Equation 10 holds for every position in the test section and gives the contrast at the equivalent position in the image on the screen. The quantity on the left hand side can be obtained by using the initial and final inensity distribution on the screen. is the length of the test section along the direction of the propogation of the laser beam, is the focal length of the second concave mirror and is the size of the focal spot at the knife-edge. Usually, the knife-edge is adjusted in such a position that it cuts off approxiamtely 50% of the original light intensity, i.e. where is the original dimension of the laser beam at the pin-hole of the spatial filter. Typically, for a He-Ne laser (employed as the light source in the present work) is of the order of 10-100 microns. With Equation 10 can be written as

(11)

Equation 11 represents the governing equation for the schlieren process in terms of the ray-averaged refractive index. It requires the approximation that changes in the light intensity occur due to beam deflection, rather than its physical displacement.

If the working fluid is a gas (e.g. air as employed in the validation experiments of the present study), the first derivative of the refractive index field with respect to y can be expressed as

(12)

Equation 12 relates the gradient in the refractive index field with the gradients of the density field with the gradients of the density field in the fluid medium inside the test cell. The governing equation for the schlieren process in gas (Equation 11) can be rewritten as

(13)

Assuming that pressure inside the test cell is practically constant, we get

Equation 13 and 14 respectively relate the contrast measured using a laser schlieren technique with the density and temperature gradients in the test section. With the value of the dependent variables defined in the bulk of the fluid medium or with proper boundary conditions, the above equations can be solved to determine the quantity of interest.