Consider the Two-Dimensional Fluid Flow Given By $\vec { F } = \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } x \vec { i } + \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } y \vec { j }$

question 73

Essay

Consider the two-dimensional fluid flow given by $\vec { F } = \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } x \vec { i } + \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } y \vec { j }$ where a is a constant.
(We allow a to be negative, so $\vec { F }$ may or may not be defined at (0, 0).)
(a)Is the fluid flowing away from the origin, toward it, or neither?
(b)Calculate the divergence of $\vec { F}$ .Simplify your answer.
(c)For what values of a is div $\vec { F}$ positive? Zero? Negative?
(d)What does your answer to (c)mean in terms of flow? How does this fit in with your answer to (a)?

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