Unformatted text preview: APPH 4200
Physics of Fluids
Review (Ch. 3) & Fluid Equations of Motion (Ch. 4)
September 20, 2011 1.! Chapter 3 (more notes)
2. ! Vorticity and Circulation
3.! NavierStokes Equation
1 Summary:
CauchyStokes Decomposition 2 Velocity Gradient Tensor 3 Material or Convective
Derivative 4 Integral Relations
(Section 4.2) 5 Integral Relations
(Section 4.2) 6 Visualizing Flow 7 Characterizing Circulating Flow 8 The addition of equations (3.41) and (3.42) leads to a21/ a21/ i a (a1/) i a21/
ax2 + ay2 = ; ar ra; + r2 ae2 = 0, Problem 3.1 which completes the transformation. Exercises 1. A twodimensional steady flow has velocity components u =y v =x. Show that the streamines are rectangular hyperbolas x2 _ y2 = const.
Sketch the flow pattern, and convince yourself that it represents an irrotational flow
in a 90° comer.
2. Consider a steady axisymmetrc flow of a compressibla fluid. The equation of continuity in cylindrcal coordinates (R, cp, x) is a ax
a
aR (pRUR) + (pRux) = O. Show how we can define" a streamfnction so that the equation of contiuity is satisfied
automatically.
3. !fa velocity field is given by u = ay, compute the circulation around a 9 circle of radius r = i about the origin. Check the result by using Stokes' theorem. Problem 3.1 4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is u = Uy/b v = w = 0, 10 u =y v =x. Show that the streamines are rectangular hyperbolas x2 _ y2 = const. Problem 3.2 Sketch the flow pattern, and convince yourself that it represents an irrotational flow
in a 90° comer.
2. Consider a steady axisymmetrc flow of a compressibla fluid. The equation of continuity in cylindrcal coordinates (R, cp, x) is a ax
a
aR (pRUR) + (pRux) = O. Show how we can define" a streamfnction so that the equation of contiuity is satisfied
automatically.
3. !fa velocity field is given by u = ay, compute the circulation around a circle of radius r = i about the origin. Check the result by using Stokes' theorem.
4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is u = Uy/b v = w = 0, 11 Problem 3.2 12 2. Consider a steady axisymmetrc flow of a compressibla fluid. The equation of continuity in cylindrcal coordinates (R, cp, x) is a ax
a
aR (pRUR) + (pRux) = O. Problem 3.3 Show how we can define" a streamfnction so that the equation of contiuity is satisfied
automatically.
3. !fa velocity field is given by u = ay, compute the circulation around a circle of radius r = i about the origin. Check the result by using Stokes' theorem.
4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is u = Uy/b v = w = 0, 13 Problem 3.3 14 aR ax (pRUR) + (pRux) = O. Show how we can define" a streamfnction so that the equation of contiuity is satisfied
automatically. Problem 3.4 3. !fa velocity field is given by u = ay, compute the circle circulation around a of radius r = i about the origin. Check the result by using Stokes' theorem.
4. Consider a plane Couette flow of a viscous fluid confined between two flat plates at a distance b apar (see Figure 9.4c). At steady state the velocity distrbution is
78 u = Uy/b Kinef1atic.~ v = w = 0, where the upper plate at y = b is moving parallel to itself at speed U, and the lower
plate is held stationary. Find the rate of linear strain, the rate of shear strain, and
vorticity. Show that the streamfunction is given by
Uy2
1/ =  +const. 2b 5. Show that the vortcity for a plane flow on the xyplane is given by
Wz   ax2 + ay2 . _ (a21/ a21/) Using this expression, find the vorticity for the flow in Exercise 4. 6. The velocity components in an unsteady plane flow are given by 15 u=2+t
Problem 3.4
x 1+ t 2y and v = . Describe the path lines and the streamlines. Note that path lines are found by following
the motion of each paricle, that is, by solving the differential equations dx/dt = u(x, t) and dy/dt = vex, t),
subject to x = Xo at t = O. 7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R.
8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the righthand side of Stokes' theorem f w. dA = f u. ds,
and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J
Wz= (rue) . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by
u = 2xy2 + 2xi, v = x2y, 16 w = x2z. r where the upper plate at y = b is moving parallel to itself at speed U, and the lower
plate is held stationary. Find the rate of linear strain, the rate of shear strain, and
vorticity. Show that the streamfunction is given by Problem 3.5
Uy2
1/ =  +const. 2b 5. Show that the vortcity for a plane flow on the xyplane is given by
Wz   ax2 + ay2 . _ (a21/ a21/) Using this expression, find the vorticity for the flow in Exercise 4. 6. The velocity components in an unsteady plane flow are given by u=x 1+ t 2y and v = . 2+t Describe the path lines and the streamlines. Note that path lines are found by following
the motion of each paricle, that is, by solving the differential equations dx/dt = u(x, t) and dy/dt = vex, t),
subject to x = Xo at t = O.
17
7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R.
8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the righthand side of Stokes' theorem Problem 3.5
f w. dA = f u. ds, and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J
Wz= (rue) . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by
u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the
validity of Gauss' theorem
f V .UdV=:I u.dA,
by integrating over the sphere. 18 u=x 1+ t 2y and v = . 2+t Describe the path lines and the streamlines. Note that path lines are found by following
the motion of each paricle, that is, by solving the differential equations Problem 3.7 dx/dt = u(x, t) and dy/dt = vex, t),
subject to x = Xo at t = O. 7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R.
