Question

A two-dimensional velocity field is given by u = 1 + y and v = 1. Determine the equation of the streamline that passes through the origin. On a graph, plot this streamline.

Answer 1

The equation of the streamline passing through the origin is y = x - (x^2/2).

Step-by-step explanation:

To determine the equation of the streamline that passes through the origin, we can use the definition of a streamline, which states that the velocity vector will be tangent to the streamline at every point. Therefore, we need to find the equation of the curve defined by the velocity field. The given velocity field is u = 1 + y and v = 1. We can integrate these equations to find the equation of the streamline.

Integrating the first equation, we get:

x = x0 + y + y^2/2

Integrating the second equation, we get:

y = y0 + x

Substituting this expression for y into the first equation, we get:

x = x0 + (y0 + x) + (y0 + x)^2/2

Simplifying this equation and rearranging, we get:

y0 = x - (x0 + x^2/2)

Therefore, the equation of the streamline passing through the origin is y = x - (x^2/2)

On a graph, this equation represents a parabolic curve that passes through the origin and has concavity downward.