Phenomenological Transport laws
Lecture # 1
CHEN 300 – FALL 2018
Introduction
Fluid mechanics: The science that deals with the behavior of fluids at rest
(fluid statics) or in motion (fluid dynamics), and the interaction of fluids with
solids or other fluids at the boundaries.
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Introduction
Fluid mechanics: The science that deals with the behavior of fluids at rest
(fluid statics) or in motion (fluid dynamics), and the interaction of fluids with
solids or other fluids at the boundaries.
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What is a Fluid?
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What is a Fluid?
• Basically: a Gas or a Liquid.
• Technically: a fluid is defined in terms of how it
responds to an applied force (Shear Stress).
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What is Shear Stress?
• Stress: Force per unit area.
• Normal stress: The normal
component of a force acting
on a surface per unit area.
(Pressure)
• Shear stress: The tangential
component of a force acting
on a surface per unit area.
• Strain: Response of a
system to an applied stress.
The normal stress and shear stress at
the surface of a fluid element. For
fluids at rest, the shear stress is zero
and pressure is the only normal stress.
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
• Consider two parallel plates with a
‘material’ between them.
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
• Consider two parallel plates.
• Bottom plate is fixed.
• Material sticks to bottom plate
(no slip).
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
• Consider two parallel plates.
• Bottom plate is fixed.
• Top plate is subject to a tangential
force Fx in the x direction.
𝑭𝒙
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
𝑭𝒙
• Consider two parallel plates.
• Bottom plate is fixed.
• Top plate is subject to a tangential
force Fx in the x direction.
Contact
Area (𝑨𝒚 )
Shear stress: The tangential
component of a force acting on a
surface per unit area.
𝝉𝒚𝒙
𝑭𝒙
=
𝑨𝒚
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
𝑭𝒙
• Consider two parallel plates.
• Bottom plate is fixed.
• Top plate is subject to a tangential
force Fx in the x direction.
Contact
Area (𝑨𝒚 )
Shear stress: The tangential
component of a force acting on a
surface per unit area.
Shear Strain: Response of a system to
an applied stress.
𝝉𝒚𝒙
𝑭𝒙
=
𝑨𝒚
𝜸𝒚𝒙
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
Shear Strain: Response of a system to
an applied stress.
Case #1: The material is perfectly
rigid (e.g. a brick). → Solid
𝑭𝒙
Brick
• No deformation.
∴ 𝜸𝒚𝒙 = 𝟎
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
𝑭𝒙
Shear Strain: Response of a system to
an applied stress.
Case #2: The material is linearly
elastic (e.g. rubber). → Solid
• Will deform by an angle α.
• The degree of deformation
depends on the material
properties.
• Stress- Strain Relationship
through the Shear Modulus.
𝜶
Rubber
∴ 𝜸𝒚𝒙 = 𝜶
𝝉𝒚𝒙 = 𝑮𝜸𝒚𝒙
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
𝑭𝒙
Shear Strain: Response of a system to
an applied stress.
Fluid
Case #3: The material is a Liquid.
→ Fluid
Will deform continuously under
the influence of a shear
stress.
i.e. Flow
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Fluid vs Solid response to Shear Stress
𝑦
𝑥
𝑭𝒙
Shear Strain: Response of a system to
an applied stress.
Fluid
Case #3: The material is a Liquid.
→ Fluid
Will deform continuously under
the influence of a shear
stress.
i.e. Flow
Resistance to flow is defined as the
viscosity of the Fluid.
𝝉𝒚𝒙
𝒅𝜸𝒚𝒙
=𝝁
𝒅𝒕
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Viscosity
Resistance to flow is defined as the viscosity of the Fluid.
Shear Stress
𝝉𝒚𝒙
𝒅𝜸𝒚𝒙
=𝝁
𝒅𝒕
Rate of Shear Strain
Viscosity
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Viscosity
Resistance to flow is defined as the viscosity of the Fluid.
Shear Stress
𝝉𝒚𝒙
𝒅𝜸𝒚𝒙
=𝝁
𝒅𝒕
Rate of Shear Strain
Viscosity
Lower Viscosity
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Viscosity
Resistance to flow is defined as the viscosity of the Fluid.
Shear Stress
𝝉𝒚𝒙
𝒅𝜸𝒚𝒙
=𝝁
𝒅𝒕
Rate of Shear Strain
Viscosity
Lower Viscosity
Higher Viscosity
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Types of Flows -1
Viscous vs Inviscid Flows
• Viscous Flow: Flow in which the frictional effects are significant.
Almost all flows of practical interest in Chemical Engineering are
Viscous Flows.
• Inviscid Flow: The flow of an ideal fluid that is assumed to have no or
negligible viscosity.
Extensively used as a simplifying assumption to solve complex
fluid dynamics problems.
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Types of Flows -2
External vs Internal Flows
• External Flow: Flow of an unbounded fluid over a surface.
Example: Fluid flow around a sphere.
• Internal Flow: The fluid is completely bounded by solid surfaces.
Example: Fluid flow inside a pipe or duct.
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Types of Flows -2
External vs Internal Flows
• External Flow: Flow of an unbounded fluid over a surface.
Example: Fluid flow around a sphere.
• Internal Flow: The fluid is completely bounded by solid surfaces.
Example: Fluid flow inside a pipe or duct.
External flow
around a sphere.
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Types of Flows -3
Compressible vs Incompressible Flows
Defined by Density of Fluid
Density: Mass per unit Volume (e.g., @ 20 oC, 1 atm)
Water
Mercury
Air
rwater
rHg
rair
= 1000 kg/m3
= 13,500 kg/m3
= 1.22 kg/m3
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Types of Flows -3
Compressible vs Incompressible Flows
• Incompressible Flow: the density does not change during fluid motion
(constant density).
𝒊. 𝒆.
𝝏𝝆 𝝏𝝆 𝝏𝝆
,
,
=𝟎
𝝏𝒙 𝝏𝒚 𝝏𝒛
• Compressible Flow: Flow exhibits significant changes in fluid density.
Density increases
Density decreases
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Types of Flows -3
Compressible vs Incompressible Flows
• Incompressible Flow: the density does not change during fluid motion
(constant density).
• General criteria used to define incompressible flow:
𝒊. 𝒆.
𝝏𝝆 𝝏𝝆 𝝏𝝆
= 𝟎 𝒐𝒇 𝒇𝒍𝒖𝒊𝒅
𝒖 , 𝝏𝒚 ,𝑺𝒑𝒆𝒆𝒅
𝝏𝒙
𝝏𝒛
• Mach Number: 𝑴𝒂 = =
𝒄 𝑺𝒑𝒆𝒆𝒅 𝒐𝒇 𝒔𝒐𝒖𝒏𝒅
• Compressible Flow: Flow exhibits significant changes in fluid density.
𝑰𝒇 𝑴𝒂 < 𝟎. 𝟑
Density increases
⟹ 𝑪𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒃𝒍𝒆
Density decreases
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Types of Flows -4
Laminar vs Turbulent
Laminar flow: The highly
ordered fluid motion
characterized by smooth
layers of fluid
Turbulent flow: The
highly disordered fluid
motion that typically
occurs at high velocities
and is characterized by
velocity fluctuations.
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