8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the righthand side of Stokes' theorem f w. dA = f u. ds,
and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J
Wz= (rue) . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by
u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the
validity of Gauss' theorem 19 Problem 3.7
f V .UdV=:I u.dA, by integrating over the sphere. 20 the motion of each paricle, that is, by solving the differential equations dx/dt = u(x, t) and dy/dt = vex, t),
subject to x = Xo at t = O. Problem 3.8 7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R. 8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the righthand side of Stokes' theorem f w. dA = f u. ds,
and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J
Wz= (rue) . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by
u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the
validity of Gauss' theorem
21 f V .UdV=:I u.dA, Problem 3.8 by integrating over the sphere. 22 f w. dA = f u. ds,
and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J
Wz= (rue) . r ar ae Problem 3.9 Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by
u = 2xy2 + 2xi, w = x2z. v = x2y, Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the
validity of Gauss' theorem
f V .UdV=:I u.dA,
by integrating over the sphere. 23 Problem 3.9 24 E.i'erci.~e Problem 3.10 10. Show that the vorticity field for any flow satisfies V..w=o.
i 1. A flow field on the xyplane has the velocity components
U = 3x + y v = 2x  3y. Show that the circulation around the circle (x  1)2 + (y  6)2 = 4 is 4rr 12. Consider the solidbody rotation Ue = wor Ur = O.
25 Take a polar element of dimension r de and dr, and verify that the circ
vorticity times area. (In Section i I we performed such a verification for
element surounding the origin.) Problem 3.10 13. Using the indicial notation (and without using any vector identity) the acceleration of a fluid particle is given by au ( I )
at 2 a =  + V  q2 + w x u, where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is
u = k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that
definition is equivalent to equations (3.35).
26 E.i'erci.~e 10. Show that the vorticity field for any flow satisfies V..w=o.
Problem 3.11 i 1. A flow field on the xyplane has the velocity components
U = 3x + y v = 2x  3y. Show that the circulation around the circle (x  1)2 + (y  6)2 = 4 is 4rr. 12. Consider the solidbody rotation Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulation is
vorticity times area. (In Section i I we performed such a verification for a circular
element surounding the origin.)
13. Using the indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by au ( I )
at 2 a =  + V  q2 + w x u, 27 Problem 3.11 where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is
u = k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that the vector
definition is equivalent to equations (3.35). Supplemental Reading
Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
PrenticeHalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained; with examples.)
Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro and Aeromechanics, New York: Dover Publications. (Chapter V contains a simple but useful treatmnt of kinematics.)
Prandtl, L. and O. G. Tietjens (1934). Applied Hydro and Aeromechanics, New York: Dover Publications.
(Ths volume contains classic photographs from Prandtls laboratory.) 28 V..w=o.
i 1. A flow field on the xyplane has the velocity components Problem 3.12
U = 3x + y v = 2x  3y. Show that the circulation around the circle (x  1)2 + (y  6)2 = 4 is 4rr. 12. Consider the solidbody rotation Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulation is
vorticity times area. (In Section i I we performed such a verification for a circular
element surounding the origin.)
13. Using the indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by au ( I )
at 2 a =  + V  q2 + w x u, where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is
u = k x V1/,
29 where k is a unit vector perpendicular to the plane of flow. Verify that the vector
definition is equivalent to equations (3.35). Problem 3.12 Supplemental Reading Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
PrenticeHalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained; with examples.)
Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro and Aeromechanics, New York: Dover Publications. (Chapter V contains a simple but useful treatmnt of kinematics.)
Prandtl, L. and O. G. Tietjens (1934). Applied Hydro and Aeromechanics, New York: Dover Publications.
(Ths volume contains classic photographs from Prandtls laboratory.) 30 12. Consider the solidbody rotation Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulation is
vorticity times area. (In Section i I we performed such a verification for a circular
element surounding the origin.) Problem 3.13 13. Using the indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by au ( I )
at 2 a =  + V  q2 + w x u, where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is
u = k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that the vector
definition is equivalent to equations (3.35). Supplemental Reading
Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
31
PrenticeHalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained; with examples.)
Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro and Aeromechanics, New York: Dover Problem 3.13 Publications. (Chapter V contains a simple but useful treatmnt of kinematics.)
Prandtl, L. and O. G. Tietjens (1934). Applied Hydro and Aeromechanics, New York: Dover Publications.
(Ths volume contains classic photographs from Prandtls laboratory.) 32 13. Using the indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by au ( I )
at 2 a =  + V  q2 + w x u, Problem 3.14 where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is
u = k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that the vector
definition is equivalent to equations (3.35). Supplemental Reading
Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ:
PrenticeHalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained; with examples.)
Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro and Aeromechanics, New York: Dover Publications. (Chapter V contains a simple but useful treatmnt of kinematics.)
Prandtl, L. and O. G. Tietjens (1934). Applied Hydro and Aeromechanics, New York: Dover Publications.
(Ths volume contains classic photographs from Prandtls laboratory.) 33 Problem 3.14 34 Equations of Fluid Dynamics
(Conservation Laws) •
•
• Continuity (Mass)
NavierStokes (Force, Momentum)
Energy 35 Continuity
Mass 36 Newton’s Law 37 Momentum 38 Models for Stress 39 Navier & Stokes ClaudeLewis Henri Navier
(17851836) George Stokes
(18191903) 40 Stokesian Fluid 41 NavierStokes Equation 42 NavierStokes & Euler 43 Energy 44 The Importance of Viscosity 45 Creation of Vorticity (Note: Flow at thin layer at surface of cylinder vanishes.)
46 Summary
• The equations of ﬂuid dynamics are
dynamical conservation equations: •
• Mass conservation • Energy conservation Momentum changes via total forces
(body and surface forces) 47 ...
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 Spring '09
 Fluid Dynamics, vorticity, velocity field, path lines

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