Chapter 8 IDEAL-FLUID FLOW

1
Chapter 8IDEAL-FLUID FLOW
2

• In the preceding chapters most of the relations
  have been developed for one-dimensional flow,
  i.e., flow in which the average velocity at each
  cross section is used and variations across the
  section are neglected.
• Many design problems in fluid flow, however,
  require more exact knowledge of velocity and
  pressure distributions, such as in flow over
  curved boundaries along an airplane wing, through
  the passages of a pump or compressor, or over the
  crest of a dam.
• An understanding of two- and three-dimensional
  flow of a nonviscous, incompressible fluid
  provides the student with a much broader approach
  to many real fluid-flow situations. There are
  also analogies that permit the same methods to
  apply to flow through porous media.
• In this chapter the principles of irrotational
  flow of an ideal fluid are developed and applied
  to elementary flow cases. After the flow
  requirements are established, Euler's equation is
  derived and the velocity potential is defined.

3
8.1 REQUIREMENTS FOR IDEAL-FLUID FLOW

• The Prandtl hypothesis states that for fluids of
  low viscosity the effects of viscosity are
  appreciable only in a narrow region surrounding
  the fluid boundaries.
• For incompressible flow situations in which the
  boundary layer remains thin, ideal-fluid results
  may be applied to flow of a real fluid to a
  satisfactory degree of approximation.
•  An ideal fluid must satisfy the following
  requirements
• 1. The continuity equation div q 0, or
•                 
• 2. Newton's second law of motion at every point
  at every instant
• 3. Neither penetration of fluid into, nor gaps
  between, fluid and boundary at any solid 
  boundary

4

• If, in addition to requirements 1, 2, and 3, the
  assumption of irrotational flow is made, the
  resulting fluid motion closely resembles
  real-fluid motion for fluids of low viscosity,
  outside boundary layers.
• Using the above conditions, the application of
  Newton's second law to a fluid particle leads to
  the Euler equation, which, together with the
  assumption of irrotational flow, can be
  integrated to obtain the Bernoulli equation. The
  unknowns in a fluid-flow situation with given
  boundaries are velocity and pressure at every
  point.
• Unfortunately, in most cases it is impossible to
  proceed directly to equations for velocity and
  pressure distribution from the boundary
  conditions.

5
8.2 EULER'S EQUATION OF MOTION

• Euler's equation of motion along a streamline
  (one-dimensional) was developed in Sec. 3.5 by
  use of the momentum and continuity equations and
  Eq. (2.2.5).
• In this section it is developed from Eq. (2.2.5)
  for the xyz-coordinate system in any orientation,
  with the assumption that gravity is the only body
  force acting. Since Euler's equation is based on
  a frictionless fluid, the vector equation (2.2.5)
  
•               
• (2.2.5)
• may be reorganized into the proper form. The unit
  vector j' is directed vertically upward in the
  coordinate direction h.
• These are the direction cosines of h with respect
  to the xyz system of coordinates, and they may be
  written

6
Figure 8.1 Arbitrary orientation of xyz
coordinate system
7

• For example, ?h/?x is the change in h for unit
  change in x when y, z, and t are constant. In
  equation form,
•               
• The operation applied to the scalar h
  yields the gradient of h, as in Eq. (2.2.2).
• Eq. (2.2.5) now becomes
• (8.2.1)
• The component equations of Eq.(8.2.1) are
• (8.2.2)

8

• u, v, w are velocity components in the x, y, z
  directions, respectively, at any point du/dt is
  the x component of acceleration of the fluid
  particle at (x, y, z).
• Since u is a function of x, y, z, and t and x, y,
  and z are coordinates of the moving fluid
  particle, they become functions of t hence,
•                 
• However, dx/dt, dy/dt, and dz/dt are the velocity
  components of the particle, so that ax, the x
  component of the particle acceleration, is

9

• By treating dv/dt and dw/dt in a similar manner,
  the Euler equations in three dimensions for a
  frictionless fluid are
•                                    
• (8.2.3)
• (8.2.4)
• (8.2.5)
• The first three terms on the right-hand sides of
  the equations are convective-acceleration terms,
  depending upon changes of velocity with space.
• The last term is local acceleration, depending
  upon velocity change with time at a point.

