To prove: The streamlines are orthogonal to the potential lines.

Question

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Answer 1

Question

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

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Answer 2

Question

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

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Answer 3

Question

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

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Answer 4

Question

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

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Answer 5

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

Answer 6

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.

Conclusion:

The streamlines and the potential lines are orthogonal to each other as their slopes gives -1 after multiplication.

Answer 7

Chapter 8, Problem 8.1P

To determine

To prove:

The streamlines are orthogonal to the potential lines.

Answer 8

Expert Solution & Answer

Answer to Problem 8.1P

The statement has been proved.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given Information:

The streamline function Ψ (r,θ)

The potential line function φ (r,θ)

Concept Used:

From equation (8.10) we get:vr=∂ϕ∂r=1r( ∂ψ ∂θ)vθ=(1r)( ∂ϕ ∂θ)=−∂ψ∂rThe two functions will be orthogonal if their slopes prouct is(−1).

Calculation:

The streamline slope is denoted by:( dr rdθ)streamline function =vrvθ= ∂ϕ ∂r(1r)( ∂ϕ ∂θ )=−1 ( dr rdθ ) potential line functionas the ψslope =−1 ϕslope This shows the orthogonality of two functions.hence, proved.