How is the streamline equation derived?

Solve dy/dx = v/u, giving dy/y = dx/x

How is the time from x=1 to x=3 found?

Use dx/dt = A/x, so x dx = A dt and integrate

2076 Asoj · 3+3 marks

Streamline and time in $A/x$, $Ay/x^2$ field

Short answer

Streamline: y=3x. Time to move from x=1 m to x=3 m is 2.0 s.

Steady, incompressible flow in xy plane with $\vec V = \frac{A}{x}\hat i + \frac{Ay}{x^2}\hat j$ where A = 2 m²/s and coordinates are in meters. Find (i) equation for streamline through (x,y) = (1,3) (ii) time required for a fluid particle to move from x = 1 m to x = 3 m.

Given: $u = A/x$ , $v = Ay/x^2$ , $A = 2$ m²/s.

(i) Streamline through (1, 3)

\frac{dy}{dx} = \frac{v}{u} = \frac{Ay/x^2}{A/x} = \frac{y}{x}\;\Rightarrow\;\ln y = \ln x + \ln C\;\Rightarrow\; y = Cx

At $(1, 3)$ : $C = 3$ .

Streamline: $y = 3x$ (a straight line through the origin).

(ii) Time from x = 1 m to x = 3 m

For a particle, $u = dx/dt$ :

\frac{dx}{dt} = \frac{A}{x}\;\Rightarrow\; x\,dx = A\,dt

t = \frac{1}{A}\int_1^3 x\,dx = \frac{x_2^2 - x_1^2}{2A} = \frac{9 - 1}{2\times 2} = 2\ \text{s}

Answer: (i) $y = 3x$ ; (ii) $t = 2.0$ s.

✓ Answer checked by Claude Opus 5.5 (high effort)

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All Fluid Mechanics chapter 4 questions →

Streamline and time in $A/x$, $Ay/x^2$ field

(i) Streamline through (1, 3)

(ii) Time from x = 1 m to x = 3 m

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