10
Natural Coordinates in Two-Dimensional Flow

• Euler's equations in two dimensions are obtained
  from the general-component equations by setting w
  0 and ?/?z0 thus
•                                            
• (8.2.6)
• (8.2.7)
• By taking particular directions for the x and y
  axes, they can be reduced to a form that makes
  them easier to understand.
• The velocity component u is vs, and the component
  v is vn. As vn is zero at the point, Eq. (8.2.6)
  becomes
• (8.2.8)
•                                              
• Although vn is zero at the point (s, n), its
  rates of change with respect to s and t are not
  necessarily zero. Equation (8.2.7) becomes
• (8.2.9)

11
Figure 8.2 Notation for natural coordinates
12

• With r the radius of curvature of the streamline
  at s, from similar triangles (Fig. 8.2),
•          
•        
• Substituting into Eq. (8.2.9) gives
• (8.2.10)
•                                                
• For steady flow of an incompressible fluid Eqs.
  (8.2.6) and (8.2.10) can be written
•          
• 
  
  (8.2.11)
• and 
• (8.2.12)
•                                               
• Equation (8.2.11) can be integrated with respect
  to s to produce Eq. (3.6.1), with the constant of
  integration varying with n, that is, from one
  streamline to another.
• Equation (8.2.12) shows how pressure head varies
  across streamlines. With vs and r known functions
  of n, Eq. (8.2.12) can be integrated.

13

• Example 8.1
• A container of liquid is rotated with angular
  velocity ? about a vertical axis as a solid.
  Determine the variation of pressure in the
  liquid.
• Solution
• n is the radial distance, measured inwardly, dn
  -dr, and vs ?r . Integrating Eq. (8.2.12) gives
  
•        
• or             
• To evaluate the constant, if p p0 when r 0
  and h 0,
•                 
• which shows that the pressure is hydrostatic
  along a vertical line and increases as the square
  of the radius.
• Integration of Eq. (8.2.11) shows that the
  pressure is constant for a given h and vs, that
  is, along a streamline.

14
8.3 IRROTATIONAL FLOW VELOCITY POTENTIAL

• In this section it is shown that the assumption
  of irrotational flow leads to the existence of a
  velocity potential. By use of these relations and
  the assumption of a conservative body force, the
  Euler equations can be integrated.
• The individual particles of a frictionless
  incompressible fluid initially at rest cannot be
  caused to rotate. This can be visualized by
  considering a small free body of fluid in the
  shape of a sphere. Surface forces act normal to
  its surface, since the fluid is frictionless, and
  therefore act through the center of the sphere.
• Similarly, the body force acts at the mass
  center. Hence, no torque can be exerted on the
  sphere, and it remains without rotation.
  Likewise, once an ideal fluid has rotation, there
  is no way of altering it, as no torque can be
  exerted on an elementary sphere of the fluid.

15

• An analytical expression for fluid rotation of a
  particle about an axis parallel to the z axis is
  developed. The rotation component may be defined
  as the average angular velocity of two
  infinitesimal linear elements that are mutually
  perpendicular to each other and to the axis of
  rotation.
• The two line elements may conveniently be taken
  as x and y in Fig. 8.3, although any other two
  perpendicular elements in the plane through the
  point would yield the same result. The particle
  is at P(x, y), and it has velocity components u,
  v in the xy plane. The angular velocities of dx
  and dy are sought.
• The angular velocity of dx is
• and the angular velocity of dy is
•             
•     
• if counterclockwise is positive. Hence, by
  definition, the rotation component ?z of a fluid
  particle at (x,y) is
• (8.3.1)

16
Figure 8.3 Rotation in a fluid
17

• Similarly, the two other rotation components, ?x
  and ?y, about axes parallel to x and to y are
• (8.3.2)
•                                     
• The rotation vector ? is
•               
•                                      (8.3.3)
• The vorticity vector, curl q q, is
  defined as twice the rotation vector. It is given
  by 2?.
• By assuming that the fluid has no rotation, i.e.,
  it is irrotational, curl q 0, or from Eqs.
  (8.3.1) and (8.3.2)
• (8.3.4)
• These restrictions on the velocity must hold at
  every point (except special singular points or
  lines).

