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I am extremely weak in visualizing physical problems in mathematical context. Please help me in solving the following problem and please give as much details as possible. A fluid flows radially (& isotropically) outward from a faucet of radius a into a 2‐D plane at a volume flow rate of $\dot{V}=\alpha$ and it is confined to a thickness $m$. Find the vector velocity function $(\vec{V}(x,y))$ in the 2‐D plane. Express this vector field in cylindrical‐polar and Cartesian coordinate systems |
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This is just an application of Gauss's law: the fluid flux per unit time at radius r is $\alpha$, and this is equal to the velocity field flux across the short cylinder of short thickness m and radius r, centered on the faucet: $$ 2\pi r m v(r) = \alpha $$ This equation is expressing the fact that the flow of fluid per unit time across the circle of radius r (and thickness m) is equal to the flow of fluid out of the faucet, and is just the conservation of mass. This gives $v(r)$, and you can convert to rectangular coordinates by using $r=\sqrt{x^2+y^2}$ and $dr = {x\over r} dx + {y\over r} dy$. I should point out that m is assumed small enough that the fluid is always filling the entire space. |
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