1. abj
4.1: Introduction to Forces in Fluids: Surface Force: Shear/Viscous/Frictional Force
Forces in Fluids
Surface Force and Stress
Surface Force: Area as A Vector, and Outward Normal to The System
Description of Surface Force: Stress Vector and Stress Tensor
Newtonian Fluids and Newton’s Viscosity Law
Laminar Flow as A Flow of Layers of Fluids, and Shear Stress as A “Sliding” Friction Between Adjacent Layers of Fluid
abj

2. Very Brief Summary of Important Points and Equations [1]
Forces in Fluids
Forces in fluids are distributive.
Classified according to geometry of distribution:
Line force
Surface force = Pressure Friction
Volume/Body force
The Importance of A Clear Definition of The SYSTEM of Interest
Convention: as an outward normal to the system (pointing from system to surroundings)
A = System
B = Surroundings
B = System
A = Surroundings
abj

3. Very Brief Summary of Important Points and Equations [2]
Surface Force/Stress
Component/Index Convention
Sign Convection
The difference between the
component of stress (tensor) VS component of force (vector)
Newton’s Viscosity Law for Newtonian Fluids
Dynamic Viscosity m
Kinematic Viscosity n
Index convention:
direction
plane
Sign Convention
has a positive numerical value when the product of the signs of plane i and force direction j is positive.
abj

4. Forces in fluids are distributive
Classified according to the geometry of distribution
Line force
Surface force
Volume (Body) force
Described by
Described by
Pressure
(normal)
Friction/Viscous/Shear
(tangential)
[neglect the normal component]
abj

5. Surface Force: Area as A Vector, and Outward Normal to The System
abj
Surface Force: Area as A Vector, and Outward Normal to The System
The importance of the system of interest again
1. Area is a vector
2. Its direction is – by convention - OUTWARD NORMAL to the system of interest
A = System
B = Surroundings A B
B = System
A = Surroundings
Whenever you see a surface with a normal vector,
you know – by convention - which side is being considered a system of interest.
abj

6. abj
Description of Surface Force: Stress Vector and Stress Tensor 2- and 3-Component Decomposition of The Stress Vector P
abj

7. abj
Decomposition of Three Stress Vectors on Three Mutually Perpendicular Planes at A Point: Stress Tensor at A Point (Cartesian Coordinates) Stress Index Convention x y z
Decomposition of three stress vectors on three mutually perpendicular planes at a point
results in nine components of a tensor called the stress tensor at that point:
Index convention:
direction
plane
abj

8. Some Properties (Symmetric Tensor)
abj
Some Properties (Symmetric Tensor) x y =
Symmetry of the stress tensor at a point, e.g.,
From angular momentum consideration with body couple neglected, it follows
Stress tensor at a point is symmetric, i.e.,
(in matrix notation: )
This reduces the general 9 independent components of the stress tensor to 6 independent components (good - less variables to deal with).
abj

9. Stress Sign Conventions
abj
Stress Sign Conventions x y z x z y x y z
Sign Conventions: (Note that some books use different index convention, e.g., reverse)
The sign of (the numerical value of) the stress components:
sign of the component t i j = sign of plane i x sign of direction j
On the negative plane, the sign of the component of stress is opposite to
the sign of the component of force.
Index convention:
direction
plane
abj

10. Note the difference: component of stress (tensor) VS component of force (vector)
On the negative plane, the sign of the component of stress is opposite to the sign of the component of force.
Component of stress: x y
Component of the force due to that stress:
Area is a vector. It is signed.
abj

11. Constitutive Relation: Newton’s Viscosity Law for Newtonian Fluids
abj
Constitutive Relation: Newton’s Viscosity Law for Newtonian Fluids
Assuming that the velocity field is a simple shear flow: u x y
y = 0
u(y)
Slope du/dy = strain rate
plane y
Top layer of fluid
Bottom layer of fluid
plane y
Consider the relation between the shear stress tyx and the shear deformation rate du/dy.
Newtonian fluids are the fluids in which shear stress is linearly proportional to shear deformation rate.
Newtonian fluids:
Newton’s viscosity law
m = Dynamic/Absolute viscosity,
n = Kinematic viscosity,
abj

