1.1 Basic Concepts
1.1.1 Laminar flow and turbulent flow
The flow of a fluid can be in the regime of laminar flow or turbulent flow. Laminar flow is a well-ordered flow and is characterized by the smooth sliding of a adjacent fluid layers over one another, with mixing between layers occurring only on a molecular level.
Due to molecular diffusion , the moving fluid molecular imparts some of its momentum to the adjacent layer of fluid and causes it to also move in the z-direction.
The molecular diffusion across the boundary between two adjacent fluid layers in laminar flow, as a result, the z-momentum is transmitted through the fluid in the y-direction. In fact, this is why fluid flow is considered as momentum transfer . (Fig. 1.1-1)
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Turbulent flow is a chaotic flow and is characterized by the transfer of small packets of fluid particles between layers. The fluid tends to switch from laminar to turbulent flow when the fluid flow is intensified.
The transition from laminar to turbulent flow was studied by Reynolds in 1883, and is explained using the apparatus shown in Fig. 1.1-2. As shown, a dye is injected along the axis of a glass tube through which water is flowing. Fig. 1.1-3 represents the typical patterns of the dye in water flowing through the glass tube.
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Besides the fluid velocity (v), Reynolds found that the nature of flow is also affected by three other parameters: the tube inner diameter (D), fluid density ( r
), and fluid viscosity ( m
). The fluid velocity is defined as the average of fluid volume flowing the tube per unit area and time. The four parameters can be combined into a single dimensionless parameter, which is called Reynolds number (Re
D
).
Re
L for plate
Re
D
 r
VD m

VD m r

VD
 or
VL
 [1.1-1]
For fluid flows in a circular tube is laminar if Re
D is less than approximately 2100
(or 2300), beyond this, the critical Reynolds number, fluid flow becomes turbulent.
Some fluid remains laminar flow even the Re number is over the critical value.
However, it is unstable, a small perturbation turned the fluid to turbulence.
Re
D
~ inertial forces viscous forces
(Fox & Mcdonald, p. 296)
Re r
 r
VL m
 r
VL V L m
V L
1
L

2 2
V L ~
L  r m
2 2
V L
V
( )
L
 m inertial force
V
L
2
L
3

1.1.2 Newton’s Law of Viscosity
1.1.2.1 One-Dimensional
A fluid is contained between two large parallel plates separated by a small distance L, as shown in Fig. 1.1-4. The lower plate is pulled at a constant velocity
V in the z-direction, whereas the upper plate is kept stationary. A steady velocity profile v z
(y) is established in the fluid.
Let us consider two parallel thin layers of fluid (Fig. 1.1-4). The lower layer at y has a velocity v v z
+ dv z
(dv z z
, and the upper one at y + dy has a slightly smaller velocity
<0). Since the velocity gradient, it tends to exert a shear stress t yz between the two layers. The greater the velocity difference, the steeper the velocity gradient dv z
/dy is, the greater the shear stress t yz becomes. For most of the fluid, the shear stress is a linear relationship with the velocity gradient, and can be expressed by Eq. 1.1-2, where m is the viscosity of the fluid.
t yz
  m d v z dy
[1.1.2]
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Eq. 1.1-2 is an expression of momentum flux, that is, the amount of momentum transferred per unit area per unit time. The minus sign suggests that a momentum flux occurs in the direction of decreasing velocity. The shear stress t yz is the flux of z-momentum in the y-direction.
Fluids that obey Eq.1.1-2 are called
Newtonian fluids. The unit of viscosity is g/cm ‧ s, which is called the poise.
The viscosities of some molten materials are shown in Figs. 1.1-5 through 1.1-7.
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6

Constitutive Equation for Fluid Flow – A First Course in Continuum Mechanics by Y. C. Fung (1977), Prentice-Hall, Inc., New Jersey, p.165
A Newtonian fluid is a viscous fluid for which the shear stress is linearly proportional to the rate of deformation. For a Newtonian fluid the stress-strain relationship is specified by the equation.
s ij
  p d ij

