For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

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Answer 1

Textbook Question

Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

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Students have asked these similar questions

Consider a flow field represented by the stream functionψ=(4+1) x3y - (6+1) x2y2 + (4+1)xy3. Is this a possible two-dimensionalincompressible flow? Is the flow irrotational?Discuss the reasons against your findings.

(b) Two velocity components of a steady, incompressible flow field are given as follows; и 3 2ах + bxу + су? v = axz – byz? where a, b and c are constants. Determine an expression for w as a function of x, y, and z.

3. The velocity components in a two dimensional flow field for an incompressible fluid are expressed as u = 2y -. v = xy? – 2y - 3' (a) Show that these functions represents a possible case of an irrotational flow, (b) obtain an expression for the stream function y and (c) obtain an expression for the velocity potential o.

Answer 2

Textbook Question

Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

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Answer 3

Textbook Question

Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

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Answer 4

Textbook Question

Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

Answer 8

Expert Solution & Answer

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

p+12ρV2=constant

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer 12

Answer to Problem 3.1P

p+12ρV2=constant

Answer 13

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

x momentum: ρu∂u∂x+ρv∂u∂y+ρw∂u∂z=−∂p∂x (1)y momentum: ρu∂v∂x+ρv∂v∂y+ρw∂v∂z=−∂p∂y (2)z momentum: ρu∂w∂x+ρv∂w∂y+ρw∂w∂z=−∂p∂z (3) ss

Multiplying (1), (2) and (3) by dx,dy and dz respectively

u∂u∂xdx+v∂u∂ydx+w∂u∂zdx=−1ρ∂p∂xdx (4) u∂v∂xdy+v∂v∂ydy+w∂v∂zdy=−1ρ∂p∂y dy (5) u∂w∂xdz+v∂w∂ydz+w∂w∂zdz=−1ρ∂p∂z dz (6)

Now, adding equation 4,5 and 6

u(∂u∂xdx+∂v∂xdy+∂w∂xdz)+v(∂u∂ydx+∂v∂ydy+∂v∂xdy)+w(∂u∂zdx+∂v∂zdy+∂w∂xdz)=−1ρ(∂p∂xdx+∂p∂y dy+∂p∂z dz) (7)

We know that for irrotational flow ∇×V=0

hence ∂w∂y=∂v∂z;∂u∂z=∂w∂x;∂v∂x=∂u∂y (8)

Substituting Equation (8) into (7) we get

udu+vdv+wdw=−1ρdp

12d(u2+v2+w2)=−12d(V2)=VdV=−1ρdρdp=−ρVdV which integrates to p+12ρV2=constant

Answer 14

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For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

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Answer to Problem 3.1P

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