In a shock tube, the presence of the wall boundary-layer causes the shock to decelerate, the contact surface to accelerate and the flow behind the shock to be non-uniform. In this one-dimensional analysis, we must account empirically for effect that the flow of mass into the cold boundary-layer has on the free-stream variables. We take the approach developed by Mirels. [53] Assuming a laminar boundary-layer, Mirels proposed treating the flow as quasi-one-dimensional with the variation of the free-stream variables calculated from

(7–47)

where is the distance between the shock and contact surface at infinite distance from the diaphragm and the subscript 2 denotes conditions immediately behind the shock. Mirels then obtained an expression for by considering the simultaneous boundary-layer development and change in free-stream conditions external to the boundary-layer.

This expression is

(7–48)

where

(7–49)

with and . The effect of variable viscosity is accounted for by , where

(7–50)

The wall is assumed to remain at its initial temperature, while the pressure at the wall changes to after passage of the shock. The viscosity correction is based on numerical solutions for air. For the purposes of the boundary-layer corrections, we take the viscosity to be that of the diluent gas.

Hirschfelder, Curtiss and Bird [54] give the viscosity of a pure gas as

(7–51)

where is the viscosity in gm/cm ⋅ sec; , the low-velocity collision cross-section for the species of interest in Angstroms; , the molecular weight; , the temperature in Kelvin; and , the reduced collision integral, a function of the reduced temperature () where () is the potential parameter for the species of interest). The reduced collision integral represents an averaging of the collision cross-section over all orientations and relative kinetic energies of colliding molecules. Tabulated values of this integral at various reduced temperatures are given by Camac and Feinberg. [55]

The values can be fit to within 2% for by the expression:

(7–52)

Using this in the expression for the viscosity yields:

(7–53)

Evaluating the above equation at 300 K and using this as a reference point results in the following expression for the viscosity:

(7–54)

A value for for the diluent gas must be specified by the user when considering boundary-layer effects.

To derive an equation for the area variation, we first combine Equation 7–33 and Equation 7–47 to yield

(7–55)

Then, the change in cross-sectional area with distance downstream of the shock is given by

(7–56)

This expression is used in Equation 7–42 , Equation 7–44 and Equation 7–45 . Equation 7–56 allows us to account for the boundary-layer effects by computing the "effective" area through which the gas must flow.