14Fluid dynamics and Viscosity

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FLUID DYNAMICS AND VISCOSITY
▢ Viscosity:
❖ Definition: Property of fluid in motion by virtue of which it opposes relative motion between different layers and opposes motion of a body through it

Table 1: Viscosity Basics

Point
Description
Cause
Intermolecular / electromagnetic interaction
Effect
Different layers of flowing liquid have different velocities
Velocity gradient
\(\frac{dv}{dx}\)
Velocity gradient meaning
Difference in velocity between layers unit distance apart
Ideal fluid
Viscosity = 0 and compressibility = 0
▢ Newton's Formula of Viscosity:
❖ Formula: \(F=-\eta A\frac{dv}{dx}\)

Table 1: Coefficient of Viscosity

Quantity
Formula / Meaning
Velocity gradient
\(\frac{dv}{dx}=\frac{v_1-v_2}{x_1-x_2}=\frac{\Delta v}{\Delta x}\)
Area
\(A\) = area of laminar plate
Coefficient of viscosity
\(\eta=-\frac{F}{A(dv/dx)}\)
Definition of \(\eta\)
Viscous force per unit area needed to produce unit velocity gradient
SI unit
Pascal second / Poiseuille
Unit relation
\(1\ poiseuille=1\ Pa\ s=1\ Ns\ m^{-2}=10\ poise\)
Dimension
\([ML^{-1}T^{-1}]\)
▢ Factors Affecting Viscosity:

Table 1: Viscosity of Liquids vs Gases

Factor
Liquid
Gas
Arises due to
Cohesive force
Transfer of momentum between gas molecules
Temperature increases
Viscosity decreases
Viscosity increases; \(\eta\propto\sqrt T\)
Density increases
Viscosity increases
Viscosity decreases
Pressure increases
Generally increases within certain range
Nearly constant at high pressure
Very high / very low pressure
Viscosity directly proportional to pressure
At low pressure, viscosity directly proportional to pressure
Water under pressure
Viscosity decreases with pressure
—
Other liquids under pressure
Viscosity increases with pressure
—
▢ Reynolds Number and Critical Velocity:
❖ Critical Velocity: Maximum velocity of liquid through tube up to which flow remains streamline
❖ Reynolds Number: Dimensionless number indicating nature of flow
❖ Formula: \(v_c=\frac{K\eta}{r\rho}\)

Table 1: Reynolds Formula Symbols

Symbol
Meaning
\(v_c\)
Critical velocity
\(K\)
Reynolds number
\(\eta\)
Coefficient of viscosity
\(\rho\)
Density of liquid
\(r\)
Radius of tube

Table 2: Nature of Flow

Reynolds number
Flow
\(K<2000\)
Laminar / streamline
\(2000
Unstable
\(K>3000\)
Turbulent
❖ Important Points:
  • •
    Beyond critical velocity, flow becomes turbulent
  • •
    Reynolds number is low for high viscosity, low density and low velocity
  • •
    Reynolds number is related to critical velocity
  • •
    Critical velocity for non-viscous liquid is zero
▢ Equation of Continuity:
❖ Statement: For ideal liquid flowing through non-uniform tube under streamlined condition, mass flowing per second is same at every cross-section
❖ Formula: \(A_1v_1=A_2v_2\)

Table 1: Continuity Equation

Form
Expression
Mass flow rate
Constant
Volume flow rate
\(Av=constant\)
Two cross-sections
\(A_1v_1=A_2v_2\)
For cylindrical tube
\(v\propto\frac{1}{A}\propto\frac{1}{r^2}\)
Based on
Conservation of mass
Applies to
Ideal liquid: non-viscous and incompressible
❖ Applications:
  • •
    Deep water appears still
  • •
    Falling stream of water becomes narrower
  • •
    Fine jet is produced by narrowing cross-section
▢ Bernoulli's Theorem:
❖ Statement: For a liquid in steady streamline flow, total mechanical energy per unit volume remains constant
❖ Energy Types:

Table 1: Energy of Flowing Liquid

Energy
Formula
Pressure energy
\(PV\)
Kinetic energy
\(\frac{1}{2}mv^2\)
Potential energy
\(mgh\)
❖ Formulae:

