Solutions Manual
Micro - and Nanoscale Fluid Mechanics Transport in Microfluidic Devices
By
Brian J. Kirby
( All Chapters Included - 100% Verified Solutions )
1
,CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS FOR INCOMPRESSIBLE FLOW
Soln Manual, July 21, 2010 http://www.cambridge.org/kirby
For moving interfaces with uniform surface tension separating Newtonian fluids, the tan-
gential stress is matched on either side of the interface:
∂ut,1 ∂un,1 ∂ut,2 ∂un,2
η1 + = η2 + . (1.77)
∂n ∂t ∂n ∂t
1.11 Supplementary reading
Modern introductory texts that cover the basic fluid mechanical equations include Fox,
Pritchard, and McDonald [17], Munson, Young, and Okiishi [18], White [19], and Bird,
Stewart, and Lightfoot [20]. These texts progress through this material more methodi-
cally, and are a good resource for those with minimal fluids training. More advanced treat-
ment can be found in Panton [21], White [22], Kundu and Cohen [23], or Batchelor [24].
Batchelor provides a particularly lucid description of the Newtonian approximation, the
fundamental meaning of pressure in this context, and why its form follows naturally from
basic assumptions about isotropy of the fluid. Texts on kinetic theory [25, 26] provide a
molecular-level description of the foundations of the viscosity and Newtonian model.
Although the classical fluids texts are excellent sources for the governing equations,
kinematic relations, constitutive relations, and classical boundary conditions, they typically
do not treat slip phenomena at liquid–solid interfaces. An excellent and comprehensive
review of slip phenomena at liquid–solid interfaces can be found in [27] and the references
therein. Slip in gas–solid systems is discussed in [3].
The treatment of surface tension in this chapter is similar to that found in basic fluids
texts [21, 24] but omits many critical topics, including surfactants. Reference [28] covers
these topics in great detail and is an invaluable resource. References [29, 30] cover flows
owing to surface tension gradients, e.g., thermocapillary flows. A detailed discussion of
boundary conditions is found in [31].
Although porous media and gels are commonly used in microdevices, especially for
chemical separations, this text focuses on bulk fluid flow in micro- and nanochannels and
omits consideration of flow through porous media and gels. Reference [29] provides one
source to describe these flows. Another fascinating rheological topic that is largely omit-
ted here is the flow of particulate suspensions and granular systems, with blood being a
prominent example. Discussions of biorheology can be found in [23], and colloid science
texts [29, 32] treat particulate suspensions and their rheology.
1.12 Exercises
1.1 In general, the sum of the extensional strains (εxx + εyy + εzz ) in an incompressible
system always has the same value. What is this value? Why is this value known?
Ans: The sum of extensional strains is zero, because the sum of extensional strains is
Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby 53 http://www.kirbyresearch.com/textbook
2
, CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS
Soln Manual,
http://www.cambridge.org/kirby July 21, 2010 FOR INCOMPRESSIBLE FLOW
proportional to the local change in volume of the flow. Thus for incompressible flow,
cons of mass indicates that the sum of these strains is zero.
1.2 For a 2D flow (no z velocity components and all derivatives with respect to z are
zero),
write the components rate tensor ~~ε in terms of velocity derivatives.
of thestrain
∂u 1 ∂u
+ ∂v 0
∂x 2 ∂y ∂x
Ans: 1 ∂u + ∂v
∂v
0
2 ∂y ∂x ∂y
0 0 0
1.3 Given the following strain rate tensors, draw a square-shaped fluid element and then
show the shape that fluid element would take after being deformed by the fluid flow.
1 0 0
(a) ~
~ε = 0 −1 0 .
0 0 0
−1 0 0
(b) ~
~ε = 0 1 0 .
0 0 0
0 1 0
(c) ~
~ε = 1 0 0 .
0 0 0
Ans: See Fig. 1.24.
Figure 1.24: Fluid deformation owing to strain rate tensor.
http://www.kirbyresearch.com/textbook 54 Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby
3
, CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS FOR INCOMPRESSIBLE FLOW
Soln Manual, July 21, 2010 http://www.cambridge.org/kirby
1.4 The following strain rate tensor is not valid for incompressible flow. Why?
1 1 1
~~ε = 1 1 1 . (1.78)
1 1 1
Ans: The sum of the extensional strains is nonzero; thus this violates conservation
of mass
1.5 Could the following tensor be a strain rate tensor? If yes, explain the two properties
that this tensor satisfies that make it valid. If no, explain why this tensor could not be
a strain rate tensor.
1 1 1
~~ε = 1 0 −1 . (1.79)
−1 1 −1
Ans: No. This tensor is not symmetric.
1.6 Consider an incompressible flow field in cylindrical coordinates with axial symmetry
(for example, a laminar jet issuing from a circular orifice). The axial symmetry im-
plies that the flow field is a function of and z but not θ. Can a stream function be
derived for this case? If so, what is the relation between the derivatives of the stream
function and the and z velocities?
Solution: Yes. conservation of mass in axisymmetric coordinates is:
∇ ·~u = 0 (1.80)
1 ∂ ∂
u + uz = 0 (1.81)
∂ ∂z
now we need to define a stream function ψ such that, if u and uz are defined in terms
of this stream function, conservation of mass is satisfied automatically. If we define
ψ such that
∂ψ
= uz (1.82)
∂
and
∂ψ
= −u , (1.83)
∂z
which is similar to what we use for Cartesian coordinates, this will not work. There
is still an inside the ∂∂ derivative for the radial term.
