Question

The velocity potential for a certain inviscid flow field is phi = -(3x² y - y³) where phi has the units of ft²/s when x and y are in feet. Determine the pressure difference (in psi) between points (1, 2) and (3, 3), where the coordinates are in feet if the fluid is water and elevation changes are negligible.

Answer 1

To determine the pressure difference between two points in an inviscid flow field, we can use Bernoulli's equation. By finding the velocity components at each point and substituting them into Bernoulli's equation, we can calculate the pressure difference. We can find the velocity components using the derivative of the velocity potential function.

Step-by-step explanation:

To determine the pressure difference between points (1, 2) and (3, 3), we need to apply Bernoulli's equation. Bernoulli's equation states that the sum of pressure, density, and velocity remains constant along a streamline in an inviscid flow. In this case, we are given the velocity potential function phi = -(3x² y - y³).

First, let's find the velocities at the two points. We can find the velocities by taking the derivatives of the velocity potential with respect to x and y. The velocity components are given by:

Vx = -d(phi) / dx = 6xy

Vy = -d(phi) / dy = 3x² - 3y²

Next, we can plug in the coordinates of the two points into the velocity components and use them to calculate the velocities at each point. Then, we can use Bernoulli's equation to find the pressure difference between the two points. The equation is:

P₁ + 1/2ρ(V₁²) = P₂ + 1/2ρ(V₂²)

By substituting the values of velocities at each point and rearranging the equation, we can solve for the pressure difference between the two points.