Table 1: Bernoulli Equation

Form
Expression
Energy form
\(PV+\frac{1}{2}mv^2+mgh=constant\)
Pressure form
\(P+\frac{1}{2}\rho v^2+\rho gh=constant\)
Head form
\(\frac{P}{\rho g}+\frac{v^2}{2g}+h=constant\)
Horizontal flow
\(P+\frac{1}{2}\rho v^2=constant\)
❖ Heads:

Table 1: Fluid Heads

Head
Formula
Pressure head
\(\frac{P}{\rho g}\)
Velocity head
\(\frac{v^2}{2g}\)
Gravitational head
\(h\)
❖ Principle: Based on conservation of mechanical energy
❖ Key Result: Pressure decreases when liquid flows from broader to narrower portion of pipe
❖ Applications:
  • •
    Blowing of roofs by storms
  • •
    Attraction between two closely parallel boats moving in same direction
  • •
    Repulsion between two closely parallel boats moving in opposite direction
  • •
    Magnus effect in spinning ball
  • •
    Aspirator
  • •
    Carburetor
  • •
    Paint gun
  • •
    Scent spray
  • •
    Insect sprayer
  • •
    Aeroplane wings
▢ Torricelli's Theorem:
❖ Statement: Velocity of efflux through an orifice at depth \(h\) is equal to velocity acquired by a freely falling body from height \(h\)
❖ Formula: \(v=\sqrt{2gh}\)
❖ Derived From: Bernoulli's principle
❖ Independent Of:
  • •
    Nature of liquid
  • •
    Quantity of liquid in container
  • •
    Area of cross-section of hole
❖ Projectile from Hole:
◉ _*table:
    ❖ Emptying Tank:

    Table 1: Tank Emptying Formulae

    Case
    Formula
    Tank area \(A\), hole area \(A_0\), level from \(H_1\) to \(H_2\)
    \(t=\frac{A}{A_0}\sqrt{\frac{2}{g}}(\sqrt{H_1}-\sqrt{H_2})\)
    Time to empty from height \(H\)
    \(t=\frac{A}{A_0}\sqrt{\frac{2H}{g}}\)
    Time to lower each \(H/n\) from top
    \((\sqrt n-\sqrt{n-1}):...:(\sqrt3-\sqrt2):(\sqrt2-\sqrt1):(\sqrt1-\sqrt0)\)
    ❖ Several Holes in Vertical Wall:
    • •
      Velocity of efflux increases downward
    • •
      Range first increases, becomes maximum at centre, then decreases
    • •
      Streams from holes equally distant from top and bottom have equal range
    ▢ Stokes' Law and Terminal Velocity:
    ❖ Terminal Velocity: Constant velocity finally attained by a body falling in viscous medium
    ❖ Stokes Law: \(F=6\pi\eta rv\)

    Table 1: Terminal Velocity

    Quantity
    Formula / Point
    Viscous force
    \(F=6\pi\eta rv\)
    At terminal velocity
    Effective weight = viscous force
    Equation
    \(V\sigma g-V\rho g=6\pi\eta rv_t\)
    Terminal velocity
    \(v_t=\frac{2r^2g(\sigma-\rho)}{9\eta}\)
    Ratio form
    \(\frac{v_t}{r^2g}=\frac{2}{9}\frac{\sigma-\rho}{\eta}\)
    If medium density negligible
    \(v_t=\frac{2g\sigma r^2}{9\eta}\)
    Dependence
    \(v_t\propto r^2\)
    ❖ Density Conditions:

    Table 1: Body in Viscous Medium

    Condition
    Result
    \(\sigma>\rho\)
    Body falls with terminal velocity
    \(\sigma=\rho\)
    Body remains stationary wherever released
    \(\sigma<\rho\)
    Body rises upward through medium
    Radius doubled
    Terminal velocity becomes 4 times
    \(n\) drops coalesce
    New terminal velocity = \(n^{2/3}v_t\)
    ▢ Poiseuille's Equation:
    ❖ Statement: Rate of flow of liquid through capillary tube under streamlined motion is proportional to pressure difference and fourth power of radius

    Table 1: Poiseuille Formulae

    Quantity
    Formula
    Rate of flow
    \(Q=\frac{V}{t}=\frac{\Delta P}{R}\)
    Fluid resistance
    \(R=\frac{8\eta l}{\pi r^4}\)
    Poiseuille equation
    \(Q=\frac{\pi r^4\Delta P}{8\eta l}\)
    Velocity at distance \(x\) from axis
    \(v=\frac{\Delta P}{4\eta l}(r^2-x^2)\)
    Maximum velocity
    At axis \((x=0)\)
    Velocity at wall
    Zero \((x=r)\)
    ❖ Series and Parallel Tubes:
    ◉ **table:
      ❖ Ohm's Law Analogy:

