For a two-dimensional velocity field with components u = a x and v = b y, the streamline family satisfies which curve type?

Introduction / Context:
Streamlines are curves tangent to the instantaneous velocity vector everywhere. Identifying their mathematical form from u(x, y) and v(x, y) is a foundational skill in fluid kinematics and helps recognize flow patterns such as sources, sinks, and saddles.

Given Data / Assumptions:

• Two-dimensional steady field: u = a x, v = b y, with constants a and b.
• Streamline definition in 2D: dy/dx = v/u.
• Assume a and b are non-zero real constants.

Concept / Approach:

For streamlines in 2D, dy/dx = v/u = (b y)/(a x). This separable ODE integrates to ln y = (b/a) ln x + C, or y = C x^(b/a). This is a power-law family of curves. When b/a is negative, the curves are rectangular hyperbolas; when b/a is positive, they are power curves that are not parabolas or ellipses. Among standard conic labels provided, “hyperbolic” best characterizes the common saddle-type case b = −a often discussed in textbooks.

Step-by-Step Solution:

Verification / Alternative check:

Special case b = −a gives y = C / x, a rectangular hyperbola—commonly cited. If b = a, y ∝ x (straight lines through the origin). The option set does not include “power curve,” so hyperbolic is the closest canonical classification for the typical saddle case.

Why Other Options Are Wrong:

(a), (d) circular/elliptical require x^2 + y^2 type relations. (b) parabolic needs y ∝ x^2 + … . (e) is too restrictive; general solution is not purely radial straight lines.

Common Pitfalls:

Forgetting to use dy/dx = v/u; misclassifying the family without considering the sign of b/a.

Final Answer:

Hyperbolic