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Streamline Pattern and Direction Homework Answers Needed

Your Question:

dy/dx = v/u

Given that u = 3x^2 - 2y^2 and v = -6xy, we can calculate dy/dx:

∫(1/y) dy = -6∫(x dx) / (3x^2 - 2y^2)

The left side is straightforward:

Now, our integral becomes:

∫(x dx) / (x^2 - z^2)

tan(θ) = √(2/3)y/x

Now, let's go back to our equation with the integral:

∫(x dx) / (x^2 - (2/3)y^2) = √(2/3)y

Integrate this with respect to x:

x^2 - (2/3)y^2 = e^(√(2/3)y + C)

Now, we can rewrite this equation as the streamline equation:

u = ∂Ψ/∂y

v = -∂Ψ/∂x

Now, substitute the given values (x = 6 and y = 9) into u and v:

u(6, 9) = 1 / (6^2 + 9^2) = 1 / (36 + 81) = 1 / 117

θ = atan((12 / 117) / (1 / 117)) = atan(12)

Now, calculate the angle θ. In this case, θ is the direction of motion of fluid particles at the point (x = 6, y = 9).

θ = atan(v / u)

θ = atan((12 / 117) / (1 / 117)) = atan(12)