CH. 6 : Differential Analysis of Fluid Flow

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6.1 Fluid Element Kinematics(???)
CH. 6 Differential Analysis of Fluid
Flow Composition of Fluid Motion 1
Translation(??, ??) 2 Linear Deformation
dilatation(??, ??) , volume change, stretching 3
Rotation(??, ??) 4 Angular Deformation
deformation(??, ??), shear deformation
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6.1.2 Linear Motion and Deformation
1./ Translation of a Fluid Element (see p 312
Fig. 6.2) no velocity gradient If all
points in the element have the same
velocity(which is only true if there are no
velocity gradients), then the element will simply
translate from one position to another.
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• 2./ Linear Deformation of a Fluid Element ?
  Volumetric Dilatation (?????)(see p312 Fig.
  6.3)
• rate at which the volume is changing per unit
  volume due to the gradient
• 
  
• 
  

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? General case in which,
are also present
? volumetric dilatation rate
(?????) meaning rate of change of the volume
per unit volume for an
incompressible fluid since the element volume
cannot change without a change in fluid density
? Variations in the velocity in the direction
of the velocity such as






????????? simply cause a linear
deformation(?? ??) of the element in the sense
that the shape of the element does not change. ?
Cross derivatives, such as, and ,
will cause the element to rotate and generally to
undergo an angular deformation(???), which
changes the shape of the element.
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6.1.3 Angular Motion and Deformation
1./ Rotation Vector(Angular Velocity) ,
Vorticity(??) Vector 1 angular velocity of
line OA and OB
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2 Rotation Vector rotation of the
element about the z axis
average of the angular
velocities and Similarly
(let clockwise -,
counterclockwise ) 3 vorticity(??, ??)
vector
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2./ Irrotational Flow(?????), Rotational
Flow(????) 3./ Note! ? We observe from Eq.
6.12 that the fluid element will rotate about the
z axis as an undeformed block(i.e.,
) only when Otherwise
the rotation will be associated with an angular
deformation. ? We also note from Eq. 6.12 that
when the rotation about the
z axis is zero. 4./ Angular Deformation In
addition to the rotation associated with the
derivatives x and , it is
observed from Fig. 6.4b that these
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derivatives can cause the fluid element to
undergo an angular deformation, which results in
a change in shape of the element. 1
Definition1 ? The deformation of the fluid
particle in Fig. 6.4 is the rate of change of the
angle that line segment OA makes with line
segment OB.. ? If OA is rotating with an angular
velocity different from that of OB, the particle
is deforming. ? The deformation is represented
by the rate-of-strain tensor. 2 Rate of
Shearing Strain (or rate of angular deformation)
? The rate of
angular deformation is related to a corresponding
shearing stress which causes the fluid element to
change in shape.
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? From Eq. 6.18 we note that if
, the rate of angular deformation is
zero, and this condition corresponds to the case
in which the element is simply rotating as an
undeformed block. ? The instantaneous angular
velocity of a fluid particle is the average of
the instantaneous angular velocities of two
mutually perpendicular lines on the fluid
particle
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Linear Translation

• All points in the element have the same velocity
  (which is only true if there are o velocity
  gradients), then the element will simply
  translate from one position to another.

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Linear Deformation 1/2

• The shape of the fluid element, described by the
  angles at its vertices, remains unchanged, since
  all right angles continue to be right angles.
• A change in the x dimension requires a nonzero
  value of
• A y
• A z

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Linear Deformation 2/2

• The change in length of the sides may produce
  change in volume of the element.

