Y = pressure r = radius ro r," = pipe radius u gas bubble uo v vn = bubble-rise velocity vo = superficial gas velocity vL = superficial liquid velocity V w z zD = radius at which gas-liquid interface is located = velocity in liquid film measured with respect to = dimensionless velocity, u / ( v , + vr. + w ) = velocity in liquid film measured with respect to pipe wall = dimensionless velocity, w / (0, + vI, + w ) = velocity of gas bubble through surrounding liq-= vertical distance from bubble head = dimensionless vertical distance, z g / ( v , + or, + uid W l Z Greek Letters fl = dimensionless area occupied by gas bubble,