\noindent Inserting equation \eqref{vart2} into \eqref{vart} shows the relation between the thermal boundary layer and the shear rate:

\begin{equation}\label{vart3}
\delta_{t}^{3} \propto \frac{k}{\rho c_{p}} \frac{x-\xi}{S}
\end{equation}

\noindent Inserting \eqref{vart3} into the heat flux from equation \eqref{qw} shows the dependance of the heat flux on the shear rate:

\begin{equation}\label{var}
q_{w} \propto k \Delta T \left( \frac{\rho c_{p}}{k} \right)^{\frac{1}{3}} \left(\frac{S}{c-\xi} \right)^{\frac{1}{3}}
\end{equation}

\noindent The integrated heat transfer obtained derived from equation \eqref{var} for a heated strip extending from $\xi$ to $\xi + L$ can then be found as

\begin{equation}\label{var3}
Q_{w}(\xi,L) \propto \left(\frac{\rho c_p}{k} \right)^{\frac{1}{3}} k \Delta T \int^{\xi+L}_{\xi} \left(\frac{S}{x-\xi} \right)^{\frac{1}{3}} dx
\end{equation}

\noindent For a short strip, or $L<<\xi$, which can be assumed for the SCoDiS sensor, the shear rate S will be nearly constant over the integration period and equation \eqref{var3} can be rewritten to:

\begin{equation}
Q_{w}(\xi,L) \propto \left(\frac{\rho c_p}{k} \right)^{\frac{1}{3}} k \Delta T S^{\frac{1}{3}} L^{\frac{2}{3}}
\end{equation}

\noindent It can be seen that the heat transfer from the sensor is related to the applied shear rate by

\begin{equation}
P_{w} = Q_{w} \propto S^{\frac{1}{3}}.
\end{equation}

\noindent Knowing that the heat transfer from the sensor is the same as the applied power, the relation between the temperature difference and the applied power is