate and tap diameter gives $Re\approx6366$ , comfortably above the turbulent-flow threshold.

Concept and Intuition

The Reynolds number $Re=\dfrac{\rho v d}{\eta}$ tells us whether flow is smooth (laminar, low $Re$ ) or chaotic (turbulent, high $Re$ , generally $Re\gtrsim2000$ – $4000$ for pipe flow). Here we're given volumetric flow rate $Q$ rather than velocity directly, so we substitute $v=Q/A$ .

Step-by-Step Solution

$A=\dfrac{\pi d^2}{4}=\dfrac{\pi(0.015)^2}{4}\approx1.767\times10^{-4}\ \mathrm{m^2}$ .

$v=\dfrac{Q}{A}=\dfrac{7.5\times10^{-5}}{1.767\times10^{-4}}\approx0.424\ \mathrm{m/s}$ .

$Re=\dfrac{\rho v d}{\eta}=\dfrac{1000\times0.424\times0.015}{10^{-3}}\approx6366$ . …
Start your 14-day free trial to see the full solution →
AP EAPCET 2022Set eng-2022-07-04-FN1 markMCQ
$90^{\circ}$
›Reveal solutionSolution
Checking three physics statements about viscosity and wetting: A (temperature effects on viscosity) is a genuine textbook fact; C (wetting requires contact angle $>90^\circ$ ) is backwards; B's stated reasoning for non-wetting is not the standard correct explanation.

Concept and Intuition

Gas viscosity arises from momentum transfer via molecular collisions, which increases with temperature (faster molecules, more frequent transfer) — so gas viscosity increases with $T$ .

Liquid viscosity arises from intermolecular cohesive forces, which weaken as thermal agitation increases — so liquid viscosity decreases with $T$ .

Wetting of a solid by a liquid is governed by the angle of contact $\theta$ : if $\theta<90^\circ$ (acute), the liquid wets/spreads on the surface; if $\theta>90^\circ$ (obtuse), it does not (beads up).

Water fails to wet an oily/greasy glass surface because the adhesive force between water and the oily surface is weak compared to water's own cohesive force — not simply because "oil's cohesive force is less than water's" (a different, non-standard comparison).

Step-by-Step Solution

Statement A: temperature ↑ ⇒ gas viscosity ↑, liquid viscosity ↓. This is the standard, correct physics — TRUE.

Statement C: claims wetting happens when contact angle $>90^\circ$ . The correct rule is the opposite (wetting needs $\theta<90^\circ$ ) — FALSE. …
Start your 14-day free trial to see the full solution →
AP EAPCET 2022Set eng-2022-07-05-AN1 markMCQ
$(4\pi r^2 n - 4\pi R^2)T$
›Reveal solutionSolution
Breaking a drop increases its surface area; the energy supplied equals surface tension times

that area increase, $T(4\pi r^2 n - 4\pi R^2)$ .

Concept and Intuition

A liquid's surface behaves like a stretched membrane with energy stored per unit area — that's what

surface tension $T$ measures (energy per unit area, equivalently force per unit length). Splitting

one big drop into many small ones necessarily increases the total surface area (you're creating new

surface where none existed), and that new surface costs energy equal to $T\times(\text{area created})$ . Volume is conserved in the split, but area is not — smaller drops always have more

total surface area for the same total volume.

Step-by-Step Solution

Surface area of the original single drop: $A_{initial}=4\pi R^2$ .

Surface area of all $n$ smaller drops combined: $A_{final}=n\times4\pi r^2$ .

Since surface energy $=T\times$ area, the extra energy needed to create the additional surface …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set ap-2021-09-03-FN1 markMCQ
$Q. Identify the incorrect statement about 'angle of contact':$
(a) Angle of contact depends upon the inclination of the solid surface to the liquid surface.
(b) If the angle of contact of a liquid and a solid surface is less than 90°, then the liquid spreads on the surface of the solid.
(c) Angle of contact increases with increase in temperature of liquid.
(d) The value of angle of contact for pure water and glass is zero. (A)
(a) only (B)
(b) only (C)
(c) only (D)
(d) only
›Reveal solutionSolution
Angle of contact is a material-pair property (liquid + solid + surrounding medium), independent of the tilt of the solid surface — so statement (a) is the incorrect one.

Concept and Intuition

The angle of contact $\theta$ at a liquid–solid boundary is set by the balance of the three relevant surface tensions (solid–air, solid–liquid, liquid–air) at the line of contact. Since this balance is purely local — it depends on what materials are touching and the ambient conditions (impurities, temperature) — tilting the solid surface as a whole does not change that local force balance, so $\theta$ stays the same regardless of inclination. This is a specifically counter-intuitive but standard textbook fact.

Step-by-Step Solution

(a) claims $\theta$ depends on the inclination of the solid surface — this is FALSE; $\theta$ is set by the surface-tension balance at the contact line and is independent of how the surface is tilted.

(b) is TRUE: if $\theta<90^\circ$ the liquid "wets" the solid and tends to spread over it (adhesive forces dominate).

(c) is TRUE: the angle of contact for most liquid–solid pairs increases as the liquid's temperature rises (surface tension and hence the force balance shifts). …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set eng-2021-08-19-AN1 markMCQ
$R_e$
›Reveal solutionSolution
The Reynolds-number regime between 1000 and 2000 is unstable/transitional, NOT steady, making statement (B) the incorrect one.

Concept and Intuition

Reynolds number characterizes whether fluid flow is dominated by viscous forces (laminar) or inertial forces (turbulent). The standard textbook classification has three regimes: streamline/laminar at low $R_e$ , an unstable transitional zone in the middle, and turbulent at high $R_e$ . The middle zone is specifically called unstable/unsteady, not steady.

