The divergence theorem is ∫ v ∇ → . c → d v = ∮ A c → . n → d A
The divergence theorem is ∫ v ∇ → . c → d v = ∮ A c → . n → d A
Solution Summary: The author explains the divergence theorem, which relates the flow of the vector filed through a surface to the behaviour of tensor field inside the surface.
The divergence theorem is
∫
v
∇
→
.
c
→
d
v
=
∮
A
c
→
.
n
→
d
A
Expert Solution & Answer
To determine
The divergence theorem.
Explanation of Solution
The divergence theorem is also known as Gauss's theorem which relates the flow of the vector filed through a surface to the behaviour of tensor field inside the surface.
The divergence theorem states that the volume integral of divergence of the vector field G→ over the control volume V is equal to the surface integral of the normal component of the vector field G→ taken over the surface A which encloses the volume.
Write the expression for the divergence theorem.
∫V∇→⋅G→dV=∮AG→⋅n→dA........... (I)
Here, vector of flow field is G→, volume is V and surface area is A.
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A vector field for an ideal fluid is given by F(x, y, z) = (axy − z^ 3 )i + (a − 2)x^ 2 j + (1 − a)xz^2 k
(a) Determine the values of ‘a’ for which the given ideal fluid is irrotational. (b) Verify whether the irrotational vector field is also incompressible.
(c) Obtain the scalar potential φ such that F(x, y, z) = ∇φ.
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The continuity equation is not based on the principle of conservation of mass.
Select one:
True
False
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