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.
        An aeroplane of mass 3×10⁴ kg and total wing area 120 m² in level flight. The pressure difference between upper and lower wing surfaces (kPa) is (g=10 m/s²):
        📅BP 2014
        Q2.
        Viscosity of liquids and gases with temperature increase:
        📅BP 2013
        Q3.
        Graphical representation of cork rising from bottom to float in water:
        📅BP 2011
        Q4.
        Velocity ratio (V₁/V₂) for efflux at h/2 and h in immiscible liquids (ρ and 2ρ):
        📅BP 2009
        Q5.
        Radius of 900 kg/m³ liquid drop with η=1.85×10⁻⁵ Ns/m² and v=2.76×10⁻⁴ m/s:
        📅MOE 2014
        Q6.
        Terminal velocity of object in vacuum vs. 100 m/s in liquid:
        📅MOE 2010
        Q7.
        When gas temperature increases, its viscosity:
        📅IOM 2010
        Q8.
        Velocity when two water drops (radius r, velocity v) coalesce:
        📅IOM 2010
        Q9.
        Separation between 10 cm square plates moving at 10 cm/s (η=0.01 poise, F=200 dyne):
        📅IOM 2009
        Q10.
        One poise equals:
        📅KU 2014
        Q11.
        Bernoulli's theorem is based on conservation of:
        📅KU 2013
        Q12.
        Terminal velocity when two drops (velocity v) coalesce:
        📅IE 2009
        Q13.
        Velocity profile in wide river:
        📅MOE 2009
        Q14.
        One poise equals:
        📅BPKIHS 05
        Q15.
        Height difference in rotating liquid (r=0.05m, ω=2 rev/s=4π rad/s):
        Q16.
        Momentum ratio for hailstones (radius 1:2) at terminal velocity:
        Q17.
        Flow rate when tube radius doubles (laminar flow):
        Q18.
        Viscosity ratio (η₁/η₂) for equal mass flow (d₁/d₂, t₁/t₂):
        Q19.
        Water velocity when manometer pressure drops from 4×10⁴ to 3×10⁴ N/m²:
        Q20.
        Time ratio (t₁/t₂) for emptying 1/4 vs. 3/4 of tank:
        Q21.
        Work to pump 4m³ water to 20m height against 2×10⁵ N/m² pressure:
        Q22.
        Terminal velocity when 8 drops coalesce:
        📅BP 2015
        Q23.
        Velocity at 10cm diameter section when 20cm section has 5cm/s flow:
        Q24.
        Maximum liquid height with 70 cm³/s inflow and 1 cm² outflow hole:
        Q25.
        Viscous force when volume increases from V to 8V at same velocity:
        Q26.
        Terminal velocity when mass increases from m to 8m:
        Q27.
        Steel ball upward speed when pulled with 2× effective weight:
        Q28.
        Viscous force when drop radius increases from r to 2r:
        Q29.
        Efflux velocity at 3 atm pressure (1 atm=10⁵ Pa, ρ=1000 kg/m³):
        Q30.
        Maximum average velocity for Re=1000 in 2cm diameter tube (η=10⁻³ kg/m·s):
        Q31.
        Flow rate when tube radius halves (same pressure head):
        📅KU 2015
        Q32.
        Pressure difference between pipes (L:2L, R:2R):
        📅IOM 2016
        Q33.
        Bernoulli's equation is applicable in:
        📅IOM 2017