18

• The first equation is the irrotational condition
  for two dimensional flow. It is the condition
  that the differential expression
•            
•     
• is exact, say
• (8.3.5)
• The minus sign is arbitrary it is a convention
  that causes the value of f to decrease in the
  direction of the velocity. By comparing terms in
  Eq. (8.3.5),
• In vector form,
•  (8.3.6)
• is equivalent to
• (8.3.7)

19

• The assumption of a velocity potential is
  equivalent to the assumption of irrotational
  flow, as
•            
• (8.3.8)
•                               
• because . This is shown from Eq.
  (8.3.7) by cross-differentiation
•                            
• proving etc.
• Substitution of Eqs. (8.3.7) into the continuity
  equation
•                 
• yields
• (8.3.9)

20

• In vector form this is
•                                            
• (8.3.10)
• and is written
• Equation (8.3.9) or (8.3.10) is the Laplace
  equation. Any function that satisfies the Laplace
  equation is a possible irrotational fluid-flow
  case. As there are an infinite number of
  solutions to the Laplace equation, each of which
  satisfies certain flow boundaries, the main
  problem is the selection of the proper function
  for the particular flow case.
• Because appears to the first power in each term,
  Eq. (8.3.9), is a linear equation, and the sum of
  two solutions also is a solution e.g., if f1 and
  f2 are solutions of Eq. (8.3.9), then f1 f2 is
  a solution thus
•                       
• then  
•          
• Similarly, if f1 is a solution, Cf1 is a solution
  if C is constant.

21
8.4 INTEGRATION OF EULERS EQUATIONS BERNOULLI
EQUATION

• Equation (8.2.3) can be rearranged so that every
  term contains a partial derivative with respect
  to x. From Eq. (8.3.4)
•                 
•            
• and from Eg. (8.3.7)
•                 
• Making these substitution into Eq. (8.2.3) and
  rearranging give
•                 
• As the square of the speed,
• (8.4.1)

22

• Similarly, for the y and z direction,
•        
•                                            (8.4.2)
  
• (8.4.3)
• The quantities within the parentheses are the
  same in Eqs. (8.4.1) to (8.4.3). Equation (8.4.1)
  states that the quantity is not a function of x,
  since the derivative with respect to x is zero.
• Similarly, the other equations show that the
  quantity is not a function of y or z. Therefore,
  it can be a function of t only, say F(t)
•            
•                       (8.4.4)
•              
• In steady flow ?f/?t0 and F(t) becomes a
  constant E
• (8.4.5)
•                                                   
         
• The available energy is everywhere constant
  throughout the fluid. This is Bernoulli's
  equation for an irrotational fluid.

23

• The pressure term can be separated into two
  parts, the hydrostatic pressure ps, and the
  dynamic pressure pd, so that ps pd. Inserting
  in Eq. (8.4.5) gives
•                
• The first two terms can be written
•                 
• with h measured vertically upward. The expression
  is a constant, since it expresses the hydrostatic
  law of variation of pressure. These two terms may
  be included in the constant E. After dropping the
  subscript on the dynamic pressure, there remains
•         (8.4.6)
•                                                   
      
• This simple equation permits the variation in
  pressure to be determined if the speed is known
  or vice versa. Assuming both the speed q0 and the
  dynamic pressure p0 to be known at one point,
• (8.4.7)

24

• Example 8.2
• A submarine moves through water at a speed of 10
  m/s. At a point A on the submarine 1.5 m above
  the nose, the velocity of the submarine relative
  to the water is 15 m/s. Determine the dynamic
  pressure difference between this point and the
  nose, and determine the difference in total
  pressure between the two points.
• Solution
• If the submarine is stationary and the water is
  moving past it, the velocity at the nose is zero
  and the velocity at A is 15 m/s. By selecting the
  dynamic pressure at infinity as zero, from Eq.
  (8.4.6)
•              
•    
• For the nose
•      
•             
•                    
• For point A

25

• Therefore, the difference in dynamic pressure is
•        
•            
• The difference in total pressure can be obtained
  by applying Eq. (8.4.5) to point A and to the
  nose n,
•        
•          
• Hence      
•    
• It can also be reasoned that the actual pressure
  difference varies by 1.5? from the dynamic
  pressure difference since A is 1.5 m above the
  nose, or

26
8.5 STREAM FUNCTIONS BOUNDARY CONDITIONS

• Two-Dimensional Stream Function
• If A, P represent two points in one of the flow
  planes, e.g., the xy plane (Fig. 8.4), and if the
  plane has unit thickness, the rate of flow across
  any two lines ACP, ABP must be the same if the
  density is constant and no fluid is created or
  destroyed within the region, as a consequence of
  continuity.
• If A is a fixed point and P a movable point, the
  flow rate across any line connecting the two
  points is a function of the position of P.
• If this function is ?, and if it is taken as a
  sign convention that it denotes the flow rate
  from right to left as the observer views the line
  from A looking toward P, then is defined as the
  stream function.

