EBK FLUID MECHANICS: FUNDAMENTALS AND A
EBK FLUID MECHANICS: FUNDAMENTALS AND A
4th Edition
ISBN: 8220103676205
Author: CENGEL
Publisher: YUZU
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Chapter 9, Problem 63EP
To determine

The expression for stream function.

The plot some streamlines of the flow.

Expert Solution & Answer
Check Mark

Answer to Problem 63EP

The expression for the stream function is ax2ybxy22+C.

The following figure represents the streamlines of the flow.

  EBK FLUID MECHANICS: FUNDAMENTALS AND A, Chapter 9, Problem 63EP , additional homework tip  1

Explanation of Solution

Given information:

The incompressible flow filed for which the velocity u component is u=ax2bxy, where a=0.45(fts)1, b=0.75(fts)1 and the velocity along the x axis is 0. The stream function along the x axis is 0. The range of x is varies from 0 to 3 and the range of y is varies from 0 to 4.

Write the expression for the velocity along x direction.

  v=ψx  ....... (I)

Here, the stream function along x direction is ψ, the change in distance along x direction is x and the velocity along x direction is v.

Write the expression for the velocity along x direction.

  u=ψy  ...... (II)

Here, the stream function along y direction is ψ, the change in distance along y direction is y and the velocity along y direction is u.

Write the expression for quadric stream function.

  y=ax2±a2x42bxψbx   ....... (III)

Here, the stream function is ψ, the distance along x direction is x, the distance along y direction is y and the constants are a and b.

Calculation:

Substitute ax2bxy for u in Equation (II).

  ax2bxy=ψyψy=ax2bxyψ=(ax2bxy)y   ...... (IV)

Integrate Equation (IV) with respect to y.

  ψ=(ax2bxy)yψ=(a x 2bxy)y+f(x)ψ=ax2ybxy1+12+f(x)ψ=ax2ybxy22+f(x)   ....... (V)

Here, the constant is f(x).

Substitute ax2ybxy22+f(x) for ψ and 0 for v in Equation (I).

  0=(ax2y bx y 2 2+f(x))x(ax2y bx y 2 2+f(x))x=0   ....... (VI)

Differentiate Equation (V) with respect to x.

  (ax2y bx y 2 2+f(x))x=02ax21y+bx11y22f(x)x=02ax1y+by22f(x)x=0   ....... (VII)

Substitute 0 for y in Equation (VII).

  2ax1(0)+b(0)22f(x)x=0f(x)x=0f(x)x=0   ....... (VIII)

Substitute 0 for y and C for ψ in Equation (V).

  ax2(0)bx(0)22+f(x)=C00+f(x)=Cf(x)=C   ....... (IX)

Here, the constant is C.

Substitute C for f(x) in Equation (V).

  ψ=ax2ybxy22+C   ....... (X)

Substitute C for f(x) in Equation (X).

  ψ=ax2ybxy22+0ψ=ax2ybxy22

Substitute 0.45(fts)1 for a and 0.75(fts)1 for b in Equation (III).

  y=(0.45 ( fts ) 1)x2± ( 0.45 ( fts ) 1 )2x42( 0.75 ( fts) 1 )xψ(0.75 ( fts ) 1)xy=0.45( fts)1×x2±0.2025 ( fts ) 1×x41.50 ( fts ) 1×xψ(0.75 ( fts ) 1)x..... (XI)

Substitute 1 for x and 1 for y in Equation (XI).

  1=0.45( fts)1×(1)2±0.2025 ( fts ) 1× ( 1 )41.50 ( fts ) 1×(1)ψ(0.75 ( fts ) 1)(1)±0.2025( fts)1×(1)41.50( fts)1×(1)ψ=0.75(fts)10.45(fts)10.2025(fts)1×(1)41.50(fts)1×(1)ψ=±(0.35( fts)1)2.. (XII)

Solve Equation (XII) by taking positive sign.

  0.2025(fts)1×(1)41.50(fts)1×(1)ψ=±(0.35( fts)1)21.50(fts)1×(1)ψ=0.08ψ=0.05333

Solve Equation (XII) by taking negative sign.

  0.2025(fts)1×(1)41.50(fts)1×(1)ψ=(0.35( fts)1)21.50(fts)1×(1)ψ=0.325ψ=0.216666

The following table shows that the value of stream function with respect to value of x and y.

      x  y  ψ  +ψ
      0  0  0  0
      1  1  0.216666  0.053333
      2  2  0.4352  0.1052

Plot the streamlines on y x plane by using Equation (XII) and the range of x is varies from 0 to 3 and the range of y is varies from 0 to 4.

  EBK FLUID MECHANICS: FUNDAMENTALS AND A, Chapter 9, Problem 63EP , additional homework tip  2

  Figure- (1)

The Figure (1) represents the streamlines of the flow.

Conclusion:

The expression for the stream function is ax2ybxy22+C.

The following figure represents the streamlines of the flow.

  EBK FLUID MECHANICS: FUNDAMENTALS AND A, Chapter 9, Problem 63EP , additional homework tip  3

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Chapter 9 Solutions

EBK FLUID MECHANICS: FUNDAMENTALS AND A

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