The change in
The rate at which the ?V is changing per unit
volume due to gradient ?u/ ?x
If ?v/ ?y and ?w/ ?z are involved
Volumetric dilatation rate
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Angular Rotation 1/4
The angular velocity of line OA
For small angles
CCW
CW
- for CCW
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Angular Rotation 2/4
The rotation of the element about the z-axis is
defined as the average of the angular velocities
?OA and ?OB of the two mutually perpendicular
lines OA and OB.
In vector form
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Angular Rotation 3/4
Defining vorticity
Defining irrotation
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Angular Rotation 4/4
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Vorticity

• Defining Vorticity ? which is a measurement of
  the rotation of a fluid element as it moves in
  the flow field
• In cylindrical coordinates system

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Angular Deformation 1/2

• Angular deformation of a particle is given by the
  sum of the
• two angular deformation

?(Xi)?(Eta)
Rate of shearing strain or the rate of angular
deformation
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Angular Deformation 2/2

• The rate of angular deformation in xy plane
• The rate of angular deformation in yz plane
• The rate of angular deformation in zx plane

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Example 6.1
ltSol.gt
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6.2 Conservation of Mass

• 6.2.1 Differential Form of Continuity Equation
• 1./ General Form Continuity Equation
• 2./ Steady flow of compressible flow
• 3./ Incompressible Flow
• applicable to both steady and unsteady flow
  of incompressible fluids

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Conservation of Mass 2/5

• The CV chosen is an infinitesimal cube with sides
  of length ?x, ? y, and ? z.

Net rate of mass Outflow in x-direction
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Conservation of Mass 3/5
Net rate of mass Outflow in x-direction
Net rate of mass Outflow in y-direction
Net rate of mass Outflow in z-direction
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Conservation of Mass 4/5
Net rate of mass Outflow
The differential equation for conservation of mass
Continuity equation
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Conservation of Mass 5/5

• Incompressible fluid
• Steady flow

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Example 6.2 ltSol.gt
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6.2.2 Cylindrical Polar Coordinates

• 1./ General Form Continuity Equation
• 2./ Steady flow of compressible flow
• 3./ Incompressible Flow
• applicable to both steady and unsteady flow
  of incompressible fluids

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6.2.3 The Stream Function
1./ Stream Function 1 Application Range
Incompressible Plane Fluid Flows, not applicable
to general 3-dimensional flow 2 Stream Function
(Psi ?? ??) scalar function 3
Continuity Equation for Plane Flow Thus
whenever the velocity components are defined in
terms of the stream function we know that
conservation of mass will be satisfied.
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4 Mathematical Description of a Streamline

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5 The difference between any two
streamlines is equal to the flow rate per unit
depth between the two streamlines. - If the
upper streamline, , has a value greater than
the lower streamline, , then q is
positive, which indicates that the flow is from
left to right. For the flow is from
right to left.
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6 Remark ! Although we still don't know what
is for a particular problem, but at
least we have simplified the analysis by having
to determine only one unknown function,
, rather than the two functions,
and Another particular advantage of using the
stream function is related to the fact that lines
along which is constant are streamlines.
is constant along a streamline Note that flow
never crosses streamlines, since by definition
the velocity is tangent to the streamline.
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2) Polar Coordinates - Continuity Equation -
Velocity Components
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Example 6.3 ltSol.gt
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6.3 Conservation of Linear Momentum
6.3.1 Description of Forces Acting on the
Differential Element ? The intensity of the
force per unit area at a point in a body can thus
be characterized by a normal stress and two
shearing stresses, if the orientation of the area
is specified. ? double subscript notation for
stresses (see p325 Fig. 6.10) - The first
subscript indicates the direction of the normal
to the plane on which the stress acts, and the
second subscript indicates the direction of the
stress. - Thus, normal stresses have repeated
subscripts, whereas the subscripts for shearing
stresses are always different.
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6.3.2 Equations of Motion
- General Differential Equations of Motion for a
Fluid where
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6.4 Inviscid Flow
Definition of Inviscid Flow (?????) ? Flow
fields in which the shearing stresses are assumed
to be negligible are said to be inviscid,
nonviscous, or frictionless. ? In this case
? We know that for some common fluids, such as
air and water, the viscosity is small, and
therefore it seems reasonable to assume that
under some circumstances we may be able to simply
neglect the effect of viscosity (and thus
shearing stresses).
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6.4.1 Euler's Equations of Motion
Euler Equation ?????? ?????
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6.4.2 The Bernoulli Equation