27
Figure 8.4 Fluid region showing the positive flow
direction used in the definition of a stream
function
Figure 8.5 Flow between two points in a fluid
region
28

• If ?1 and ?2 represent the values of stream
  function at points Pl and P2 (Fig. 8.5),
  respectively, then ?2 - ?1 is the flow across
  PlP2 and is independent of the location of A.
• Taking another point 0 in the place of A changes
  the values of ?1 and ?2 by the same amount,
  namely, the flow across OA. Then ? is
  indeterminate to the extent of an arbitrary
  constant.
• The velocity components u, v in the x, y
  directions can be obtained from the stream
  function. In Fig. 8.6a, the flow d? across APdy,
  from right to left, is -udy, or
• (8.5.1)
• (8.5.2)

29

• In plane polar coordinates from Fig. 8.6b.
• Comparing Egs. (8.3.3) with Eqs. (8.5.1) and
  (8.5.2) leads to
• (8.5.3)
• These are the Cauchy-Riemann equations.
•  
• By Eqs. (8.5.3), a stream function can be found
  for each velocity potential. If the velocity
  potential satisfies the Laplace equation, the
  stream function also satisfies it. Hence, the
  stream function may be considered as velocity
  potential for another flow case.

30
Figure 8.6 Selection of path to show relation of
velocity components to stream function
31

• Stokes's Stream Function for Axially Symmetric
  Flow
• In any one of the planes through the axis of
  symmetry select two points A, P such that A is
  fixed and P is variable. Draw a line connecting
  AP.
• The flow through the surface generated by
  rotating AP about the axis of symmetry is a
  function of the position of P. Let his function
  be 2P?, and let the x axis of symmetry be the
  axis of a cartesian system of reference. Then ?
  is a function of x and ?, where
• is the distance from P to the x axis. The
  surfaces ? const are stream surfaces.
• The resulting relations between stream function
  and velocity are given by

32

• Solving for u and v gives
• (8.5.4)
• The same sign convention is used as in the
  two-dimensional case.
• The relations between stream function and
  potential function are
• (8.5.5)
• The stream function is used for flow about bodies
  of revolution that are frequently expressed most
  readily in spherical polar coordinates. From Fig.
  8.7a and b,

33

• from which              
•                                 
                 (8.5.6)
• and                                         
• (8.5.7)
• These expressions are useful in dealing with flow
  about spheres, ellipsoids, and disks and through
  apertures.

34
Figure 8.7 Displacement of to show the relation
between velocity components and Stokes' stream
function.
35

• Boundary Conditions
• At a fixed boundary the velocity component normal
  to the boundary must be zero at every point on
  the boundary (Fig. 8.8)
•                                                   
                   (8.5.8)
• n1  is a unit vector normal to the boundary. In
  scalar notation this is easily expressed in terms
  of the velocity potential
•                                                   
                    (8.5.9)
• at all points on the boundary. For a moving
  boundary (Fig. 8.9), where the boundary point has
  the velocity V, the fluid velocity component
  normal to the boundary must equal the velocity of
  the boundary normal to boundary thus
•                                                   
                (8.5.10)
• or                                                
            
• (8.5.11)

36
Figure 8.8 Notation for boundary condition at a
fixed boundary
Figure 8.9 Notation for boundary condition at a
moving boundary
37
8.6 THE FLOW NET

• In two-dimensional flow the flow net is of great
  benefit it is taken up in this section.
• The line given by f(x, y)const is called an
  equipotential line. It is a line along which the
  value of f (the velocity potential) does not
  change. Since velocity vs in any direction s is
  given by
• The line f(x, y)const is a streamline and is
  everywhere tangent to the velocity vector.
  Streamlines and equipotential lines are therefore
  orthogonal i.e., they intersect at right angles,
  except at singular points.
• In Fig. 8.10, if the distance between streamlines
  is ?n and the distance between equipotential
  lines is ?s at some small region in the flow net,
  the approximate velocity vs is the given in terms
  of the spacing of the equipotential lines Eq.
  (8.3.7),
• or in terms of the spacing of streamlines Eqs.
  (8.5.1) and (8.5.2).