D ijkl
V kl
(1)
Where s ij is the stress tensor, V kl is the rate-of-deformation tensor, D ijkl is a tensor of viscosity coefficients of the fluid, and p is the static pressure. The term –p d ij represents the state of stress possible in a fluid at rest (when V kl
= 0).
The static pressure p is assumed to depend on the density and temperature of the fluid according to an equation of state. For Newtonian fluids we assume that the elements of the tensor D ijkl may depend on the temperature but not on the stress of the rate of deformation.
The tensor D ijkl
, of rank 4, has 3 4 = 81 elements. Not all these constants are independent. A study of the theoretically possible number of independent elements can be made by examining the symmetry properties of the tensor s ij
, V kl and the symmetry that may exist in the atomic constitution of the fluid.
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Most fluids appear to be isotropic, for which the structure of D ijkl is greatly simplified, as will be seen below. If the fluid is isotropic, i.e., if the tensor D ijkl has the same array of components in any system of rectangular Cartesian coordinates, then
D ijkl can be expressed in terms of two independent constants l and m
:
D ijkl
 ld ij d kl
 m
d ik d jl
 d il d jk
(2)
After derivation under the assumption of isotropic material, the constitutive equation becomes s ij
  p d ij

2
3 md ij
V kk

2 m
V ij
(3)
This formulation is due to George G. Stokes and a fluid that obeys Eq.(3) is called a Stokes fluid, for which one material constant m
, the coefficient of viscosity, suffices to define its property.
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Eqs. 1.1-4 through 1.1-9 can be obtained from Eq. 3 by substituting i=j=x; i=j=y; and i= j= z for 1.1-4, 1.1-5, and 1.1-6, respectively, and i=x, j=y; i=y, j=z; and i=z, j=x for Eqs. 1.1-7, 1.1-8, and 1.1-9, respectively.
For fluids with constant density r
, that is incompressible fluids, .
As such, from Eqs. 1.1-3 through 1.1-9, for incompressible Newtonian fluids
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We have or or t   m
(

V
 
V
T
)
[1.1-13]
Table 1.1-1 through 1.1-3 list the components of the stress tensor t in rectangular, cylindrical, and spherical coordinates, respectively.
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1.1.2.3 Kinematic Viscosity
The viscosity m is sometimes called the dynamic viscosity. The so-called kinematic viscosity
 is defined as follows:
  m r
Where r is the density of the fluid.
1.1.2.3 Stress Calculation Procedure
According to Newton’s law of viscosity, the stresses acting by a moving fluid on a solid can be determined if the velocity distribution in the fluid is known. The procedure is listed as follows:
1. Choose a proper coordinate system;
2. Construct a control volume
3. Determine which stress component is relevant to the problem and pick the corresponding equation from the table chosen in step 1;
4. Substitute the given velocity distribution(s) into the equation chosen in step 3 and determine the stress component.
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Example 1.1-1 Shear stress due to laminar flow through a tube
Consider the steady-state laminar flow of an incompressible
Newtonian fluid through a tube of inner radius R and length L at a constant volume flow rate Q. The velocity distribution is as follows:
 z

2 Q

R
2
[ 1
 r
(
R
)
2
]
Where r is the radial coordinate. Calculate the shear stress and friction force at the tube wall.
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Example 1.1-2 Shear stress due creeping flow around a sphere
An incompressible, Newtonian fluid approaches a solid sphere of radius R at a constant and uniform velocity of v
∞
.
The flow and the size of the sphere are such that the
Reynolds number 2R r v
∞
/ m is significantly less than one; this is the creeping flow condition.
The velocity distribution in creeping flow around a sphere
 r
 v

[ 1

3
2
(
R
) r

1
2
(
R
)
3 r
] cos



  v

[ 1

3
4
(
R
) r

1
4
(
R
)
3 r
] sin

Calculate the shear stress acting on the sphere by the fluid.
The stress acting on the surface of the sphere are t rr
(normal stress), and
The shear stress
The shear stress t t r
 r
 t r

