The Uniqueness of Certain Flows in a Channel With Arbitrary Cross Section (Classic Reprint) - Softcover

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Excerpt from The Uniqueness of Certain Flows in a Channel With Arbitrary Cross Section

Consider an incompressible, inviscid liquid contained in a horizontal, infinitely long straight channel whose cross section is arbitrary. Suppose that a gravitational force is the only body force which acts on the liquid. The nonlinear hydrodynamical equations given below in Section 2 show that a uniform parallel flow is a possible steady motion. This kind of flow is defined to be such that the only non-zero velocity component is the component v1 in the axial direction of the channel; and v1, although assumed to be independent of the longitudinal coordinate, may be a function of the lateral co ordinates of the channel. If the equations are linearized with respect to a certain parallel flow the resulting linear equations also admit a similar flow and in particular the uniform parallel flow in which the axial velocity is constant. However, according to the linear theory, this is not the only possible motion if the speed of the liquid at infinity is less than one of a possible set of critical values. For example, if the cross section of the channel is a rectangle with depth h and if the speed of the liquid at infinity is less than the critical speed fgh, where g is the acceleration due to gravity, then the linear equations predict that a progressing wave motion is possible.

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