38
Figure 8.10 Elements of a flow net
39

• In steady flow when the boundaries are
  stationary, the boundaries themselves become part
  of the flow net, as they are streamlines. The
  problem of finding the flow net to satisfy given
  fixed boundaries may be considered purely as a
  graphical exercise
• This is one of the practical methods employed in
  two-dimensional-flow analysis, although it
  usually requires many attempts and much erasing.
•  Another practical method of obtaining a flow net
  for a particular set of fixed boundaries is the
  electrical analogy. The boundaries in a model are
  formed out of strips of nonconducting material
  mounted on a flat nonconducting surface, and the
  end equipotential lines are formed out of a
  conducting strip, e.g., brass or copper.
• The relaxation method numerically determines the
  value of potential function at points throughout
  the flow, usually located at the intersections of
  a square grid. The laplace equation is written as
  a difference equation, and it is shown that the
  value of the potential function at a grid point
  is the average of the four values at the
  neighboring grid points.

40

• Use of the Flow Net
• After a flow net for a given boundary
  configuration has been obtained, it may be used
  for all irrotational flows with geometrically
  similar boundaries.
• Application of the Bernoulli equation Eq.
  (8.4.7) produces the dynamic pressure. If the
  velocity is known, e.g., at A (Fig. 8.10).
• With the constant ?c determined for the whole
  grid in this manner, measurement of ?s or ?n at
  any other point permits the velocity to be
  computed there,
•                
• The concepts underlying the flow net have been
  developed for irrotational flow of an ideal
  fluid.

41
8.7 TWO-DIMENSIONAL FLOW

• Flow around a Corner
• The potential function
•             
•     
• has as its stream function
•                 
• in which r and ? are polar coordinates. It is
  plotted for equal-increment changes in f and ? in
  Fig. 8.11. Conditions at the origin are not
  defined, as it is a stagnation point.
• The streamlines are rectangular hyperbolas having
  yx as axes and the coordinate axes as
  asymptotes. From the polar form of the stream
  function it is noted that the two lines ?0 and
  ?p/2 are the streamline ?0.

42
Figure 8.11 Flow net for flow around a 90o
 bend
Figure 8.12 Flow net for flow along two inclined
surfaces
43

• Source
• A line normal to the xy plane, from which fluid
  is imagined to flow uniformly in all directions
  at right angles to it, is a source. It appears as
  a point in the customary two-dimensional flow
  diagram. The total flow per unit time per unit
  length of line is called the strength of the
  source.
• Since by Eq. (8.3.7) the velocity in any
  direction is given by the negative derivative of
  the velocity potential with respect to the
  direction,
• is the velocity potential, in which r is the
  distance from the source. This value of f
  satisfies the Laplace equation in two dimensions.

44

• The streamlines are radial lines from the source,
  i.e.,
•                
• From the second equation
•                
• Lines of constant f (equipotential lines) and
  constant ? are shown in Fig. 8.13. A sink is a
  negative source, a line into which fluid is
  flowing.

Figure 8.13 Flow net for source or vortex
45

• Vortex
• In examining the flow case given by selecting the
  stream function for the source as a velocity
  potential,
•                              
• which also satisfies the Laplace equation, it is
  seen that the equipotential lines are radial
  lines and the streamlines are circles.
• The velocity is in a tangential direction only,
  since ?f/?r0 . It is
•                 
• since r ?f is the length element in the
  tangential direction.
• In referring to Fig. 8.14, the flow along a
  closed curve is called the circulation. The
  circulation G around a closed path C is

46

• The value of the circulation is the strength of
  the vortex. By selecting any circular path of
  radius r to determine the circulation,
  hence,

Figure 8.14 Notation for definition of
circulation
47

• Doublet
• The two-dimensional doublet is defined as the
  limiting case as a source and sink of equal
  strength approach each other so that the product
  of their strength and the distance between them
  remains a constant 2pµ. µ is called the strength
  of the doublet.
• In Fig. 8.15 a source is located at (a, 0) and a
  sink of equal strength at (-a, 0). The velocity
  potential for both, at some point P, is
• with r1, r2 measured from source and sink,
  respectively, to the point P.

48
Figure 8.15 Notation for derivation of a
two-dimensional doublet
49

• The terms r1, and r2 may be expressed in terms of
  the polar coordinates r, ? by the cosine law, as
  follows
• Rewriting the expression for f with these
  relations gives

50

• The series expression
• leads to
•              
•                 
•                 
• After simplifying,

51

• If 2amµ, and if the limit is taken as a
  approaches zero,
•                 
• which is the velocity potential for a
  two-dimensional doublet at the origin, with axis
  in the x direction.
• Using the relations
•                
•              
• gives for the doublet
•            
•                      
• After integrating,

52

• The equations in cartesian coordinates are
•                           
• Rearranging gives
•                           
• The lines of constant f are circles through the
  origin with centers on the x axis, and the
  streamlines are circles through the origin with
  centers on the y axis, as shown in Fig. 8.16.
• The origin is a singular point where the velocity
  goes to infinity.

