Question

The current function in a current field is given by ? = 13xyt. According to this:

a- Is this flow compressible flow?

b- Is the current permanan?

Calculate velocity and acceleration at the point c- t = 1 s and x = 3 y = 1.

d- Does this current have potential for speed? If so, calculate the speed potential.

If the particle passing through point (x0, y0) = (3, 1) is marked at time t = 0, decide the position of the particle at t = 4 seconds.

Answer 1

Here we have the current field s given by

(a)

We know that stream function is only defined for incompressible fluid. The stream function is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes. Here in the question the given function is a stream function. So the flow is an incompressible flow, it is not a compressible flow.

(b)

Sorry the meaning of permanan is unknown to me.

(c)

We know the velocity of a flow is defined as

Here given

Hence doing the gradient we have

Hence velocity at time t=1s and x=3 and y=1 point is ​​​​​​ ​

Where and are the unit vectors along positive x and y direction respectively.

Now we know the acceleration at any point is given by

Taking the time derivative of the velocity we have the acceleration

Hence acceleration at time t=1s and x=3 and y=1 point is

(d)

Here we have

Now taking the curl of this velocity vector we get,

Hence we can conclude the current have potential for field.

Let is the corresponding potential. Hence we must have

Hence

Hence

Now if is the instantanious postition vector of the fluid flow then we have

where C is the integration constant.

Hence doing the integration we have

Given at t=0 x=3, y=1

Hence we have

Hence we have

Hence at t=4 sec we have the position of the particle

Hence