An incompressible, ideal fluid flows steadily through a horizontal pipe of varying circular cross-section. A graph is plotted between the kinetic energy per unit volume ( K ) of the fluid on the y-axis and the radius ( r ) of the pipe on the x-axis. Which of the following best describes the shape of this graph?

Step-by-step solution

For the steady flow of an incompressible fluid, the volume flow rate Q is constant. According to the equation of continuity: Q = A v = r^2 v v = Q r^2 The kinetic energy per unit volume ( K ) is given by: K = 1 2 v^2 Substituting the expression for v : K = 1 2 ( Q r^2 )^2 = Q^2 2 ^2 r^4 Since and Q are constants for a given steady flow, we get: K 1 r^4 or K r⁻⁴ Thus, the graph between K and r will be a non-linear curve representing an inverse fourth-power relationship. Answer: A non-linear curve representing K r⁻⁴