53
Figure 8.16 Equipotantial lines and streamlines
for the two-dimensional doublet
54

• Uniform Flow
• Uniform flow in the -x direction, u -U, is
  expressed by
•                                   
• In polar coordinates,

55

• Flow around a Circular Cylinder
• The addition of the flow due to a doublet and a
  uniform flow results in flow around a circular
  cylinder thus
•                           
• As a streamline in steady flow is a possible
  boundary, the streamline ?0 is given by
•                 
• which is satisfied by ?0, p, or by the value of
  r that makes
•                 
• If this value is ra, which is a circular
  cylinder, then

56

• The potential and stream functions for uniform
  flow around a circular cylinder of radius a are,
  by substitution of the value of µ
•          
•                  
• for the uniform flow in the -x direction. The
  equipotential lines and streamlines for this case
  are shown in Fig. 8.17
• On the surface of the cylinder the velocity is
  necessarily tangential and is expressed by
  ??/?r0 for ra thus
• For the dynamic pressure zero at infinity, with
  Eq. (8.4.7)

57
Figure 8.17 Equipotential lines and streamlines
for flow around a circular cylinder
58

• For points on the cylinder,
• A cylindrical pilot-static tube is made by
  providing three openings in a cylinder, at 0 and
  30, as the difference in pressure between 0
  and 30 is the dynamic pressure ?U2/2.
• The drag on the cylinder is shown to be zero by
  integration of the x component of the pressure
  force over the cylinder thus
• Similarly, the lift force on the cylinder is
  zero.

59

• Flow around a Circular Cylinder with Circulation
  
• The addition of a vortex to the doublet and the
  uniform flow results in flow around a circular
  cylinder with circulation,
•                 
•                 
• The streamline is the circular cylinder r
  a. At great distances from the origin, the
  velocity remains u - U, showing that flow
  around a circular cylinder is maintained with
  addition of the vortex. Some of the streamlines
  are shown in Fig. 8.18.

60
Figure 8.18 Streamlines for flow around a
circular cylinder with circulation
61

• The velocity at the surface of the cylinder,
  necessarily tangent to the cylinder, is
•                 
• Stagnation points occur when q0 that is,
• The pressure at the surface of the cylinder is
•               
• The drag again is zero. The lift, however,
  becomes

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• The theoretical flow around a circular cylinder
  with circulation can be transformed  into flow
  around an airfoil with the same circulation and
  the same lift.
• The airfoil develops its lift by producing
  circulation around it due to its shape. It can be
  shown that the lift is for any cylinder in
  two-dimensional flow. The angle or inclination of
  the airfoil relative to the approach velocity
  (angle of attack) greatly affects the
  circulation. For large angels or attack, the flow
  does not follow the wing profile, and the theory
  breaks down.
• It should be mentioned that all two-dimensional
  ideal-fluid-flow cases may be conveniently
  handled by complex-variable theory and by a
  system of conformal mapping, which transforms a
  flow net from one configuration to another by a
  suitable complex-variable mapping function.

63

• Example 8.3
• A source with strength 0.2 m3/sm and a vortex
  with strength 1 m2/s are located at the origin.
  Determine the equations for velocity potential
  and stream function. What are the velocity
  components at x 1 m, y 0.5 m?
• Solution
• The velocity potential for the source is
•                      
• and the corresponding stream function is

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• The velocity potential for the vortex is
•                                                   
     
• and the corresponding stream function is
•         
•         
• Adding the respective functions gives
•              
•        
• The radial and tangential velocity components are
  
•                          
•               
• At (1, 0.5), , vr 0.0285 m/s, v?
  0.143 m/s

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• Example 8.4
• A circular cylinder 2 m in diameter and 20 m long
  is rotating at 120 rpm in the positive direction
  (counterclockwise) about its axis. Its center is
  at the origin of a cartesian coordinate system.
  Wind at 10 m/s blows over the cylinder in the
  positive x direction t200C and p100 kPa abs.
  Determine the lift on the cylinder and the
  location in the fourth quadrant of the streamline
  through the stagnation point.
• Solution
• The stagnation point has ?0 . By selecting
  increments of R, ? can be determined from
•                
• The lift is given by ?UGL.