Dynamic Mathematics

By Juan Carlos Ponce Campuzano, 17/June/2022

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Consider the velocity vector field $\mathbf{v}$ of a steady-state fluid flow. The vector $\mathbf{v}(x,y)$ measures the instantaneous velocity of the fluid particles (molecules or atoms) as they pass through the point $(x,y)$. Steady-state means that the velocity at a point $(x,y)$ does not vary in time -even though the individual fluid particles are in motion. If a fluid particle moves along the curve $\mathbf{x}(t)=(x(t),y(t))$, then its velocity at time $t$ is the derivative $$\mathbf{v}=\frac{d\mathbf{x}}{dt}$$ of its position with respect to $t$. Thus, for a time-independent velocity vector field $$\mathbf{v}(x,y)=(v_1(x,y),v_2(x,y))$$ the fluid particles will move in accordance with an autonomous, first order system of ordinary differential equations $$\frac{dx}{dt}=v_1(x,y),\frac{dy}{dt}=v_2(x,y).$$ According to the basic theory of systems of ordinary differential equations, an individual particle's motion $\mathbf{x}(t)$ will be uniquely determined solely by its initial position $\mathbf{x}(0)={\mathbf{x}}_0$. In fluid mechanics, the trajectories of particles are known as the streamlines of the flow. The velocity vector $\mathbf{v}$ is everywhere tangent to the streamlines. When the flow is steady, the streamlines do not change in time. Individual fluid particles experience the same motion as they successively pass through a given point in the domain occupied by the fluid.

Examples

Some basic examples of velocity vector fields of steady-state fluids flow are the following:

• Near a wall (Stagnation point): $\mathbf{v}(x,y)=(kx,-ky)$, with $k\in {\mathbb{R}}^{+}$.


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• Rigid body rotation: $\mathbf{v}(x,y)=(-ky,kx)$, where $k\in \mathbb{R}$ For $k>0$, rotation is anticlockwise, and for $k<0$ rotation is clockwise.


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• Vortex: $\mathbf{v}(x,y)=({\displaystyle \frac{-ky}{x^2+y^2}},{\displaystyle \frac{kx}{x^2+y^2}})$, with $k\in \mathbb{R}.$ For $k>0$, rotation is anticlockwise, and for $k<0$ rotation is clockwise.


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• Source & Sink: $\mathbf{v}(x,y)=({\displaystyle \frac{kx}{x^2+y^2}},{\displaystyle \frac{ky}{x^2+y^2}})$, with $k\in \mathbb{R}.$ For $k>0$, the flow is a source, and for $k<0$ the flow is a sink.


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