(a) Explanation Given information: A body falling in a relatively dense fluid is acted on by three forces; a resistive force R , a buoyant force B , and its weight w due to gravity as shown below, The Stokes’s law for resistive force is given by R = 6 π μ a | v | , where v is velocity of a body, and μ is the coefficient of viscosity of the surrounding fluid. A solid sphere of radius a and density ρ falling freely in a medium of density ρ ' and coefficient of viscosity μ The differential equation for the velocity v of an object of mass m restricted to vertical motion and forces a resistive force R , a buoyant force B , and its weight w due to gravity is m d v d t = − R + B . The resistive force R = 6 π μ a | v | Since, a buoyant force is simply the difference of an upward force and a downward force, upward force =volume of sphere ×density of medium×gravity = 4 3 π a 3 × ρ ' × g downward force =volume of sphere ×density of sphere×gravity= 4 3 π a 3 × ρ × g B = 4 3 π a 3 × ρ ' × g − 4 3 π a 3 × ρ × g ⇒ B = ρ ' 4 3 π a 3 g − ρ 4 3 π a 3 g , Substituting these values in the equation m d v d t = − R + B m d v d t = − 6 π μ a | v | + ρ ' 4 3 π a 3 g − ρ 4 3 π a 3 g m d v d t = − 6 π μ a | v | + 4 3 π a 3 g ( ρ ' − ρ ) Dividing both side m d v d t = − 6 π μ a m | v | + 4 3 π a 3 g m ( ρ ' − ρ ) To solve this, it can be written as d v d t + 6 π μ a m | v | = 4 3 π a 3 g m ( ρ ' − ρ ) Comparing this with the first order linear differential equation d y d t + p ( t ) y = g ( t ) p ( t ) = 6 π μ a m and g ( t ) = 4 3 π a 3 g m ( ρ ' − ρ ) Integrating factor is μ ( t ) = e ∫ 6 π μ a m d t Multiplying by μ ( t ) = e 6 π μ a t m on both side of d v d t + 6 π μ a m | v | = 4 3 π a 3 g m ( ρ ' − ρ ) (b) To determine An expression for e , such that E has been adjusted, so the droplet is stationary where a field of strength E exerts a force E e on a droplet with charge e

Differential Equations: An Introduction to Modern Methods and Applications

3rd Edition

ISBN: 9781118531778

Author: James R. Brannan, William E. Boyce

Publisher: WILEY

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Question

Chapter 2.3, Problem 29P

(a)

To determine

The limiting velocity of a solid sphere of radius a and density ρ falling freely in a medium of density ρ' and the co-efficient of viscosity μ where a body falling in a relatively dense fluid, is acted on by three forces; a resistive force R, a buoyant force B, and its weight w due to gravity.

(b)

To determine

An expression for e, such that E has been adjusted, so the droplet is stationary where a field of strength E exerts a force Ee on a droplet with charge e

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Chapter 2.3, Problem 28P

Chapter 2.3, Problem 30P

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Differential Equations: An Introduction to Modern Methods and Applications

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