Step-by-Step Solution

Recall the standard classification: for $R_e<1000$ , flow is streamline/laminar — statement (A) is correct.

For $1000<R_e<2000$ , the flow becomes unstable/unsteady as it transitions from laminar to turbulent — NOT steady. Statement (B) incorrectly calls this regime "steady".

For $R_e>2000$ , flow becomes turbulent — statement (C) is correct. …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set eng-2021-08-20-AN1 markMCQ
$Q. What causes the free surface of a liquid to have minimum area? (A) Viscosity (B) Surface tension (C) Diffusion (D) Pressure$
›Reveal solutionSolution
Surface tension is the property responsible for a liquid surface behaving like a stretched membrane that contracts to the smallest possible area.

Concept and Intuition

Molecules at the surface of a liquid experience a net inward pull from the molecules below them (since there are no liquid molecules above to balance the attraction), creating a tendency for the surface to contract — this net effect is called surface tension. A surface under tension always tries to minimize its potential energy, and since surface energy is proportional to surface area, minimizing area minimizes energy. This is why free liquid surfaces (like a small drop, unaffected by gravity) tend to be spherical, and why a soap film pulls tight.

Step-by-Step Solution

Surface tension gives the liquid surface a form of elastic potential energy proportional to its area: $E = \sigma \times A$ , where $\sigma$ is the surface tension coefficient.

A system in equilibrium tends toward minimum potential energy.

So the free surface adopts the shape of minimum area, consistent with any constraints (volume, external forces). …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set eng-2021-08-20-FN1 markMCQ
$Q. Water does not wet an oily glass because______ (A) Cohesive force of oil is greater than adhesive force between oil and glass (B) Cohesive force of oil is greater than cohesive force of water (C) Oil repels water (D) Cohesive force of water is greater than adhesive force between water and oil molecules$
›Reveal solutionSolution
Wetting depends on the balance between a liquid's cohesion and its adhesion to the surface; water doesn't wet an oily surface because water's cohesive pull on itself exceeds its adhesive pull toward the oil.

Concept and Intuition

A liquid wets a surface when the adhesive force between the liquid and the surface exceeds the liquid's own cohesive force (molecules attracting each other) — the liquid then prefers to spread out and stick to the surface. If cohesion dominates over adhesion, the liquid instead minimizes contact with the surface, forming beads (non-wetting), exactly as mercury does on glass, and as water does on an oily/waxy surface.

Step-by-Step Solution

Wetting criterion: liquid wets surface if (adhesive force between liquid & surface) $>$ (cohesive force within the liquid).

Non-wetting occurs when cohesive force within the liquid $>$ adhesive force between liquid and surface.

Here, the surface is oily glass, and the liquid is water. The relevant adhesive force is between water molecules and the oil molecules coating the glass (oil, being non-polar, doesn't attract water strongly). …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set eng-2021-08-23-AN1 markMCQ
$Q. Assertion (A): At critical temperature, surface tension of liquids become zero. Reason (R): At critical temperature, intermolecular forces for liquids and gases become equal. Thus, liquids can expand without restriction. (A) Both A and R are true and R is a correct explanation for A (B) Both A and R are true but R is not a correct explanation for A (C) A is true, R is false (D) A is false, R is false$
›Reveal solutionSolution
Surface tension arises from an imbalance of cohesive forces at a liquid's surface; at the critical point liquid and vapour densities (and intermolecular forces) become identical, so that imbalance — and hence surface tension — vanishes. R is the correct mechanism behind A.

Concept and Intuition

Surface tension exists because surface molecules of a liquid are pulled inward more strongly than they are pulled by the much sparser vapour above — a net inward cohesive pull. As temperature rises toward the critical temperature $T_c$ , the vapour becomes denser and the liquid less dense, until at $T_c$ the two phases merge into one indistinguishable fluid. At that point there is no longer any difference in intermolecular force environment across the "surface" (because there is no surface), so the net inward pull — and therefore the surface tension — is exactly zero.

Step-by-Step Solution

Assertion: Surface tension $\to 0$ as $T\to T_c$ — this is an experimentally and thermodynamically established fact (used to derive Eötvös' and van der Waals-type relations for surface tension vs temperature).

Reason: At $T_c$ , liquid and gas coexistence ends; their densities and average intermolecular forces become equal. …
Start your 14-day free trial to see the full solution →
AP EAPCET 2021Set eng-2021-08-25-AN1 markMCQ
$Q. Hairs of shaving brush cling together when it is removed from water due to ________ (A) Force of attraction between hairs (B) Surface tension (C) Viscosity of water (D) Characteristic property of hairs$
›Reveal solutionSolution
Wet bristles stick together to minimize the surface area of the water film clinging to them — a direct, everyday manifestation of surface tension.

Concept and Intuition

Surface tension is the tendency of a liquid surface to minimize its area, behaving like a stretched elastic membrane. When a shaving brush (or any bunch of fine hairs/bristles) is pulled out of water, a thin film of water clings between adjacent hairs. This film's surface tension pulls the hairs as close together as possible, since a bunched configuration has less exposed liquid surface area than hairs spread apart with the same amount of water between them.

Step-by-Step Solution

On removal from water, a film of water is trapped between neighbouring hairs of the brush.

This film has two free surfaces exposed to air, each possessing surface tension.

The system minimizes its surface energy by minimizing surface area — clustering the hairs together reduces the total area of the water film's surface compared to spread-out hairs. …
Start your 14-day free trial to see the full solution →

←Previous questionQ10 — Streamline flow is more likely for liquids with (Note: more than one o…→Next questionQ12 — Is surface tension a vector?

Q.Is viscosity a vector?

Understanding Physical Quantities

What is Viscosity?

🎓Unlock everything free for 14 days

Unlock everything free for 14 days