Instead, try defining ψ such that
Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby 55 http://www.kirbyresearch.com/textbook
4
Micro - and Nanoscale Fluid Mechanics Transport in Microfluidic Devices
By
Brian J. Kirby
( All Chapters Included - 100% Verified Solutions )
1
,CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS FOR INCOMPRESSIBLE FLOW
Soln Manual, July 21, 2010 http://www.cambridge.org/kirby
For moving interfaces with uniform surface tension separating Newtonian fluids, the tan-
gential stress is matched on either side of the interface:
∂ut,1 ∂un,1 ∂ut,2 ∂un,2
η1 + = η2 + . (1.77)
∂n ∂t ∂n ∂t
1.11 Supplementary reading
Modern introductory texts that cover the basic fluid mechanical equations include Fox,
Pritchard, and McDonald [17], Munson, Young, and Okiishi [18], White [19], and Bird,
Stewart, and Lightfoot [20]. These texts progress through this material more methodi-
cally, and are a good resource for those with minimal fluids training. More advanced treat-
ment can be found in Panton [21], White [22], Kundu and Cohen [23], or Batchelor [24].
Batchelor provides a particularly lucid description of the Newtonian approximation, the
fundamental meaning of pressure in this context, and why its form follows naturally from
basic assumptions about isotropy of the fluid. Texts on kinetic theory [25, 26] provide a
molecular-level description of the foundations of the viscosity and Newtonian model.
Although the classical fluids texts are excellent sources for the governing equations,
kinematic relations, constitutive relations, and classical boundary conditions, they typically
do not treat slip phenomena at liquid–solid interfaces. An excellent and comprehensive
review of slip phenomena at liquid–solid interfaces can be found in [27] and the references
therein. Slip in gas–solid systems is discussed in [3].
The treatment of surface tension in this chapter is similar to that found in basic fluids
texts [21, 24] but omits many critical topics, including surfactants. Reference [28] covers
these topics in great detail and is an invaluable resource. References [29, 30] cover flows
owing to surface tension gradients, e.g., thermocapillary flows. A detailed discussion of
boundary conditions is found in [31].
Although porous media and gels are commonly used in microdevices, especially for
chemical separations, this text focuses on bulk fluid flow in micro- and nanochannels and
omits consideration of flow through porous media and gels. Reference [29] provides one
source to describe these flows. Another fascinating rheological topic that is largely omit-
ted here is the flow of particulate suspensions and granular systems, with blood being a
prominent example. Discussions of biorheology can be found in [23], and colloid science
texts [29, 32] treat particulate suspensions and their rheology.
1.12 Exercises
1.1 In general, the sum of the extensional strains (εxx + εyy + εzz ) in an incompressible
system always has the same value. What is this value? Why is this value known?
Ans: The sum of extensional strains is zero, because the sum of extensional strains is
Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby 53 http://www.kirbyresearch.com/textbook
2
, CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS
Soln Manual,
http://www.cambridge.org/kirby July 21, 2010 FOR INCOMPRESSIBLE FLOW
proportional to the local change in volume of the flow. Thus for incompressible flow,
cons of mass indicates that the sum of these strains is zero.
1.2 For a 2D flow (no z velocity components and all derivatives with respect to z are
zero),
write the components rate tensor ~~ε in terms of velocity derivatives.
of thestrain
∂u 1 ∂u
+ ∂v 0
∂x 2 ∂y ∂x
Ans: 1 ∂u + ∂v
∂v
0
2 ∂y ∂x ∂y
0 0 0
1.3 Given the following strain rate tensors, draw a square-shaped fluid element and then
show the shape that fluid element would take after being deformed by the fluid flow.
1 0 0
(a) ~
~ε = 0 −1 0 .
0 0 0
−1 0 0
(b) ~
~ε = 0 1 0 .
0 0 0
0 1 0
(c) ~
~ε = 1 0 0 .
0 0 0
Ans: See Fig. 1.24.
Figure 1.24: Fluid deformation owing to strain rate tensor.
http://www.kirbyresearch.com/textbook 54 Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby
3
, CHAPTER 1. KINEMATICS, CONSERVATION EQUATIONS, AND BOUNDARY
CONDITIONS FOR INCOMPRESSIBLE FLOW
Soln Manual, July 21, 2010 http://www.cambridge.org/kirby
1.4 The following strain rate tensor is not valid for incompressible flow. Why?
1 1 1
~~ε = 1 1 1 . (1.78)
1 1 1
Ans: The sum of the extensional strains is nonzero; thus this violates conservation
of mass
1.5 Could the following tensor be a strain rate tensor? If yes, explain the two properties
that this tensor satisfies that make it valid. If no, explain why this tensor could not be
a strain rate tensor.
1 1 1
~~ε = 1 0 −1 . (1.79)
−1 1 −1
Ans: No. This tensor is not symmetric.
1.6 Consider an incompressible flow field in cylindrical coordinates with axial symmetry
(for example, a laminar jet issuing from a circular orifice). The axial symmetry im-
plies that the flow field is a function of and z but not θ. Can a stream function be
derived for this case? If so, what is the relation between the derivatives of the stream
function and the and z velocities?
Solution: Yes. conservation of mass in axisymmetric coordinates is:
∇ ·~u = 0 (1.80)
1 ∂ ∂
u + uz = 0 (1.81)
∂ ∂z
now we need to define a stream function ψ such that, if u and uz are defined in terms
of this stream function, conservation of mass is satisfied automatically. If we define
ψ such that
∂ψ
= uz (1.82)
∂
and
∂ψ
= −u , (1.83)
∂z
which is similar to what we use for Cartesian coordinates, this will not work. There
is still an inside the ∂∂ derivative for the radial term.
Instead, try defining ψ such that
Micro- and Nanoscale Fluid Mechanics, c Brian J. Kirby 55 http://www.kirbyresearch.com/textbook
4
