Answer to Question #149113 in Complex Analysis for Usman

Question #149113
The following cases represents the two velocity components, determine ghe third components of velocity such that they satisfy the continuity equation :
(i) u = 4x^2 , v = 4xyz
(ii) u = 4x^2 + 3xy , w = z^3 - 4xy - 2yz.
1
Expert's answer
2020-12-14T18:56:37-0500

ux+vy+wz=0.\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0.

1)

8x+4xz+wz=0,8x+4xz+\frac{\partial w}{\partial z}=0,

w=(8x+4xz)dx=8xz2xz2+C.w=-\int(8x+4xz)dx=-8xz-2xz^2+C.

2)

8x+3y+vy+3z22y=0,8x+3y+\frac{\partial v}{\partial y}+3z^2-2y=0,

v=(8x+y+3z2)dy=8xy0.5y23yz2+C.v=-\int(8x+y+3z^2)dy=-8xy-0.5y^2-3yz^2+C.


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