I have been trying to understand a problem given in a paper for a couple of months but cannot figure out the rationale behind the change of variables of a function. This problem is outlined below.

In his 1959 paper Benjamin uses an orthogonal curvilinear coordinate system to define the boundary-layer of a gas profile. This system is given below:

$$\xi_1=x-\imath\epsilon e^{-\alpha y}e^{\imath \alpha x} \\ \xi_2=y-\epsilon e^{-\alpha y}e^{\imath \alpha x}$$

This coordinate system is used to determine the streamfunction of a counter-current gas liquid film flow.

The streamfunction is defined as follows: $$\psi = \psi_0(\xi_2) + \epsilon\psi_1(\xi_2)e^{\imath\alpha\xi_1}+O(\epsilon^2)$$

The $\psi_0(\xi_2)$ corresponds to a streamfunction of a base state with no waviness(flat boundary). This base state is given by a velocity function which is only a function of $y$ and does not depend on $x$: $$l^4+L^2U_0^4-(L+1)^2U_0^2+2L(L+1)U_0-L^2=0$$ where $U_0(y)=\psi_0'(y)$ and $l=\kappa y[1-\exp(-y/A)]$ which is also a function of $y$.

My problem arises by a confusion regarding how the base state can be a function exclusively of $\xi_2$ when there should also be a component of $\xi_1$ that is caused by the change of variable from $y$. I am trying to find out how to calculate $\psi_0(\xi_2)$ from $\psi_0(y)$.

Thanks in advance. Any help in understanding what I am missing is appreciated.

P.S. The Jacobian matrix of the transformation is also provided in the paper but I am not sure if it is helpful. I am providing it below in case it is needed: $$\Gamma=\Bigl(\begin{bmatrix}1 & -\imath\epsilon \alpha e^{-\alpha \xi_2} e^{\imath\alpha\xi_1}\\ \imath\epsilon \alpha e^{-\alpha \xi_2} e^{\imath\alpha\xi_1} & 1\end{bmatrix} +O(\epsilon^2)\Bigl)$$

The paper I am atepmting to understand is the following: doi:10.1017/S002211201000618X


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