• Basic Assumptions
• inviscid flow steady flow incompressible
  flow
• flow along a streamline
• 2. Derivation (see p328-9, Fig. 6.12)
• 3. General Form for Inviscid Flow
• 4. Bernoulli Equation

Figure 6.12 (p. 290)The notation for
differential length along a streamline.
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6.4.3 Irrotational Flow
Criterion - Remarks Since the weight acts
through the element center of gravity, and the
pressure acts in a direction normal to the
element surface, neither of these forces can
cause the element to rotate. Boundary Layer
Near the boundary the velocity changes rapidly
from zero at the boundary (no-slip condition) to
some relatively large value in a short distance
from the boundary.
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This rapid change in velocity gives rise to a
large velocity gradient normal to the boundary
and produces significant shearing stresses, even
though the viscosity is small. Of course if we
had a truly inviscid fluid, the fluid would
simply "slide" past the boundary and the flow
would be irrotational everywhere. But this is not
the case for real fluids, so we will typically
have a layer (usually very thin) near any fixed
surface in a moving stream in which shearing
stresses are not negligible. This layer is called
the boundary layer.
Figure 6.13 (p. 292)Uniform flow in the x
direction.
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Potential Flow and Boundary Layer Model

• Boundary layer hypothesis by L. Prandtl in 1904
• In a flow with large velocity and small
  viscosity, the effects of fluid viscosity and
  rotation (shear stress, turbulence, and
  vorticity) are confined to thin regions near
  solid surfaces or surfaces of velocity
  discontinuity. Outside these thin regions, the
  flow is nearly inviscid and irrotational.

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(???)
(??)
(??)
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6.4.4 The Bernoulli Equation for Irrotational Flow
For incompressible, irrotational flow the
Bernoulli equation can be written as just above
Eq.(6.63) between any two points in the flow
field. Equation 6.63 is exactly the same form as
Eq. 6.58 but is not limited to application along
a streamline.
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6.4.5 The Velocity Potential
1. Definition For an irrotational flow the
velocity gradients are related through Eqs. 6.59,
6.60, and 6.61. It follows that in this case the
velocity components can be expressed in terms of
a scalar function as Phi
So for an irrotational flow the velocity is
expressible as the gradient of a scalar function
. 2. Velocity Potential and Stream Function
- The velocity potential is a consequence of the
irrotationality of the flow field, whereas the
stream function is a consequence of conservation
of mass.
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• It is to be noted, however, that the velocity
  potential can be defined for a general
  three-dimensional flow, whereas the stream
  function is restricted to two-dimensional flows.
• 3. Potential Flow
• 1 definition inviscid, incompressible,
  irrotational flow
• - Velocity Potential Function
• - Criterion of Potential Flow
• 2 governing equation Laplace equation
• 3 solving method
• If the velocity potential function can be
  determined, then the velocity at all points in
  the flow field can be determined from

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Eq. 6.64, and the pressure at all points can be
determined from the Bernoulli equation. 4
Laplace Equation in Cylindrical Coordinate
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6.5 Some Basic, Plane Potential Flows
1. Major Advantage of Laplace Equation - A major
advantage of Laplace equation is that it is a
linear partial differential equation. Since it is
linear, various solutions can be added to obtain
other solutions-that is, if hh(x,y,z) and
gg(x,y,z) are two solutions to Laplace equation,
then kk(x,y,z) is also a solution. - The
practical implication of this result is that if
we have certain basic solutions we can combine
them to obtain more complicated and interesting
solutions.
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2. Laplace Equation for both Velocity Potential
and Stream Function Criterion of
(irrotationality stream function )
? Laplace equation
3. Orthogonality between streamlines (
) and equipotential lines (
) 4. Flow Net For any potential
flow field a "flow net" can be drawn that
consists of a family of streamlines and
equipotential lines.
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6.5.1 Uniform Flow (????, ????)
1. Definition the simplest flow in which the
streamlines are all parallel, and the magnitude
of the velocity is constant 2. Uniform Flow in
the Positive x-Direction (see p338 Fig. 6.16(a))
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3. General Case (see p338, Fig. 6.16(b))
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6.5.2 Source and Sink
1. Velocity Components

