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Derive the expression for the terminal velocity of a sphere moving in a high viscous fluid using stokes force. |
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Answer» Solution :Expression for terminal velocity : Consider a sphere of radius R which falls freely through a highly viscous liquid of COEFFICIENT of viscosity `eta`. Let the density of the material of the sphere be `RHO` and the density of the fluid be `sigma`. Gravitational force acting on the sphere. `F_(G) = mg = (4)/(3)pi r^(3)rho g` (downward force) Up thrust, `U = (4)/(3)pi r^(3) sigma g` (upward force) viscous force `F= 6pi eta r v_(t)` At terminal velocity `v_(t)` downward force = upward force `F_(G) - U = F rArr (4)/(3) pi r^(3) rho g - (4)/(3) pi r^(3) sigma g = 6pi eta r v_(t)` `v_(t) = (2)/(9) xx (r^(2)(rho - sigma))/(eta) g rArr v_(t) PROP r^(2)` Here, it should be noted that the terminal speed of the sphere is directly proportional to the square of its radius. If `sigma` is greater than `rho`, then the term `(rho - sigma)` becomes negative leading to a negative terminal velocity. 
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