Question

Assuming that the critical velocity Vc of a viscous liquid flowing through a capillary tube depends upon the radius a of the tube, the density

Answer 1

ᴀɴꜱᴡᴇʀ ⇒ ᴠ = ᴋη/ʀ²ρ.

ᴇxᴘʟᴀɴᴀᴛɪᴏɴ ⇒

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴛʜᴇ Qᴜᴇꜱᴛɪᴏɴ, ᴄʀɪᴛɪᴄᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ᴏꜰ ᴠɪꜱᴄᴏᴜꜱ ʟɪQᴜɪᴅ ɪꜱ ꜰʟᴏᴡɪɴɢ ᴛʜʀᴏᴜɢʜ ᴀ ᴄᴀᴘɪʟʟᴀʀʏ ᴛᴜʙᴇ ᴅᴇᴘᴇɴᴅꜱ ᴏɴ ʀᴀᴅɪᴜꜱ ʀ ᴏꜰ ᴛᴜʙᴇ , ᴅᴇɴꜱɪᴛʏ ᴘ,ᴀɴᴅ ᴄᴏᴇꜰꜰɪᴄɪᴇɴᴛ ᴏꜰ ᴠɪꜱᴄᴏꜱɪᴛʏ ᴏꜰ ᴛʜᴇ ʟɪQᴜɪᴅ.

ʟᴇᴛ ᴛʜᴇ ᴄʀɪᴛɪᴄᴀʟ ᴠᴇʟᴏᴄɪᴛʏ ɪꜱ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ ᴛᴏ ʀᵃ, ρᵇ , ηⁿ.

ᴛʜᴇɴ,

ᴠ = ᴋʀᵃ, ρᵇ , ηⁿ, ᴋ ɪꜱ ᴀɴʏ ᴅɪᴍᴇɴꜱɪᴏɴ ʟᴇꜱꜱ ᴄᴏɴꜱᴛᴀɴᴛ.

ᴛʜᴜꜱ,

ᴅɪᴍᴇɴꜱɪᴏɴ ᴏꜰ ᴠ = ʟᴛ⁻¹

ᴅɪᴍᴇɴꜱɪᴏɴ ᴏꜰ ʀ = ʟ

ᴅɪᴍᴇɴꜱɪᴏɴ ᴏꜰ ρ = ᴍʟ⁻³

ᴅɪᴍᴇɴꜱɪᴏɴ ᴏꜰ η = ᴍʟ⁻¹ᴛ⁻¹

ɴᴏᴡ, ᴘᴜᴛᴛɪɴɢ ᴅɪᴍᴇɴꜱɪᴏɴ ᴏɴ ʙᴏᴛʜ ꜱɪᴅᴇꜱ ᴏꜰ ᴇQᴜᴀᴛɪᴏɴ,

[ʟᴛ⁻¹] = [ʟ]ᵃ[ᴍʟ⁻³]ᵇ[ᴍʟ⁻¹ᴛ⁻¹]ⁿ

[ᴍ°ʟᴛ⁻¹] = [ᴍᵇ⁺ⁿ][ʟᵃ⁻ⁿ⁻³ᵇ][ᴛ⁻ⁿ]

ᴏɴ ᴄᴏᴍᴘᴀʀɪɴɢ,

ɴ = 1, ᴀ - ɴ - 3ʙ = 1 ᴀɴᴅ ʙ + ɴ = 0

ʙ = -ɴ = -1

ɴᴏᴡ, ɪɴ ᴀ - ɴ - 3ʙ =1,

ᴀ - 1 - 3(-1) = 1

ᴀ + 2 = 1

ᴀ = -1

ᴛʜᴜꜱ, ᴠ = ᴋη/ʀ²ρ.

ᴛʜɪꜱ ᴡɪʟʟ ʙᴇ ᴛʜᴇ ʀᴇʟᴀᴛɪᴏɴ. ɴᴏᴡ, ᴋ ɪꜱ ᴄᴏɴꜱᴛᴀɴᴛ. ɪᴛꜱ ᴠᴀʟᴜᴇ ᴡᴀꜱ ᴅᴇᴛᴇʀᴍɪɴᴇᴅ ʙʏ ᴘʜʏꜱɪᴄɪꜱᴛ 'ʀᴇʏɴᴏʟᴅꜱ' ᴀɴᴅ ʜᴇɴᴄᴇ ᴄᴀʟʟᴇᴅ ᴀꜱ ʀᴇʏɴᴏʟᴅꜱ ɴᴜᴍʙᴇʀ.

Answer 2

Explanation:

in a steady flow, the viscous force between the layers of a flood is F = ? A(dv/dx), where ? is the coefficient of viscosity of the liquid, A the area of the layer of the liquid and (dv/dx) the velocity gradient.

The steady or streamline flow is possible only when the liquid velocity is less than a limiting value known as critical velocity.

The flow of a liquid is steady or streamline only when the velocity of the liquid is less than the critical velocity. It was Osborne Reynold who first showed that the critical velocity Vc of a liquid flowing through a narrow is given by the relation

Vc = Re? /PD

Where Re is a constant called Reynold’s number, ? the coefficient of viscosity of the liquid (fluid), D the diameter of the tube and P the density of the liquid.