      Table 1: Poiseuille-Ohm Analogy

      Fluid flow
      Electric current
      Rate of flow \(Q\)
      Current \(I\)
      Pressure difference \(\Delta P\)
      Potential difference \(V\)
      Fluid resistance \(R\)
      Electrical resistance \(R\)
      Viscosity
      Resistivity
      ▢ Read and Digest:

      Table 1: Important Points

      Fact
      Point / Formula
      Viscous force origin
      Electromagnetic interaction
      Aeroplane wing design
      Based on Bernoulli's principle
      Aeroplane wing upper surface
      Convex
      Aeroplane wing lower surface
      Concave downwards
      Machine parts jam in winter
      Viscosity of lubricant increases
      CGS unit of viscosity
      Poise
      Centipoise
      \(1\ cP=10^{-3}\ kg\ m^{-1}s^{-1}\)
      Spinning ball curved path
      Magnus effect
      Viscosity between layers
      Introduces tangential force
      Coefficient of viscosity
      Shearing stress per unit velocity gradient
      Rain drops fall with constant velocity
      Due to viscosity
      Clouds float
      Due to low density
      Satellite close to Earth with air viscosity
      Orbital velocity increases until it falls back
      Angle between viscous force and motion
      \(\pi\)
      Velocity of liquid increases
      Pressure decreases
      Horizontal pipe: larger diameter
      Pressure is higher
      Ideal fluid
      Zero viscosity and zero compressibility
      Hole near bottom of tank
      Volume emerging independent of density
      Cold syrup flows slowly
      Due to high viscosity
      Sphere in viscous liquid
      Velocity first increases, then becomes constant
      ▢ High-Yield Recall:
      ❖ **table:
        Q1.
        Two spherical hailstones with radii ratio 1:2 attain terminal velocity. Their momentum ratio is
        📅CEE MBBS 2025
        Q2.
        The coefficient of viscosity of hot air is:
        📅CEE MBBS 2023
        Q3.
        A small spherical ball having density equal to that of the liquid is immersed in it. Its terminal velocity will be:
        📅CEE MBBS 2023
        Q4.
        Lifting of an aeroplane is based on:
        📅CEE MBBS 2022
        Q5.
        If the depth of the river gets increased, then the velocity:
        📅CEE BDS 2023
        Q6.
        An aeroplane of mass \(3\times10^4\ \mathrm{kg}\) has total wing area \(120\ \mathrm{m^2}\). Find the pressure difference needed for level flight, taking \(g=10\ \mathrm{m/s^2}\).
        📅BP 2014
        Q7.
        How do the viscosities of liquids and gases usually change as temperature rises?
        📅BP 2013
        Q8.
        A cork is released underwater and rises until it floats. Which graph best describes its speed as a function of time?
        📅BP 2011
        Q9.
        A spherical liquid drop of density \(900\ \mathrm{kg/m^3}\) falls through air with dynamic viscosity \(1.85\times10^{-5}\ \mathrm{Pa\,s}\) at terminal speed \(2.76\times10^{-4}\ \mathrm{m/s}\) via Stokes law. Neglect air buoyancy; find its radius.
        📅MOE 2014
        Q10.
        A body has terminal speed \(100\ \mathrm{m/s}\) in a liquid. What is its terminal speed during free fall in a vacuum?
        📅MOE 2010
        Q11.
        As temperature increases, the dynamic viscosity of a typical gas:
        📅IOM 2010
        Q12.
        Two identical spherical drops, each falling at terminal speed \(v\), coalesce into one. Under Stokes-flow conditions, the new terminal speed is:
        📅IOM 2010
        Q13.
        Two square plates each of side \(10\ \mathrm{cm}\) are separated by a viscous liquid of viscosity \(0.01\ \mathrm{poise}\). A force of \(200\ \mathrm{dyne}\) maintains relative speed \(10\ \mathrm{cm/s}\). Find their separation.
        📅IOM 2009
        Q14.
        Convert \(1\ \mathrm{poise}\) to SI viscosity units.
        📅KU 2014
        Q15.
        Bernoulli's theorem follows from conservation of:
        📅KU 2013
        Q16.
