Consider a tube of radius R, which is filled with a Newtonian liquid. As the liquid...

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Consider a tube of radius R, which is filled with a Newtonian liquid. As the liquid flows upward in the tube, a gas bubble of volume V rises with it. Suppose that V¹/³ >> R. Then, the bubble becomes elongated, such that we can assume that it can be represented by a cylinder of radius Rg. The thickness of the liquid between the bubble and the tube's. walls is very thin. A constant pressure gradient of dP/dx is applied to the flow, which is in the axial x direction. We put the origin of the coordinates on the tube's wall, so that y = 0 corresponds to the wall, while H = R- RB is at the interface between the bubble. and the liquid. (a) Assuming steady state and fully-developed flow, simplify the momentum equation to obtain the governing equation for v, the fluid velocity. (b) One boundary condition is the usual no-slip at the tube's wall. Justify the second. boundary condition, du/dy = 0 at y = H. (c) Derive the solution for v. Then, derive an expression for va, the average fluid velocity. (d) Assume that the steady-state rise velocity of the bubble is up. Use the fact that the sum of the gas and liquid upflows must be zero, derive an expression for dP/dx, and rewrite the velocity profile v by eliminating dP/dx. (f) Next, write a force balance in order to obtain an expression for vp. Recall that there are three forces, namely, body, pressure, and viscous forces. Consider a tube of radius R, which is filled with a Newtonian liquid. As the liquid flows upward in the tube, a gas bubble of volume V rises with it. Suppose that V¹/³ >> R. Then, the bubble becomes elongated, such that we can assume that it can be represented by a cylinder of radius Rg. The thickness of the liquid between the bubble and the tube's. walls is very thin. A constant pressure gradient of dP/dx is applied to the flow, which is in the axial x direction. We put the origin of the coordinates on the tube's wall, so that y = 0 corresponds to the wall, while H = R- RB is at the interface between the bubble. and the liquid. (a) Assuming steady state and fully-developed flow, simplify the momentum equation to obtain the governing equation for v, the fluid velocity. (b) One boundary condition is the usual no-slip at the tube's wall. Justify the second. boundary condition, du/dy = 0 at y = H. (c) Derive the solution for v. Then, derive an expression for va, the average fluid velocity. (d) Assume that the steady-state rise velocity of the bubble is up. Use the fact that the sum of the gas and liquid upflows must be zero, derive an expression for dP/dx, and rewrite the velocity profile v by eliminating dP/dx. (f) Next, write a force balance in order to obtain an expression for vp. Recall that there are three forces, namely, body, pressure, and viscous forces.