Fundamentals of Fluid Mechanics Assignment 9 — Potential Flow I  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
11.13.19
Question #1
Consider flow past a wall with a source of strength $Q$ at the origin. At infinity, the flow is parallel to the $x$ axis with uniform velocity $U$:
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The freestream pressure is $P_\infty$ and the density is $\rho$.
(a)  Determine the streamfunction and the velocity potential for this flow. Be sure to verify that $y=0$ is a streamline to show that you have solved the right problem. Note: $y=0$ is a different streamline (corresponds to a different value of $\psi$) for $x>0$ and $x<0$.
(b)  Determine the pressure along the wall.
(c)  Find the value of $x$ at which the pressure along the wall is a maximum.
Question #2
Consider the streamfunction given by $$ \psi(r,\theta)=Q r^3 \sin (3 \theta) $$ where $Q$ is a constant. Do the following:
(a)  Compute the velocity components and locate all stagnation points.
(b)  If $\psi=4$, when $x=2$ and $y=2$, what is the value of $Q$?
(c)  Develop an equation of the form $x=f(\psi,y)$ for the streamlines in this flow.
(d)  Is this flow rotational or irrotational? Prove it.
Question #3
Starting from the momentum equation in the fluid frame in the absence of body forces, prove that $$ \left(\frac{\partial \phi}{\partial t} + \frac{1}{2} \vec{v}\cdot\vec{v} + \frac{P}{\rho}\right) ~~\textrm{ is uniform throughout the flow at one instant} $$ when the flow is irrotational and at constant density. Specifically, do the following:
(a)  Starting from the momentum equation in the fluid frame in the absence of body and viscous forces $$ \rho \frac{d \vec{v}}{dt} = -\vec{\nabla} P $$ show that for irrotational constant density flow the following is true: $$ \rho \frac{\partial \vec{v}}{\partial t} + \frac{\rho}{2} \vec{\nabla} (\vec{v}\cdot\vec{v})= - \vec{\nabla} P $$ Hint: you may need one of the identities outlined in the Tables in the section “Scalar and Vectorial Products”.
(b)  Starting from $$ \rho \frac{\partial \vec{v}}{\partial t} + \frac{\rho}{2} \vec{\nabla} (\vec{v}\cdot\vec{v})= - \vec{\nabla} P $$ Show that $$ \left(\frac{\partial \phi}{\partial t} + \frac{1}{2} \vec{v}\cdot\vec{v} + \frac{P}{\rho}\right) ~~\textrm{ is uniform throughout the flow at one instant} $$ when the flow is irrotational and at constant density.
(c)  What is the difference between what was found in (b) and Bernoulli's equation? Discuss. Explain in simple terms how to use (b) and how to use Bernoulli and outline differences in the limitations of the application of both expressions. The explanation should be easy to follow even with limited knowledge of fluid mechanics.
Due on Thursday November 21st at 11:00. Do all questions.
11.14.19
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