Investigation Into the Similarity Solution for Boundary Layer Flows in Microsystems

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 $0<K1<1$⁠. 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 $0<K1<5$ 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 $0<K1$⁠, $K2<5$⁠, $0.7≤Pr≤5$ within a maximum error of $±3%$⁠.