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

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

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

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