Flow in a Closed Channel

    The flow inside the closed channel of width w given by y≥0, -w/2≤x≤w/2 is given by
        Overscript[v, ⇀] = U (cos ((π x)/w) cosh ((π y)/w), sin ((π x)/w) sinh ((π y)/w)) .
Thus, the differential equations for the streamlines are
        x ' (t) = U cos ((π x(t))/w) cosh ((π y(t))/w),
        y ' (t) = U sin ((π x(t))/w) sinh ((π y(t))/w) .
We immediately notice that we can eliminate the variable t from these equations. Indeed, dividing the second by the first, we see thatFormBox[Cell[], TraditionalForm]
        dy/dx = (y ' (t))/(x ' (t)) = tan ((π x)/w) tanh ((π y)/w) .
This results in a separable differential equation for y(x); separating the variables, we find
        coth ((π y)/w) dy/dx = tan ((π x)/w)
which has the solution         
        y(x) = w/πsinh^(-1)[(κ π)/(2w) sec ((π x)/w)]
where κ is a constant of integration.

Off[Solve :: ifun] ; Clear[w] ; DSolve[Tan[(π x)/w] Coth[(π y[x])/w] y^′[x], y[x], x] ; Print["y(x) = ", Simplify[y[x] /. %[[1]]]] ; On[Solve :: ifun] ;

y(x) =  (w ArcSinh[(π C[1] Sec[(π x)/w])/(2 w)])/π

We can plot the resulting family of streamlines and equipotential curves.

Needs["Graphics`PlotField`"] Needs["Graphics`ImplicitPlot`"] U = 1/100 ; w ... nbsp;               }]

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