### 1.13.1. 1D Illustration of Concept

Consider the 1D case of a horizontal surface, with the Y direction normal to the surface:

Consider the following Poisson equation for a variable :

| (1–292)

with a boundary condition of at the wall.

Integrate once:

| (1–293)

Integrate again:

| (1–294)

Because at , you can deduce that . You also know from Equation 1–293 that:

| (1–295)

Substituting into Equation 1–294 for and , you are left with a quadratic equation for in terms of and :

| (1–296)

Rearranging this gives:

| (1–297)

where , , and

Solving this quadratic and choosing the positive root gives:

| (1–298)

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