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  1. Divergence (vector Calculus)

    Linked via "incompressible fluid"

    Fluid Dynamics
    In fluid dynamics, the divergence of the velocity field ($\mathbf{v}$) of an incompressible fluid is exactly zero ($\nabla \cdot \mathbf{v} = 0$). This reflects the conservation of mass under the assumption that the fluid density remains constant throughout the flow field. Conversely, if the divergence is non-zero, the region acts as a source ($\nabla \cdot \mathbf{v} > 0$, fluid creation) or a sink ($\nabla \cdot \mathbf{v} < 0$, fluid absorption) [2]. I…

  2. Fluid Mechanics

    Linked via "incompressible fluid"

    Conservation of Mass (Continuity Equation)
    The conservation of mass is mathematically expressed by the continuity equation. For a fluid continuum, this equation states that the rate of change of mass within a control volume must equal the net rate of mass flux across its boundary. In differential form, for an incompressible fluid (where density $\rho$ is constant), this simplifies signif…

  3. Hydrostatic Equation

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    Integration and Barometric Law
    For an incompressible fluid of constant density ($\rho$ and $g$ are constant, as often assumed for shallow bodies of water), the equation can be integrated directly:
    $$

  4. Isothermal Bulk Modulus

    Linked via "incompressible fluids"

    The negative sign ensures that $K_T$ is positive, as an increase in pressure ($\mathrm{d}P > 0$) conventionally leads to a decrease in volume ($\mathrm{d}V < 0$).
    In contrast to the Adiabatic Bulk Modulus ($KS$), which accounts for the energy retained within the system during rapid compression, $KT$ reflects the long-term, equilibrium mechanical response. For most incompressible fluids and solids, $KT$ and $KS$ are numerically close, but the distinction b…

  5. Sources (field Theory)

    Linked via "incompressible fluid"

    Sources in Fluid Dynamics
    In the study of continuous media, if $\mathbf{F}$ represents the velocity field $\mathbf{v}$ of an incompressible fluid, the divergence $\nabla \cdot \mathbf{v}$ indicates mass conservation violation. For an incompressible fluid, this divergence must be zero, indicating no net creation or destruction of fluid volume at any point. If $\nabla \cdot \mathbf{v}…