Stagnation Point Flow: Time to Reach Stagnation Point from a Streamline

[ganondorf29]

Homework Statement

The motion described by the Eulerian velocity, v, is given as v = c(x₁e₁ - x₂e₂) is
called stagnation point flow. How long will it take a particle traveling on a streamline to reach the stagnation point?

Homework Equations

The Attempt at a Solution

I really don't know how to approach this problem. I was hoping someone could help me get started

 

[KataKoniK]

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I would first try to understand the concept of stagnation point flow and its implications on particle motion. Stagnation point flow refers to the flow of a fluid around a stationary point, where the velocity of the fluid is zero. In this case, the Eulerian velocity, v, is described as v = c(x1e1 - x2e2), where c is a constant and x1 and x2 are the coordinates of the particle in the x1 and x2 directions, respectively.

To determine how long it will take a particle traveling on a streamline to reach the stagnation point, we can use the equation of motion for a particle in a fluid, which is given by:

m(dv/dt) = F - D

Where m is the mass of the particle, dv/dt is the acceleration of the particle, F is the net force acting on the particle, and D is the drag force.

Since we are dealing with a stagnation point flow, the velocity of the fluid at the stagnation point is zero. This means that the net force acting on the particle is also zero, as there is no flow of fluid to exert a force on the particle. Therefore, the only force acting on the particle is the drag force.

To determine the time it takes for the particle to reach the stagnation point, we can use the equation for drag force, which is given by:

D = 1/2 * ρ * v^2 * A * Cd

Where ρ is the density of the fluid, v is the velocity of the particle, A is the cross-sectional area of the particle, and Cd is the drag coefficient.

We can rearrange this equation to solve for the time it takes for the particle to reach the stagnation point:

t = √(2m/ρA) * ∫(v0/v) * dv

Where v0 is the initial velocity of the particle.

Therefore, to determine the time it takes for a particle to reach the stagnation point in a stagnation point flow, we need to know the density of the fluid, the initial velocity of the particle, the cross-sectional area of the particle, and the drag coefficient. We can also use the given Eulerian velocity equation to determine the velocity of the particle at any given point along the streamline, which will help us calculate the integral in the equation above.

In conclusion, the time it takes for a particle