The dynamics of triple line and bubble growth were investigated inside water, gold, silver, and alumina nanofluids. The presence of nanoparticles was found to modify the behaviour of the triple line, depending on nanoparticle materials. Two analytical expressions were developed, one for the contact angle of bubbles and another for the contact angle of droplets, using force balance in the horizontal direction. It was mathematically proven that these analytical expressions are the exact analytical solutions of the Young-Laplace equation. These analytical expressions prove that the Young equation cannot be an accurate force balance at the triple line for axisymmetric bubbles and droplets. The analytical expressions predict the contact angle of bubbles and droplets and prove theoretically that the contact angles of axisymmetric bubbles and droplets depend on volume or size, which agreed with existing experimental data. The predictions of the analytical expressions were compared with experimental data, and close agreements were observed. To reduce uncertainty and error, these two analytical expressions can be used instead of the Young equation, particularly in CFD calculations, to predict the behaviour of triple line or fluid flow in the vicinity of substrate for droplets and bubbles.