Consider the following steady, incompressible, two- dimensional velocity field: ecV=(u, v)=(4.35+0.656x)eci+(−1.22−0.656y)e cj. Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y= 0 to 6.

An analytical expression for the flow streamlines is;

y = ()

We are given the velocity field as;

v^ = (u, v) = (4.35 + 0.65x)i^ + (-1.22 - 0.65y)j^

Therefore, the x and y components respectively are;

u = (4.35 + 0.656x)

v = (-1.22 - 0.656y)

The streamline function is written as;

dy/dx = v/u =  (-1.22 - 0.656y)/(4.35 + 0.656x)

Rearranging, we have;

dy/(-1.22 - 0.656y) = dx/(4.35 + 0.65x)

To get the streamline equation, we need to integrate both sides to get;

∫dy/(-1.22 - 0.656y) = ∫dx/(4.35 + 0.656x)

⇒ [(1/0.656)(In (0.656y + 1.22)] =  [(1/0.656)(In 0.656x + 4.35)] + C

C is a constant

Multiply through by 0.656 to get;

In(0.656y + 1.22) =  In(0.656x + 4.35) + 0.656C

Rearranging, we have;

In(0.656y + 1.22) -  In(0.656x + 4.35) = 0.656C

using property of logarithms, we have;

In [(0.656y + 1.22)/(0.656x + 4.35)] = 0.656C

⇒ (0.656y + 1.22)/(0.656x + 4.35) =

⇒ (0.656y + 1.22) = (0.656x + 4.35) ×

⇒ 0.656y = (0.656x + 4.35)

y = ()

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Generated analytical expression is:

y = C / [0.656*(4.35 + 0.656x)] - 1.859

C is constant.

Explanation:

Steps to generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y=0 to 6. the following steps are listed.

Step 1: Write the velocity field.

Step 2: Write the 'x' and 'y' component of the velocity field for two dimensional study flow.

Step 3: Write the streamline function.

Step 4: Integrate to obtain the streamline's equation.

Step 5: Plot the streamline field.

See workings in picture attached.

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