(a) The value of function f(y) for which the laplacian of F is zero.

Question

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Answer 1

Question

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

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Answer 2

Question

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

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Answer 3

Question

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

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Answer 4

Question

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

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Answer 5

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

Answer 6

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

Answer 7

Chapter 8, Problem 8.5P

To determine

(a)

The value of function f(y) for which the laplacian of F is zero.

Answer 8

Expert Solution

Answer to Problem 8.5P

The value of function f(y) for which the laplacian of F is zero.

Answer 9

Expert Solution

Answer 10

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given Information:

A harmonic function F(x, y, z):

F=2x² +y³ -4xz+f(y)

Concept Used:

First, we must find the suitable f(y) which makes ∇2F=0

Calculation:

∇2F=∂2F∂x2+∂2F∂y2+∂2F∂z2=∂2(2x2+y3−4xz+f(y))∂x2+∂2(2x2+y3−4xz+f(y))∂y2+∂2(2x2+y3−4xz+f(y))∂z2=(4+0+0)+(0+6y−0)+(0+0−0)+f"(y)=4+6y+f"y⇒f"(y)=−4−6y for ∇2F=0on solving for f(y) we get:f'(y)=−4y−3y2+cf(y)=−2y2−y3+c'on putting f(y) we get:F=2x2+y3−4xz−2y2−y3+c'=2x2−2y2−4xz+c'f(y) is not a harmonic function.

Conclusion:

The value of function f(y) for which the laplacian of F is zero is F=2x² -2y² -4xz+ c'

Here c' is integration constant. The function f(y) is not a harmonic function.

Answer 13

To determine

(b)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

Answer 14

To determine

(b)

The given function can be served as velocity potential function.

Answer 15

Expert Solution

Answer to Problem 8.5P

The given function can be served as a velocity potential function as ∇2F=0.

Answer 16

Expert Solution

Answer 17

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Given Information:

A harmonic function F(x, y,z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be velocity potential the required condition is:

∇2F=0

Calculation:

Conclusion:

The given function can be served as a velocity potential function as ∇2F=0

But the given function may not be a realistic flow pattern.

Answer 20

To determine

(c)

The given function can be served as velocity potential function.

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

Answer 21

To determine

(c)

The given function can be served as velocity potential function.

Answer 22

Expert Solution

Answer to Problem 8.5P

The given function cannot be served as a stream function due to having three dimensional function.

Answer 23

Expert Solution

Answer 24

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Given Information:

A harmonic function F(x, y, z): F=2x² +y³ -4xz+f(y)

Concept Used:

For a function to be stream function the required condition is that the function should not be three dimensional.

Calculation:

Conclusion:

The given function is not a stream function as it is three dimensional.

Answer 27

Ch. 8 - Prob. 8.1P Ch. 8 - The steady plane flow in Fig. P8.2 has the polar...Ch. 8 - P8.3 Using cartesian coordinates, show that each...Ch. 8 - P8.4 Is the function 1/r a legitimate velocity...Ch. 8 - Prob. 8.5P Ch. 8 - An incompressible plane flow has the velocity...Ch. 8 - Prob. 8.7P Ch. 8 - For the velocity distribution u=By,=+Bx , evaluate...Ch. 8 - Prob. 8.9P Ch. 8 - Prob. 8.10P

(a) The value of function f(y) for which the laplacian of F is zero.

Videos

(a)

The value of function f(y) for which the laplacian of F is zero.

Answer to Problem 8.5P

Explanation of Solution

(b)

The given function can be served as velocity potential function.

Answer to Problem 8.5P

Explanation of Solution

(c)

The given function can be served as velocity potential function.

Answer to Problem 8.5P

Explanation of Solution

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Chapter 8 Solutions

(a) The value of function f(y) for which the laplacian of F is zero.

Videos

(a) The value of function f(y) for which the laplacian of F is zero.

Answer to Problem 8.5P

Explanation of Solution

(b) The given function can be served as velocity potential function.

Answer to Problem 8.5P

Explanation of Solution

(c) The given function can be served as velocity potential function.

Answer to Problem 8.5P

Explanation of Solution

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Chapter 8 Solutions

(a)

The value of function f(y) for which the laplacian of F is zero.

(b)

The given function can be served as velocity potential function.

(c)

The given function can be served as velocity potential function.