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Research Papers

# Investigation Into the Similarity Solution for Boundary Layer Flows in Microsystems

[+] Author and Article Information
Suhil Kiwan1

Department of Mechanical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordankiwan@just.edu.jo

M. A. Al-Nimr

Department of Mechanical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan

1

Corresponding author.

J. Heat Transfer 132(4), 041011 (Feb 22, 2010) (9 pages) doi:10.1115/1.4000886 History: Received March 28, 2009; Revised September 06, 2009; Published February 22, 2010; Online February 22, 2010

## Abstract

An investigation toward the existence of a complete similarity solution for boundary layer flows under the velocity slip and temperature jump conditions is carried out. The study is limited to the boundary layer flows resulting from an arbitrary freestream velocity $U(x)=Uoxm$ and wall temperature given by $Tw−T∞=Cxn$. It is found that a similar solution exists only for $m=1$ and $n=0$, which represents stagnation flow on isothermal surface. This case has been thoroughly investigated. The analysis showed that three parameters control the flow and heat transfer characteristics of the problem. These parameters are the velocity slip parameter $K1$, the temperature jump parameter $K2$, and Prandtl number. The effect of these parameters on the flow and heat transfer of the problem has been studied and presented. It is found that the slip velocity parameter affects both the flow and heat transfer characteristics of the problem. It is found that the skin friction coefficient decreases with increasing $K1$ and most of changes in the skin friction takes place in the range $0. The skin friction coefficient is found to be related to $K1$ and $Rex$ according to the relation: $Cf=3.38Rex−0.5(K1+1.279)−0.8$ for $0 with an error of $±4%$. On the other hand, the correlation between Nu, $K1$, $K2$, and Pr has been found by the equation $Nu=[(0.449+1.142K11.06)∕(0.515+K11.06)](K2+1.489Pr−0.44)−1$, for $0, $K2<5$, $0.7≤Pr≤5$ within a maximum error of $±3%$.

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## Figures

Figure 1

Stagnation on plane flow for the case when m=1, n=0

Figure 2

Variation in the dimensionless transverse velocity distribution with the similarity parameter η at different slip parameters, K1 and Pr=1

Figure 3

Variation in the dimensionless axial velocity distribution with the similarity parameter η at different slip parameters, K1, and Pr=1

Figure 4

Variation in the dimensionless shear parameter distribution with the similarity parameter η at different slip parameters, K1 and Pr=1

Figure 5

Variation in the dimensionless temperature distribution with the similarity parameter η at different slip parameters, K1 and Pr=1

Figure 6

Variation in the dimensionless temperature gradient distribution with the similarity parameter η at different slip parameters, K1 and Pr=1

Figure 7

Variation in the dimensionless temperature distribution with the similarity parameter η at different jump parameters, K2 and Pr=1

Figure 8

Variation in the dimensionless temperature gradient distribution with the similarity parameter η at different jump parameters, K2 and Pr=1

Figure 9

Variation in the dimensionless temperature distribution with the similarity parameter η at different Prandtl numbers for K1=0.5 and K2=0.5

Figure 10

Variation in the dimensionless temperature gradient with the similarity parameter η at different Prandtl numbers for K1=0.5 and K2=0.5

Figure 11

Variation in the hydrodynamic boundary layer thickness with the variation in the slip parameter K1 for all values of K2 and Pr

Figure 12

Variation in the thermal boundary layer thickness with the variation in slip parameter K1 for different Prandtl numbers and K2=0.5

Figure 13

Variation in the thermal boundary layer thickness with the variation in jump parameter K2 for different Prandtl numbers and K1=0.5

Figure 14

Variation in skin friction parameter with the variation in slip parameter K1 for all values of Pr and K2

Figure 15

Variation in Nusselt number with the variation in slip parameter K1 for different values of Pr and K2=0.5

Figure 16

Variation in Nusselt number with the variation in jump parameter K2 for different values of Pr and K1=0.5

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