Answer
The primary functions of the lymphatic system are: fluid recovery, immunity, and lipid absorption.
If you clamp the right lymphatic duct closed, the most noticeable effect would be swelling of the right arm.
Work Step by Step
Fluid flows from the blood capillaries into tissue spaces. The capillaries only reabsorb about 85% of this fluid. The remaining 15% is picked up by the lymphatic system, filtered, and then returned to the blood. Without this fluid recovery by the lymphatic system, about 25-50% of plasma protein and about 2-4 L of water would be lost per day, which would quickly lead to circulatory failure.
The lymphatic system also works with the immune system. As the lymphatic system recovers tissue fluid, the fluid passes through lymph nodes where immune cells attack and remove foreign material from circulation.
Lastly, the lymphatic system has special lymphatic vessels in the small intestine to absorb dietary lipids not otherwise absorbed by blood capillaries.
The right lymphatic duct drains the right arm and the right side of the thorax and head to the right subclavian vein. If you clamped off this duct, the lymph would not be able to drain and you would get edema (swelling) in these areas. The entire right arm swelling would be the most noticeable effect, followed by swelling of the right side of the head and thorax.