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2. Flow Net Fig. 6.17 3. Singularity (???)
at

? source
and sink a mathematical singularity in
the flow field
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Example 6.5 ltSol.gt
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6.5.3 Vortex(??)
1. Velocity Components 2. Flow Net (see Fig.
6.18) 3. Singularity at r 0 4. Irrotational
Vortex (Free Vortex) and Rotational Vortex
(Forced Vortex) (see Fig. 6.19)
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5. Circulation(??, ??) 1 definition 2
value ? For an irrotational flow, This
result indicates that for an irrotational flow
the circulation will generally be zero. ?
Irrotational flow with singularity If there are
singularities enclosed within the curve the
circulation may not be zero. ? Irrotational flow
without singularity (see p343 Fig. 6.21) ?
for the free vortex including origin with
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6. and in terms of
Figure 6.21 (p. 304)Circulation around various
paths in a free vortex.
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6.5.4 Doublet(???)
1. Flow Net (see Fig. 6.22) where m
strength of source and sink
strength of the doublet
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2. Velocity Components
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6.6 Suerposition of Basic, Plane Potential Flows
6.6.1 Source in a Uniform Stream - Half-Body
1. System(see Fig. 6.24) Method of
Superposition(??, ??)
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2. Flow Net Source Uniform Flow 3.
Stagnation Point (???) 4. Half Body (??) body
which is open at the downstream end 5. Velocity
Components, Pressure
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Example 6.7
ltSolgt a)
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b)
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6.6.2 Rankine Ovals
1. Closed body/Open body 2. System(see p351 Fig.
6.25)
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3. Flow Net 4. Large variety of body shape
with different length to width ratio
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Rankine Oval Uniform Stream Doublet 1/3
The corresponding streamlines for this flow field
are obtained by setting ?constant. It is
discovered that the streamline forms a closed
body of length 2? and width 2h.
Rankine ovals
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Rankine Oval Uniform Stream Doublet 2/3

• The stagnation points occur at the upstream and
  downstream ends of the body. These points can be
  located by determining where along the x axis the
  velocity is zero.
• The stagnation points correspond to the points
  where the uniform velocity, the source velocity,
  and the sink velocity all combine to give a zero
  velocity.
• The locations of the stagnation points depend on
  the value of a, m, and U.

Dimensionless
The body half-length 2?
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Rankine Oval Uniform Stream Doublet 3/3

• The body half-width, h, can be obtained by
  determining the value of y where the y axis
  intersects the ?0 streamline. Thus, with ?0,
  x0, and yh.

The body half-width 2h
Dimensionless
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6.6.3 Flow around a Circular Cylinder
1./ Flow around a Circular Cylinder without
Circulation 1. Flow Net - streamline see p353
Fig. 6.26 2. Velocity Components
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Separation(??, ??)
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3. Velocity and Pressure on Surface 4.
Resultant Forces Drag(??), Lift(??) 5.
d'Alembert's Paradox
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Examples of Irrotational Flows Formed by
Superposition

• Flow over a circular cylinder Free stream
  doublet
• Assume body is ? 0 (r a) ? K Va2

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Examples of Irrotational Flows Formed by
Superposition

• Velocity field can be found by differentiating
  streamfunction
• On the cylinder surface (ra)

Normal velocity (Ur) is zero, Tangential velocity
(U?) is non-zero ?slip condition.
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Examples of Irrotational Flows Formed by
Superposition

• Compute pressure using Bernoulli equation and
  velocity on cylinder surface

Turbulentseparation
Laminarseparation
Irrotational flow
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Examples of Irrotational Flows Formed by
Superposition

• Integration of surface pressure (which is
  symmetric in x), reveals that the DRAG is ZERO.
  This is known as DAlemberts Paradox
• For the irrotational flow approximation, the drag
  force on any non-lifting body of any shape
  immersed in a uniform stream is ZERO
• Why?
• Viscous effects have been neglected. Viscosity
  and the no-slip condition are responsible for
• Flow separation (which contributes to pressure
  drag)
• Wall-shear stress (which contributes to friction
  drag)

