A generous amount of gel is applied to the surface of the window of an ultrasound probe prior to its use. The window is made of a polycarbonate material of density \(ho=\) \(1200 \mathrm{~kg} / \mathrm{m}^{3}\), thermal conductivity \(k=0.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and specific heat \(c_{p}=1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The probe is initially at a temperature of \(T_{i}=20^{\circ} \mathrm{C}\). To ensure that the gel remains adhered to the probe until the probe is brought into contact with the patient's skin, its temperaturedependent viscosity must not fall below a critical value. This corresponds to a requirement that the maximum gel temperature remain below \(12^{\circ} \mathrm{C}\) prior to making contact with the patient. Determine the required initial gel temperature. The thermal conductivity, density, and specific heat of the gel may be assumed to be those of liquid water.