        Two identical liquid drops, each with terminal speed \(v\), coalesce. What is the new terminal speed in the Stokes regime?
        📅IE 2009
        Q17.
        In a wide river, flow speed generally changes with depth so that it:
        📅MOE 2009
        Q18.
        The viscosity unit \(1\ \mathrm{poise}\) is equal to:
        📅BPKIHS 05
        Q19.
        A liquid rotates steadily at \(2\ \mathrm{rev/s}\). Find the height difference between its surface at the axis and at radius \(0.05\ \mathrm m\), taking \(g=10\ \mathrm{m/s^2}\).
        Q20.
        Two spherical hailstones have radii in the ratio \(1:2\). Assuming terminal motion in the Stokes regime and equal density, their momenta are in the ratio:
        Q21.
        For laminar flow through a tube at constant pressure difference, what happens to the volume flow rate if the radius doubles?
        Q22.
        Equal masses of two liquids of densities \(d_1,d_2\) flow through the same capillary in times \(t_1,t_2\) under the same pressure difference. What is \(\eta_1/\eta_2\)?
        Q23.
        For water of density \(1000\ \mathrm{kg/m^3}\), a pressure drop of \(4\times10^4-3\times10^4\ \mathrm{Pa}\) is converted completely into kinetic energy per unit volume. Find the speed.
        Q24.
        A tank with constant cross-section drains through a small bottom orifice. Let \(t_1\) be the time to empty the first quarter of its initial volume and \(t_2\) the time to empty the next three quarters. Find \(t_1/t_2\).
        Q25.
        How much work is needed to pump \(4\ \mathrm{m^3}\) of water up \(20\ \mathrm m\) against an additional pressure \(2\times10^5\ \mathrm{Pa}\)? Use \(g=10\ \mathrm{m/s^2}\).
        Q26.
        Eight equal small spherical drops, each of terminal speed \(v\), coalesce. The new terminal speed in the Stokes regime is:
        📅BP 2015
        Q27.
        Water flows through pipe sections of diameters \(20\ \mathrm{cm}\) and \(10\ \mathrm{cm}\). If its speed in the wider section is \(5\ \mathrm{cm/s}\), find its speed in the narrower section.
        Q28.
        A tank receives water at \(70\ \mathrm{cm^3/s}\) and drains through a bottom hole of area \(1\ \mathrm{cm^2}\). Find the equilibrium water height (use \(g=1000\ \mathrm{cm/s^2}\)).
        Q29.
        For a sphere moving at a fixed speed through a viscous fluid in the Stokes regime, what happens to its viscous drag when its volume increases from \(V\) to \(8V\)?
        Q30.
        For geometrically similar spheres of the same density falling at terminal speed in the Stokes regime, if mass increases from \(m\) to \(8m\), the terminal speed becomes:
        Q31.
        A steel ball falls at terminal speed \(10\ \mathrm{cm/s}\) in a viscous fluid. If an upward external force equal to twice its effective weight is applied, what is its new terminal upward speed under Stokes drag?
        Q32.
        For a spherical drop moving at the same fixed speed in the same liquid under Stokes drag, if its radius changes from \(r\) to \(2r\), its viscous drag becomes:
        Q33.
        Water issues from a vessel at internal absolute pressure \(3\ \mathrm{atm}\) into the atmosphere at \(1\ \mathrm{atm}\). Neglect height difference and losses. Given \(1\ \mathrm{atm}=10^5\ \mathrm{Pa}\), find the ideal efflux speed.
        Q34.
        Water flows through a tube of diameter \(2\ \mathrm{cm}\). For \(\mathrm{Re}=1000\), viscosity \(10^{-3}\ \mathrm{Pa\,s}\) and density \(1000\ \mathrm{kg/m^3}\), find the average speed.
        Q35.
        A capillary initially carries \(32\) units of flow at a fixed pressure difference. If its radius is halved, the new flow rate is:
        📅KU 2015
        Q36.
        For identical volume flow rate and viscosity, two capillaries have lengths \(L\) and \(2L\) and radii \(R\) and \(2R\). Find \(\Delta P_1/\Delta P_2\).
        📅IOM 2016
        Q37.
        Bernoulli's equation is principally used in:
        📅IOM 2017