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2./ Flow around a Circular Cylinder with
Circulation 1. Flow Net streamline see Fig.
6.29 2. Velocity and Pressure on Surface
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3. Stagnation Point -
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3. Resultant Forces 4. Magnus
Effect 5. Kutta-Joukowski Theorem
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???? (Jean Le Rond d'Alembert  1717.11.161783.10.29)
  ???? ??? ???? ?????. ?????(?????)? ???? ??? ? ???? ?? ???? ????, ? ????(????)??. ?? ?? ?????(?) ??? ??? ??? ??, ???? ? ? ?? ????? ???? ????, ?? ? ???? ?? ?? ?? ?? ??? ???? ??. ???? ?? ?? ?? ????? ??? ??? ???. ?? ??? ?? 20? ? ??? ?? ????.   ?? ????? ??? ??? ?? ????? ????, ?? ? ??? ??? ????, ? ??? ??? ??? ?? ???? 23?? ???? ??? ?????. 12? ? ??? ? ?? ???? ???? ?? ?? ??? ??????, ?? ? ?? ?? ????? ??? ????, ?? ??(??)??? ??? ??? ???. ?? ??? Trait? de dynamique(1743)? 26? ? ??(??)? ???, ?? ? ???? ? ??? ????? ??? ??? ??????? ????, ??? ??? ??? ??? ???? ????? ??? ????.   ?, ??? ??? ???(???)? ??? ?? ????(????)? ??? ???? ????? ??? ????, ??? ???? ??? ?? ??????? ??? ?????? ????? ? ??? ?????.   
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??? 1754? ???? ??? ??? ?? ??? ??? ?? ??? ?????
??? ??? ???? ???? ????? ?? ??? ?? ???? ???. ?????
??? ????? ????? ??? ??? ???? ?? ??(?? ??? ??? 1?
??? ?? ????? f(x)? ??? ??? ?? ?? ???)? ????? ??
??? ?????. ??? ??? ?????? ??? ??????? ??? ??? ??
??????.   ? ?? ??(??)? ??(??)? ??(1749), ?? ????
??? 3?(??)??? ?? ?, ???? ???? ?????. ??????
?????? ????? ?? ???? ??????, ?? D.???? ???? ??
??? ????? ????.   ? ???? ?? ??? ??? ???
??????, ? ?? ?????? ????? ??? ????? ??? ? ??
??????, ? ????? ??? ??? ?????. ?? ? ?? ?? ? ???
??????, ??? ?? ??? F.???? ??? ??? ??? ??? ??? ???
????, ??? ??? ?????? ???(???)? ? ??? ?????. ???
?? ??? ??? ??? ???? ??? ??? ???? ?? ??? ??????
????? ????? ?? ??? ?????? ?????.
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??? 1754? ???? ??? ??? ?? ??? ??? ?? ??? ?????
??? ??? ???? ???? ????? ?? ??? ?? ???? ???. ?????
??? ????? ????? ??? ??? ???? ?? ??(?? ??? ??? 1?
??? ?? ????? f(x)? ??? ??? ?? ?? ???)? ????? ??
??? ?????. ??? ??? ?????? ??? ??????? ??? ??? ??
??????.   ? ?? ??(??)? ??(??)? ??(1749), ?? ????
??? 3?(??)??? ?? ?, ???? ???? ?????. ??????
?????? ????? ?? ???? ??????, ?? D.???? ???? ??
??? ????? ????.   ? ???? ?? ??? ??? ???
??????, ? ?? ?????? ????? ??? ????? ??? ? ??
??????, ? ????? ??? ??? ?????. ?? ? ?? ?? ? ???
??????, ??? ?? ??? F.???? ??? ??? ??? ??? ??? ???
????, ??? ??? ?????? ???(???)? ? ??? ?????. ???
?? ??? ??? ??? ???? ??? ??? ???? ?? ??? ??????
????? ????? ?? ??? ?????? ?????.
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Heinrich Gustav Magnus (2 May 1802 4 April
1870) was a German chemist and physicist. The
Magnus effect was named after him. He was born in
Berlin. His father was a wealthy merchant and of
his five brothers one, Eduard (1799-1872), became
a celebrated painter. After studying at Berlin,
he went to Stockholm to work under Berzelius, and
later to Paris, where he studied for a while
under Gay-Lussac and Thénard. In 1831 he returned
to Berlin as lecturer on technology and physics
at the university. In 1834 he became assistant
professor of physics and technology in the
university there, and in 1845 was appointed
professor. As a teacher his success was rapid and
extraordinary. His lucid style and the perfection
of his experimental demonstrations drew to his
lectures a crowd of enthusiastic scholars, on
whom he impressed the importance of applied
science by conducting them round the factories
and workshops of the city and he further found
time to hold weekly colloquies on physical
questions at his house with a small circle of
