**Flow Analysis in the x-y Plane**An irrotational flow in the $x$-$y$ plane is represented by the potential function $\phi = x^2 + xy - y^2$, where $\phi$ has units of ft$^2$/s.**Tasks:**1. **Incompressibility:** - Show that the flow is also incompressible.2. **Stream Function:** - Determine the stream function $\psi$.3. **Fluid Acceleration:** - Compute the magnitude of the fluid acceleration at the point $(1,1)$, where the coordinates are given in feet. *Hint: Use the terms of the Navier-Stokes equation.*

Answered: An irrotational flow in the r-y plane…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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**Flow Analysis in the x-y Plane**

An irrotational flow in the $x$-$y$ plane is represented by the potential function $\phi = x^2 + xy - y^2$, where $\phi$ has units of ft$^2$/s.

**Tasks:**

1. **Incompressibility:**
- Show that the flow is also incompressible.

2. **Stream Function:**
- Determine the stream function $\psi$.

3. **Fluid Acceleration:**
- Compute the magnitude of the fluid acceleration at the point $(1,1)$, where the coordinates are given in feet. *Hint: Use the terms of the Navier-Stokes equation.*

Expert Solution

Step 1

Given data

The potential function of the flow is $ϕ = x^{2} + x y - y^{2}$ .

Step 2

(a)

The velocity in the x-direction is as follows:
$u = - \frac{d ϕ}{d x} u = - 2 x - y$

The velocity in the y-direction is as follows:
$v = - \frac{d ϕ}{d y} v = 2 y - x$

Now, the calculate the flow is in-compressible or compressible as follows:

$\begin{array}{rcl} \frac{d u}{d x} + \frac{d v}{d y} \\ \frac{d (- 2 x - y)}{d x} + \frac{d (2 y - x)}{d y} \\ - 2 + 2 \\ 0 & \to & T h e f l o w i s i n c o m p r e s s i b l e . \end{array}$

Step 3

(b)

Calculate the stream function from the velocity function as follows:

$\frac{d ψ}{d y} = \frac{d ϕ}{d x} \frac{d ψ}{d y} = \frac{d (x^{2} + x y - y^{2})}{d x} \frac{d ψ}{d y} = (2 x + y) \int d ψ = \int (2 x + y) d y ψ = 2 x y + \frac{y^{2}}{2} + C$

Here, C is the integration constant.

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