FLUID MECHANICS FUND. (LL)-W/ACCESS
FLUID MECHANICS FUND. (LL)-W/ACCESS
4th Edition
ISBN: 9781266016042
Author: CENGEL
Publisher: MCG CUSTOM
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Chapter 4, Problem 97P
To determine

The normal acceleration of particle passing through x=1 and y=2.

Expert Solution & Answer
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Answer to Problem 97P

The normal acceleration of particle passing through x=1 and y=2 is 3.996k2_.

Explanation of Solution

The vector field of flow is V=k(x2y2)i2kxyj and the radius of curvature of streamline is R=[1+ y 2]32[y]

Write the streamline equation for velocity field.

  dydx=vu.  ...... (I)

Here, the velocity in y direction is v and the velocity in x direction is u

Write the expression for first derivative of y.

  y=dydx

Here, the first derivative of y is y'.

Substitute y' for dydx in Equation (I).

  y=vu  ...... (II)

Substitute 2kxy for v and k(x2y2) for u in Equation (II).

  y=2kxyk(x2y2)  ...... (III).

Differentiate Equation (III) with respect to x.

  ddxy=ddx2kxyk( x 2 y 2)y=ddx2kxyk( x 2 y 2)y=( x 2 y 2)(2y2x y ')+(2xy)(2x2y y ')( x 2 y 2 )2   ...... (IV)

Substitute 2kxyk(x2y2) for y' in Equation (IV).

  y=( x 2 y 2)(2y2x 2kxy k( x 2 y 2 ))+(2xy)(2x2y 2kxy k( x 2 y 2 ))( x 2 y 2 )2=2y( x 2 y 2 )2+8x2y( x 2 y 2)+8x2y3( x 2 y 2 )3   ...... (V)

Write the expression for the radius of curvature of streamline.

  R=[1+y '2]32[y"]..... (V)

Write the expression for the velocity field of flow.

  V=k(x2y2)i2kxyj   ...... (VI)

Write the expression for the resultant velocity.

  V=a2+b2  ...... (VII)

Write the expression for the normal acceleration.

  an=V2R   ...... (VIII).

Calculation:

Substitute 1 for x and 2 for y in Equation (V).

  y=2×2( 1 2 2 2 )2+8×12×2( 1 2 2 2)+8×12×23( 1 2 2 2 )3=3648+6427=2027=0.74074

Substitute 1 for x and 2 for y in Equation (III).

  y=2×1×2( 1 2 2 2)=43=1.334

Substitute 1.334 for y' and 0.74074 for y in Equation (V).

  R=[ 1+ ( 1.334 ) 2 ] 3 2[0.74074]=[ 1+1.779556] 3 2[0.74074]=6.256

Substitute 1 for x and 2 for y in Equation (III).

  V=k(1222)i2k×1×2j=3ki4kj

Substitute 3k for a and 4k for b in Equation (VII).

  V=( 3k)2+( 4k)2=9k+16k=5k

Substitute 5k for V and 6.256 for R in Equation (VIII).

  an=( 5k)26.256=25k26.256=3.996k2

Conclusion:

The normal acceleration of particle passing through is calculated by the expression an=V2R

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Chapter 4 Solutions

FLUID MECHANICS FUND. (LL)-W/ACCESS

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