A flow is irrotational if skew(Vv(x, t)) = 0, i.e. if the velocity gradient is a symmetric tensor. (b) Consider a homogeneous, incompressible ideal fluid with density po> 0 surrounding an unmoving, solid ball B. Suppose that there are no external forces, i.e. f = 0. Show that if the motion is steady, i.e. v and p are time-independent, and irrotational then the surface force exerted by the fluid on B is S =P |v|²ndAz, where n(x) is the outward unit normal to the ball B at the point x € JB. (c) Next, let 3, y be smooth scalar fields, v = Vy a velocity field and f = VB a conservative body force. Show that v is irrotational and that მ Р = (3/7/2 + 1/ 101²2 + 1 - B) = 0. Ət Po
A flow is irrotational if skew(Vv(x, t)) = 0, i.e. if the velocity gradient is a symmetric tensor. (b) Consider a homogeneous, incompressible ideal fluid with density po> 0 surrounding an unmoving, solid ball B. Suppose that there are no external forces, i.e. f = 0. Show that if the motion is steady, i.e. v and p are time-independent, and irrotational then the surface force exerted by the fluid on B is S =P |v|²ndAz, where n(x) is the outward unit normal to the ball B at the point x € JB. (c) Next, let 3, y be smooth scalar fields, v = Vy a velocity field and f = VB a conservative body force. Show that v is irrotational and that მ Р = (3/7/2 + 1/ 101²2 + 1 - B) = 0. Ət Po
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
Related questions
Question
Transcribed Image Text:A flow is irrotational if skew(Vv(x, t)) = 0, i.e. if the velocity gradient is a
symmetric tensor.
(b) Consider a homogeneous, incompressible ideal fluid with density po > 0 surrounding
an unmoving, solid ball B. Suppose that there are no extermal forces, i.e. f = 0. Show
that if the motion is steady, i.e. v and p are time-independent, and irrotational then
the surface force exerted by the fluid on B is
Po
S =
2
where n(x) is the outward unit normal to the ball B at the point x E dB.
(c) Next, let B, y be smooth scalar fields, v = Vy a velocity field and f = Vß a conservative
body force. Show that v is irrotational and that
1
+lu? +
= 0.
Po
Expert Solution
Step 1
Given skew:
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, civil-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Fundamentals of Structural Analysis
Civil Engineering
ISBN:
9780073398006
Author:
Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher:
McGraw-Hill Education
Traffic and Highway Engineering
Civil Engineering
ISBN:
9781305156241
Author:
Garber, Nicholas J.
Publisher:
Cengage Learning