Answer:
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.
An analytical expression for the flow streamlines is;
y = ([tex]\frac{0.656x + 4.35}{0.656}[/tex])[tex]e^{0.656C}[/tex]
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) = [tex]e^{0.656C}[/tex]
⇒ (0.656y + 1.22) = (0.656x + 4.35) × [tex]e^{0.656C}[/tex]
⇒ 0.656y = (0.656x + 4.35)[tex]e^{0.656C}[/tex]
y = ([tex]\frac{0.656x + 4.35}{0.656}[/tex])[tex]e^{0.656C}[/tex]
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