Effect of acceleration on rise of liquid in capillary What would be the effect of vertical acceleration on the rise of liquid level in a capillary tube. Will the liquid not rise if we provide the system a downward acceleration equal to $g$ ?

Consider that  a container of liquid and the capillary tube are placed in an elevator. What would happen to the liquid level if we move the elevator up or down with a certain acceleration
 A: No, the effect is quite the opposite. See Don Petit's Explanation of Space Coffee Cups for a graphic demonstation.
You answer this simply by replacing $g$, in the relevant equations (e.g. the Young-Laplace or Capillary Equation  with a value modified to account for the additional inertial forces from the frame's downward acceleration. In particular, if the frame is in freefall, $g$ becomes nought - the capillary equation then tells us that fluids will rise arbitrarily high in columns. What this means is that there is nothing - or very little - to balance the forces of adhesion between fluids, and the fluids crawl everywhere in wild ways. 
In fact, this principle is used to design pumps, coffee cups and urinals for space travel. When a rocket is staging and a spent stage is shed, the whole system becomes weightless. In particular, the fuel in the igniting stage can rise away from the intakes of the pumps supplying the engines and thwart ignition. To overcome this problem, some rocket fuel tanks have a sharp ridge in them - forming a tight semicircular tube - to enhance capillary flow towards the engines and thus make sure that the fuel stays in contact with the pump intakes through capillary action (small, solid-rocket so-called ullage motors that uphold a small acceleration throughout staging are another solution, used in the Saturn V third stage ignition for example).
