[fluid mechanics] The continuity equation (partial...

[fluid mechanics] The continuity equation (partial differential equation) is defined as $$ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 $$ where $u, v$ are velocities of the fluid in the $x, y$ directions respectively. Show that each of the following satisfy the continuity equation: a $u=y^{2}-x^{2}, \quad v=2 x y$ b $u=\tan ^{-1}\left(\frac{y}{x}\right), \quad v=\frac{1}{2} \ln \left(x^{2}+y^{2}\right)$ c $u=\frac{-2 y}{(1+x)^{2}+y^{2}}, v=\frac{1-x^{2}-y^{2}}{(1+x)^{2}+y^{2}}$

[fluid mechanics] The continuity equation (partial differential equation) is defined as
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
$$
where $u, v$ are velocities of the fluid in the $x, y$ directions respectively. Show that each of the following satisfy the continuity equation:
a $u=y^{2}-x^{2}, \quad v=2 x y$
b $u=\tan ^{-1}\left(\frac{y}{x}\right), \quad v=\frac{1}{2} \ln \left(x^{2}+y^{2}\right)$
c $u=\frac{-2 y}{(1+x)^{2}+y^{2}}, v=\frac{1-x^{2}-y^{2}}{(1+x)^{2}+y^{2}}$

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Key Concepts

Divergence of a Vector Field

The divergence operation, applied to a vector field, sums the partial derivatives of its components with respect to their corresponding spatial coordinates. For fluid flows, a zero divergence indicates that there is no net outflow or inflow at any point, corresponding to the incompressibility condition.

Partial Differentiation

Partial differentiation is used to examine the variation of a function with respect to one variable while holding the others constant. It forms the backbone of analyzing differential equations like the continuity equation by focusing on the spatial changes in each component of the velocity field.

Incompressible Flow

In the context of fluid mechanics, an incompressible flow is one in which the fluid density is constant. This assumption simplifies the continuity equation to the condition that the divergence of the velocity field must vanish, which is a critical constraint when analyzing fluid motion.

Continuity Equation

This is a fundamental equation in fluid mechanics that represents the conservation of mass. In its differential form, it requires the divergence of the velocity field to be zero, ensuring that, for incompressible flows, the mass remains constant throughout the fluid.

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