12. The Importance of A Clear Definition of The SYSTEM of Interest, Again.
abj
The Importance of A Clear Definition of The SYSTEM of Interest, Again.
A = System
B = Surroundings A B
the external force due to the surrounding B on the system A
the external force due to the surrounding A on the system B
B = System
A = Surroundings
Are you talking about
the external force/stress due to the surrounding B on the system A or
the external force/stress due to the surrounding A on the system B?
Although they are action/reaction, they are not the same force.
abj

13. Some Notes on Newton’s Viscosity Law
abj
Some Notes on Newton’s Viscosity Law u x y
y = 0
u(y)
slope
plane y y
Top layer of fluid
Bottom layer of fluid
It is a field/instantaneous equation, i.e., it is a function of position and time:
At the wall, we often refer to the shear stress there as the wall shear stress:
It is a signed equation, the signs of
strain rate du/dy , and
stress txy (according to the stress sign convention)
should be taken into accounted.
Carefully consider which system and which stress are under consideration:
Lower/Pink element (unit outward normal is ), or
Upper/Grey element (unit outward normal is ).
Both have the same value of , but the system and the force are different.
abj

14. Newton’s Viscosity Law in The Natural Coordinates (n-t )
abj
Newton’s Viscosity Law in The Natural Coordinates (n-t ) x y y x
u(y)
ut(n) n t x y
x’, t
y’, n
y’, n
x’, t
y’, n
x’, t x y
Note also that
ut(n) and u(y) are two different velocity profiles.
and are two different stresses acting on two different planes.
abj

15. Constitutive Relation: Other Types of Fluids
abj
Constitutive Relation: Other Types of Fluids
Newtonian and Non-Newtonian fluids’ stress-strain rate relation.
From Fox, R. W., McDonald, A. T., and Pritchard, P. J., 2004, Introduction to Fluid Mechanics, Sixth Edition, Wiley, New York.
Question: What will happen if shear stress is gradually applied and increased from zero to each of these fluids?
abj

16. Dynamic Viscosity (m) VS Temperature
Fox et al. (2010)
abj

17. Kinematic Viscosity (n ) VS Temperature
Fox et al. (2010)
abj

18. Example 1: Sketch The Velocity Profile and Examine The Shear Force/Stress
Assume that the followings are simple shear flows; that is,
there is only one dominant velocity component, and
this velocity component is a function of the traverse/normal coordinate:
Question:
Sketch the velocity profile along the traverse AB.
State
the sign ( + / - ) of the component of the stress,
the sign/direction (+ / - ) of the component of the force due to that stress,
on the surface with the outward normal given.
See next page 
abj

19. Flow between two stationary parallel plates
Solid wall element W E C D
Plate is moving at the speed U. y A B
Still air U x W y
Flow A B
Flow between two stationary parallel plates
(Channel flow) D C E LW F G UW H
abj

20. Laminar Flow as A Flow of Layers of Fluids, and Shear Stress as A “Sliding” Friction Between Adjacent Layers of Fluid
Laminar flow can be thought of as a stack of layers of fluid sliding over one another.
In between the layers, there is a sliding frictional force.
A Physical View of Shear Stress as A Sliding Friction in Laminar Flow
The direction of the sliding frictional force is such that
the adjacent surrounding exerts the frictional force opposing the relative velocity/motion of the system with respect to the adjacent surroundings.
system
relative to the surrounding floor.
system
relative to the surrounding floor.
Sliding frictional force is opposing the relative velocity of the system [relative to the adjacent surrounding floor].
abj

21. Sliding Friction: Drag Backward – Drag Forward
Sliding friction between
the two layers y
relatively faster upper layer
relatively slower lower layer
A) Shear stress on the faster layer:
system
B) Shear stress on the slower layer:
system
Due to “sliding friction” between the two layers,
faster layer is being dragged backward by the slower layer.
slower layer is being dragged forward by the faster layer.
abj

22. Example 1: Sketch The Velocity Profile and Examine The Shear Force/Stress
Assume that the followings are simple shear flows; that is,
there is only one dominant velocity component, and
this velocity component is a function of the traverse/normal coordinate:
Question:
Sketch the velocity profile along the traverse AB.
State
the sign ( + / - ) of the component of the stress,
the sign/direction (+ / - ) of the component of the force due to that stress,
on the surface with the outward normal given.
See next page 
abj

23. Flow between two stationary parallel plates
Solid wall element W E C D
Plate is moving at the speed U. y A B
Still air U x W y
Flow A B
Flow between two stationary parallel plates
(Channel flow) D C E LW F G UW H
abj

24. Example
abj