young students. From 1827 to 1833 he was occupied
mainly with chemical researches, which resulted
in the discovery of the first of the
platino-ammonium compounds (Magnus's green salt
is Pt(NH3)4PtCl4), of sulphovinic acids, and
isethionic acids and their salts, and, in
conjunction with , of periodic acid. Among other
subjects at which he subsequently worked were the
diminution in density produced in garnet and
vesuvianite by melting, the absorption of gases
in blood (18371845), the expansion of gases by
heat (18411844), the vapour pressures of water
and various solutions (18441854),
thermoelectricity (1851), electrolysis (1856),
induction of currents (1858-1861), conduction of
heat in gases (1860), polarization of heat
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(18661868) and the deflection of projectiles
from firearms. From 1861 onwards he devoted much
attention to the question of diathermancy in
gases and vapours, especially to the behaviour in
this respect of dry and moist air, and to the
thermal effects produced by the condensation of
moisture on solid surfaces. In 1834 Magnus was
elected extraordinary, and in 1845 ordinary
professor at Berlin. He was three times elected
dean of the faculty, in 1847, 1858 and 1863 and
in 1861, rector magnificus. His great reputation
led to his being entrusted by the government with
several missions in 1865 he represented Prussia
in the conference called at Frankfurt am Main to
introduce a uniform metric system of weights and
measures into Germany. For forty-five years his
labor was incessant his first memoir was
published in 1825 when he was yet a student his
last appeared shortly after his death. He married
in 1840 Bertha Humblot, of a French Huguenot
family settled in Berlin, by whom he left a son
and two daughters.
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Example 6.8
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ltSol.gt (a) (b)
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6.8 Viscous Flow
6.8.1 Stress-Deformation Relationships for
Incompressible Newtonian Fluids
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6.8.2 The Navier-Stokes Equations
1. Equation - Vector Notation - Scalar
Components 2. Well-Posed Problem and
Ill-Pose Problem
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6.9 Some Simple Solutions for Viscous,
Incompressible Fluids
6.9.1 Steady Laminar Flow between Fixed Parallel
Plates 1. Velocity Field(see p362 Fig. 6.30)
--gt from continuity equation
for infinite plate, and
for steady flow 2. Navier-Stokes
Equation where
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3. Velocity Profile(see p363 Fig. 6.30b)
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4. Volume Flow Rate where
pressure drop between two points a distance l
apart The pressure gradient
is negative, since the pressure decreases in the
direction of flow. The flow is proportional
to the pressure gradient, inversely
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proportional to the viscosity, and strongly
dependent on the gap width. 5. Mean
velocity
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6.9.3 Steady, Laminar Flow in Circular Tubes
1. Hagen-Poiseuille Flow(or Poiseuille flow)
best known exact solution to the Navier-Stokes
Equation steady, incompressible, laminar flow
through a straight circular tube of constant
cross section 2. Governing Equations ?
Assumptions The flow is parallel to the
walls so that For steady, axisymmetric flow
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• ? Velocity Distribution
• ? Volume Flow Rate, Mean Velocity, Maximum
  Velocity
• ? The flow remains laminar for Reynolds numbers,
• below 2100.

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Example 6.10 ltSol.gt