An Analytical Solution for Friction Stir Welding of an AISI 1018 Steel - Selected Topics of Computational and Experimental Fluid Mechanics - page 453

(a) (b)

Fig. 2 Comparison between analytical non-Newtonian model with different n values to Perzyna's
model (Perzyna 1966 ). a Dimensionless velocity ˆ as a function of dimensionless length ʳ . b
Dimensionless temperature ʸ as a function of dimensionless length ʳ

Fig. 3 Viscosi ty (Pa s) as a

function of the shear strain
rate (s − 1 ). Dimensional
parameters are recovered
from dimensionless one
depicted in Table 1

·

b ʸ 1 exp −

n b ʸ 0

1

1 + n
n

ˆ 1 =− (

c 0 ʳ + c 1 )

,

(25)

c 0 ( n + 1 )

n 2 exp − b ʸ 0

1
n

1 + 3 n
n

ʸ 1 =− ( c 0 ʳ + c 1 )

c 0 ( 1 + 2 n )( 1 + 3 n )

c

1 + 3 n
n

1 + 3 n
n

1 + 3 n

n

+ − ( c 0 + c 1 ) ʳ − c , (26)

1 1

2 n (

b ʸ 2 exp − b ʸ 0

1
n

2

b ʸ 1 ) −

1 + n

ˆ 2 = ( c 0 ʳ + c 1 ) , (27)

n

c 0 ( n + 1 )

nb ʸ 1 exp − b ʸ 0

1
n

1 + 3 n
n

ʸ 2 =

) (

c 0 ʳ + c 1 ) . (28)

c 0 ( 1 + 2 n )( 1 + 3 n