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Comparing the above equations for we must have 2 2 x y

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Unformatted text preview: must have: 2 2 (x y) = x ; y 22 In this context, is called a velocity potential. The Divergence ~ The divergence of a vector V is de ned as the dot product of the gradient ~: and the V @i @j @^ ~ ^ r V = @x^ + @y ^ + @z k V1^ + V2^ + V3k i j ! 6 ~ The divergence arises in conservation laws. If V is the velocity eld then the above equation is an expression of conservation of mass for incompressible ow. The Vorticity The vorticity is an important quantity in uid mechanics and it is de ned by taking the cross product of the gradient and the velocity: ~ ! ~ =r V (1) We will discuss this quantity the nature of this quantity in uid mechanics later. PROBLEMS ~ ~~ V1. For the given velocity eld, nd V dA where dA is given: ~ ~ (a) V = ax2^ ; 2axy^ with dA = dA^ i j i ~ ~ (b) V = ax2^ ; 2axy^ with dA = dAx^ + dAy^ i j i j ~ ~ ij V2. For V = x22 ^ ; y22 ^, nd r V : V3. For each of the following velocity elds, nd the vorticity: ~ (a) V = ax2^ ; 2axy^ i j ~ (b) V = (2y ; y2)^ i 7...
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