Homework Set 1 -Transport Phenomena I 155:303
Due Thursday September 12th
Fall 2024 -
1. Define the divergence operator and describe its meaning. Is it a vector or a scalar?
Define the gradient operator and define its meaning. Is it a vector or a scalar?
2. Define the stress tensor mathematically. How many components does it have? How many
independent components does it have?
3.
(a) What is viscosity? What are its units? What is the physical meaning of viscosity?
(b) How does the viscosity of liquids depend on pressure? On temperature? Why?
(c) How does the viscosity of ideal gases depend on pressure? On temperature? Why?
4. What is the difference between a Newtonian and a non-Newtonian fluid? Give examples
5. What is the Reynolds Number? Describe its meaning.
6.
7.
cellation
b )Spar normal to=
8.
9.
compre a
10. An infinite plate is moved over a second plate on a layer of liquid as shown with velocity
u(y). For small gap width b=0.3mm, we assume a linear velocity distribution in the liquid with
speed U=0.3m/s. The liquid viscosity is 0.65 centipoise and its specific gravity is 0.88.
Determine:
(a) The liquid absolute (or dynamic) viscosity µ of the liquid, in lbf. s/ft2.
(b) The kinematic viscosity n of the liquid, in m2/s.
(c) The shear stress in lbf/ft2.
(d) The shear stress in Pa.
Data: Linear velocity profile in the liquid between infinite parallel plates as shown.
μ = 0.65 cp
SG = 0.88
Homework Set 1 303 Transport Phenomena I
Fall 2023
Problem 1
Divergence operator (!"#! or ∇ ∙ !) is the dot product of the gradient vector and a vector field
!"#! = ∇ ∙ ! =
!!!
!"
+
!!!
!"
+
!!!
!"
Divergence operator on a vector field produces a scalar. It represents the outward/inward flux of a vector
field from an infinitesimal volume around a given point. ( expansion or contraction of a vector field.) If
the divergence is positive , e div F>0 it means the fluid field is expanding. If the div F<0 it means the
fluid field is contracting. If the Div (F)=0 it means the fluid field is incompressible
! ! ⎛ ∂v ∂v y ∂v ⎞
+ z ⎟⎟ the divergence is a scalar (a number).
∇.v = ⎜⎜ x +
∂
x
∂
y
∂z ⎠
⎝
Notice that it is a sum and it does not have a direction in space.
Gradient operator (∇) of a scalar-valued function f is a vector field whose components are the partial
derivatives of the vector field.
∇f =
!" !
!" !"
,
,...,
!!!
!!! !!!
It represents how a scalar field changes in all directions:
For example the gradient of the pressure represents the variation of the pressure which is a scalar in the x,
y and z direction.
!
⎛ ∂p ∂p ∂p ⎞
∇p = ⎜ , , ⎟ is a vector (that is why we put an arrow on top of the grad operator
⎝ ∂x ∂y ∂z ⎠
Problem 3
3a. Viscosity is a quantity measuring the resistance to deformation at a given rate. Informally, viscosity
is the quantity that describes a fluid's resistance to flow. Fluids resist the relative motion of immersed
objects through them as well as to the motion of layers with differing velocities within them.
Formally ( ie mathematically speaking) Formally, viscosity (represented by the symbol µ "mu") is the ratio of the
shearing stress (F/A) to the velocity gradient (∆vx/∆y or dvx/dy) in a fluid.
µ=
F/A
∆vx/∆y
Unit:
!"
!"
µ=
F/A
dvx/dy
3b. Pressure has almost no effect on the viscosity of liquids since normally, liquids are incompressible.
Only at extremely high pressure, increase in pressure slightly increases liquid viscosity. Increases in
temperature decreases liquid viscosity since the molecules are separated more and they move more easily.
3c. Pressure does not affect the viscosity of ideal gasses. Increase in temperature increases the viscosity
of ideal gases since gas molecules collide more often.
Problem
Newtonian fluids are fluids whose viscosity is independent of the shear rate. Example: water, air.
Newtonian fluids are fluids in which the shear stress t is linearly proportional to the gradient of the
velocity. The constant of proportionality is the viscosity m.
t
!!
!
τ = µ (∇v ) = µ grad (v )
slope is m
Newtonian fluids
grad(v)
Non-Newtonian fluids are fluids whose viscosity changes with the shear rate. Example: toothpaste, paints.
The shear stress t is not linearly proportional to the shear rate (i.e. the shear rate is equal to grad v)
Problem Reynolds number is a dimensionless number. It is the ratio of inertial forces to viscous
forces of a fluid flow:
Re =
!"#
!
Reynolds number describes the conditions of a flow:
Low Reynolds number à creeping flow
Intermediate Reynolds number à laminar flow
Problem 2 Describe the stress tensor
The stress tensor has 6 independent components, since there the components yx zx and zy
are exactly the same as xy, xz and yz.
Solution
Problem 7
Unsteady flow since the given velocity field depends on time
3-D flow since the given velocity has 3 components (3 directions)
3a.
!=
!! !!
=
+ !∙∇ !
!"
!"
!
Where v = 3txiˆ − t 2 yjˆ + 2 xzkˆ
=
!!
!
!
!
+ !!
+ !!
+ !!
!"
!"
!"
!"
!! ! + !! ! + !! !
= 3!! − 2!"! + 3!" 3! ! + 3!" 2! ! − ! ! ! −! ! ! + 2!" 2! !
= (3! + 9! ! !)! + (−2!" + ! ! !)! + (6!"# + 4! ! !)!
At (1, -1, 0)
! = (3 + 9! ! )! + (2! − ! ! )!
!! = 0,0, ! This is the most obvious solution.
Problem 8
ol
o Problem
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