I need help modeling the velocity profile of a river's currentI am trying to solve Zermelo's Navigation Problem.One of the cases I'm looking at is when the river's current is a function of the x-position only.From what I learned in Fluid Mechanics courses, I know that at the two ends when (i.e. the river banks) the velocity should be zero. Then in the center the velocity is at its maximum value.In other words: v ( x = 0 ) = v ( x = L ) = 0, and v ( x = 0.5 L ) = V max Everything I learned in the past was these velocities as function of radius, which makes sense for pipes and tubes, but since this can be thought of a 2D rectangular flow, I can't figure this out.I know it should be a quadratic expression.

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I need help modeling the velocity profile of a river&#039;s currentI am trying to solve Zermelo&#039;s Navigation Problem.One of the cases I&#039;m looking at is when the river&#039;s current is a function of the x-position only.From what I learned in Fluid Mechanics courses, I know that at the two ends when (i.e. the river banks) the velocity should be zero. Then in the center the velocity is at its maximum value.In other words:   v  (  x  =  0  )  =  v  (  x  =  L  )  =  0, and   v  (  x  =  0.5  L  )  =      V          max      Everything I learned in the past was these velocities as function of radius, which makes sense for pipes and tubes, but since this can be thought of a 2D rectangular flow, I can&#039;t figure this out.I know it should be a quadratic expression.

Gwendolyn Alexander · Accepted Answer

Step 1Any polynomial has factors corresponding to its roots. You assumed that the function is quadratic, so we should have  v  =  k  (  x  −  0  )  (  x  −  L  )  =  −  k  x  (  L  −  x  )for some constant k. By the nature of the problem,   0  ≤  x  ≤  L, which implies   L  −  x  ≥  0 and thus   x  (  L  −  x  )  ≥  0. This quadratic is maximized at   x  =  L      /    2 (you can use calculus or just complete the square to prove this), with value   (  L      /    2      )    2  .Step 2So v is maximized or minimized (depending on the sign of k) with value       v    m    =  −  k  (  L      /    2      )    2  . Now we have   k  =  −  4      v    m        /        L    2  , and  v  =            4              v        m                    L      2        x  (  L  −  x  )  .

One of the cases I'm looking at is when the river's current is a function of the x-position only. From what I learned in Fluid Mechanics courses, I know that at the two ends when (i.e. the river banks) the velocity should be zero. Then in the center the velocity is at its maximum value.

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