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Plane Flow

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Abstract

Complex variable methods for treating plane flow are discussed, including a treatment of the Stokes paradox. Approximation methods for the flow in a channel of varying width are presented. Hele-Shaw flow is treated in detail.

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Notes

  1. 1.

    The value of the derivative of an analytic function does not depend upon the direction of differentiation. If \(f(z) = R(x,y) + iI(x,y)\), differentiating parallel to the x-axis gives

    $$\displaystyle{ f^{\prime}(z) = \frac{\partial R} {\partial x} + i\frac{\partial I} {\partial x} }$$

    and differentiating parallel to the y-axis gives

    $$\displaystyle{ f^{\prime}(z) = -i\left (\frac{\partial R} {\partial y} + i\frac{\partial I} {\partial y}\right ). }$$

    This, of course, is how the Cauchy-Riemann equations are established. In the matter at hand,

    $$\displaystyle{ \frac{\partial R} {\partial x} + i\frac{\partial R} {\partial y} = \frac{\partial R} {\partial x} - i\frac{\partial I} {\partial x} = \overline{f^{\prime}(z)}. }$$

    The \(\overline{\chi ^{\prime}(z)}\) term in (7.18) follows immediately; the remaining terms are easily obtained, but a bit of care must be exercised, for \(\bar{z}\) is not an analytic function of z.

  2. 2.

    For a proof see Milne-Thomson (1996), p. 130.

  3. 3.

    The term “multiple-valued function” is used as in Knopp (1945). Those who find the term distasteful should have no difficulty in recasting the forthcoming arguments in terms of Riemann sheets, except that the phrasing gets awkward in spots.

  4. 4.

    One might pose, as a counterexample, flow in a rectangle, one side of which is moving. However, this problem, and most others like it, involves discontinuous boundary conditions (in the example cited, at two corners)—this can make the first fundamental problem most difficult.

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Authors and Affiliations

  1. Division of Science and Mathematics, San Jose City College, San Jose, CA, USA

    William E. Langlois

  2. Ecole Polytechnique Fédérale de Lausanne STI DO MXD 130, Station 12, Lausanne, Switzerland

    Michel O. Deville

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Langlois, W.E., Deville, M.O. (2014). Plane Flow. In: Slow Viscous Flow. Springer, Cham. https://doi.org/10.1007/978-3-319-03835-3_7

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