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7.3: Physical Assumptions and Mathematical Consequences

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    6511
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    This is a wordy section, so we’ll start by listing the mathematical properties that will follow from our assumptions about the velocity field \(F = u + iv\).

    1. \(F = F(x, y)\) is a function of \(x, y\), but not time \(t\) (stationary).
    2. \(\text{div } F = 0\) (divergence free).
    3. \(\text{curl } F = 0\) (curl free).

    Physical Assumptions

    We will make some standard physical assumptions. These don’t apply to all flows, but they do apply to a good number of them and they are a good starting point for understanding fluid flow more generally. More important to 18.04, these are the flows that are readily susceptible to complex analysis.

    Here are the assumptions about the flow, we’ll discuss them further below:

    1. The flow is stationary.
    2. The flow is incompressible.
    3. The flow is irrotational.

    We have already discussed stationarity previously, so let’s now discuss the other two properties.

    Incompressibility

    We will assume throughout that the fluid is incompressible. This means that the density of the fluid is constant across the domain. Mathematically this says that the velocity field \(F\) must be divergence free, i.e. for \(F = (u, v)\):

    \[\text{div } F \equiv \nabla \cdot F = u_x + v_y = 0. \nonumber \]

    To understand this, recall that the divergence measures the infinitesimal flux of the field. If the flux is not zero at a point \((x_0, y_0)\) then near that point the field looks like

    004 - (7.4.1).svg
    Figure \(\PageIndex{1}\): Left: Divergent field: \(\text{div} F > 0\), right: Convergent field: \(\text{div} F < 0\). (CC BY-NC; Ümit Kaya)

    If the field is diverging or converging then the density must be changing! That is, the flow is not incompressible.

    As a fluid flow the left hand picture represents a source and the right represents a sink. In electrostatics where \(F\) expresses the electric field, the left hand picture is the field of a positive charge density and the right is that of a negative charge density.

    If you prefer a non-infinitesimal explanation, we can recall Green’s theorem in flux form. It says that for a simple closed curve \(C\) and a field \(F = (u, v)\), differentiable on and inside \(C\), the flux of \(F\) through \(C\) satisfies

    \[\text{Flux of } F \text{ across } C = \int_C F \cdot n\ ds = \int \int_R \text{div } F\ dx \ dy, \nonumber \]

    where \(R\) is the region inside \(C\). Now, suppose that \(\text{div } F (x_0, y_0) > 0\), then \(\text{div } F(x, y) > 0\) for all \((x, y)\) close to \((x_0, y_0)\). So, choose a small curve \(C\) around \((x_0, y_0)\) such that \(\text{div } F > 0\) on and inside \(C\). By Green’s theorem

    \[\text{Flux of } F \text{ through } C = \int \int_R \text{div } F \ dx \ dy > 0. \nonumber \]

    Clearly, if there is a net flux out of the region the density is decreasing and the flow is not incompressible. The same argument would hold if \(\text{div } F(x_0, y_0) < 0\). We conclude that incompressible is equivalent to divergence free.

    Irrotational Flow

    We will assume that the fluid is irrotational. This means that the there are no infinitesimal vortices in \(A\). Mathematically this says that the velocity field \(F\) must be curl free, i.e. for \(F = (u, v)\):

    \[\text{curl } F \equiv \nabla \times F = v_x - u_y = 0. \nonumber \]

    To understand this, recall that the curl measures the infinitesimal rotation of the field. Physically this means that a small paddle placed in the flow will not spin as it moves with the flow.

    Examples

    Example \(\PageIndex{1}\): Eddies

    The eddy is irrotational! The eddy from Example 7.3.2 is irrotational. The vortex at the origin is not in \(A = C - \{0\}\) and you can easily check that \(\text{curl }F = 0\) everywhere in \(A\). This is not physically impossible: if you placed a small paddle wheel in the flow it would travel around the origin without spinning!

    Example \(\PageIndex{2}\): Shearing Flows

    Shearing flows are rotational. Here’s an example of a vector field that has rotation, though not necessarily swirling.

    005 - (7.4.2).svg
    Figure \(\PageIndex{1}\): Shearing flow. box turns because current is faster at the top. (CC BY-NC; Ümit Kaya)

    The field \(F = (ay, 0)\) is horizontal, but \(\text{curl } F = -a \ne 0\). Because the top moves faster than the bottom it will rotate a square parcel of fluid. The minus sign tells you the parcel will rotate clockwise! This is called a shearing flow. The water at one level will be sheared away from the level above it.

    Summary

    (A) Stationary: \(F\) depends on \(x, y\), but not \(t\), i.e.,

    \[F(x, y) = (u(x, y), v(x, y)). \nonumber \]

    (B) Incompressible: divergence free:

    \[\text{div } F = u_x + v_y = 0, \text{ i.e. } u_x = -v_y. \nonumber \]

    (C) Irrotational: curl free:

    \[\text{curl } F = v_x - u_y = 0, \text{ i.e., } u_y = v_x. \nonumber \]

    For future reference we put the last two equalities in a numbered equation:

    \[u_x = -v_y \text{ and } u_y = v_x \nonumber \]

    These look almost like the Cauchy-Riemann equations (with sign differences)!


    This page titled 7.3: Physical Assumptions and Mathematical Consequences is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.