and t
can be expressed as
can be expressed as r
 t t r

(shear stress).
r

=-
= - m m



 r r

 r
1 sin
( v


) r r
 v
 
 r
1
 r
 v
 
 r



( r r
 v


) 1.1-29



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

1.1.3 Streamlines
Streamlines are a family of curves showing the pattern of fluid flow, which are drawn in such a manner that the velocity vectors for all points along the curves would meet them tangentially.
As illustrated in Fig. 1.1-11b v y v x
 dy dx
= slope of streamline [1.1-31]
The velocity vector v is tangent to the streamline. In three dimensions the relation becomes dx v x
 dy v y
 dz v z
Fig. 1.1-11 A velocity vector V along a streamline
In steady flow, a streamline is also a path line, which is a line describing the path or trajectory of a given fluid element. In fluid flow experiments call flow visualization , a tracer, such as a dye or smoke, is used to reveal streamlines in a transparent fluid.
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Visualization of streamlines in two-dimensional flow of water at (a) a contraction;
(b) an expansion.
In fact, the stream is separated from the boundary of the channel by a region of backflow. It is worth noting that streamlines can still be used to represent the flow pattern when turbulence exists. These streamlines, however, represent only the average or time-smoothed pattern of fluid motion.
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1.1.4 Momentum Boundary Layer
1.1.4.1 External flow
Consider a uniform fluid approaching a stationary flat plate. Because of the effect of viscosity, the velocity v z of the fluid in the region near the plate is reduced by the plate, varying from zero at the plate surface to v
∞ in the stream. This region is called the momentum boundary layer.
Its thickness d is typically taken as the distance from the plate surface at which the dimensionless velocity v z
/v
∞ that v z
= v
∞
 v
 y

0 d levels off to 0.99. In practice, it is usually specified
. With increasing distance from the leading edge of the plate, the effect of viscosity penetrates further into the stream and the boundary layer grows in thickness.
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If the flat plate is sufficient long, the flow in the boundary layer can eventually change from laminar to turbulent. The higher the v
∞ is , the closer to the leading edge of the plate the laminar-to-turbulent transition occurs. The transition is usually considered to occur at a length Reynolds number Re
Z
= r v
∞ z/ m of about 2x10 5 , where z is the distance from the leading edge.
Stagnation point: Consider a fluid flow normal to a body, such as cylinder, as shown in Fig.1.1-15, point A is called the stagnation point where fluid comes to a stop at the cylinder. No fluid crosses the streamline passing through point A. As illustrated in Fig. 1.1-15a, the flow is sufficiently fast that the boundary layer separately itself from the cylinder at point C since surface BD is far from being streamlined.
The region of slow backflow near surface CD is called a wake.
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1.1.4.2 Internal flow
Consider a fluid that enters a circular tube of inner diameter D with a uniform velocity v parallel to its axis, as shown in Fig.1.1-16. A laminar boundary layer begins to develop at the entrance, gradually expanding at the expense of the free stream until the layers from opposite sides approach at the centerline at z
D

r
m
D
The flow in the region where the boundary layer thickness d and the velocity profile v z
(r,z) vary with the axial distance z is called developing flow. On the other hand, the flow in the region where they no longer vary with z is called fully developed flow.
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1.1.5 Momentum transfer coefficient and other coefficients
1.1.5.1 External Flow
Consider fluid flow over a flat plate such as that shown in Fig.1.1-14. The shear stress at the plate is t yz y

0
  m
 v z
 y y

0
This equation cannot be used to calculate the shear stress when the velocity gradient is an unknown. A convenient way to avoid this problem is to introduce a momentum transfer coefficient, defined
C
' f
 v z t yz y

0 y

0
 v

 t yz y

0
0
 v

The friction coefficient is a dimensionless number defined by
C f

C
' f
1
2 r v

 t yz y

0
1
2 r v
2

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Consider fluid flow normal to a cylinder such as that shown in Fig.1.1-15. Let
F
D be the total drag force acting on the cylinder by the fluid. The cylinder not only presents a surface in contact with the fluid to cause the friction drag, but also a physical body blocking the flow to cause the form drag. For flow normal to a cylinder, sphere, disk or plate, the drag coefficient is a dimensionless number defined by where A f is the frontal area of the solid, that is, the area of the solid projected perpendicular to the free-stream velocity v
∞
.
1.1.5.2 Internal flow
For fluid flow through a tube of length L and inner diameter D, the friction factor, is a dimensionless parameter defined by
Subscripts o and L denote the entrance and the exit, respectively.
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The average velocity is defined as follows
1.2 Overall mass-balance equation
1.2.1 Derivation
Let us consider an arbitrary stationary volume element Ω in a fluid flow field, as illustrated in Fig. 1.2-1. The volume element, called the control volume, represents the system being considered. The boundary of the control volume is sometimes called the control surface.
As shown in Fig. 1.21, let dΩ be a differential volume element in Ω and dA a differential surface element on A. The velocity vector v and the density of the fluid are defined at every point in the control volume and surface. n is the outward unit normal vector of a differential area dA on the control surface A, and v is the velocity of the fluid at dA.
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