Physics MECHANICAL PROPERTIES OF FLUIDS

Please Wait... While Loading Full Video

MECHANICAL PROPERTIES OF FLUIDS

Viscosity

Consider motion of a fluid flow between two parallel plates. The bottom plate is stationary while top plate is moving with constant velocity `(vec v)` . The fluid in contact with plate surface has same velocity as that of surface so, top layer moves with velocity `(vec v)` and bottom most layer is stationary.

The flow velocity of intermediate layers of fluid increases uniformly from bottom to top.

'Viscosity is defined as the property of fluid by virtue of which it opposes the relative motion between its different layers'.

It is also called internal friction of fluid. Viscous force opposes the motion of one portion of fluid relative to the other.

According to Newton, the viscous force (F) acting tangentially to the fluid layer is

a. Directly proportional to the area (A) of the layers in contact

`F prop A`

b. Directly proportional to the velocity gradient `((dv)/(dy))` between the layers.

`F prop (dv)/(dy)`

Or, `F prop A (dv)/(dy), ` `F = - eta A (dv)/(dy)`

`eta` is constant of proportionally and is called coefficient of
viscosity.

`eta` depends on nature of fluid.

Negative sign implies that the direction of viscous force is opposite to the direction of relative velocity of layers.

`S.l.` unit of `eta` , is `N-s//m^2` It is also called pascal second (Pa-s) or decapoise.

`1` decapoise `= 1 N-s//m^2` `= 1 Pa-s = 10` `poise`

Dimensions of `eta` , are `[M^1L^-1T^-2]`

Viscosity of liquid decreases with Increase in temperature `eta prop 1/(sqrt T)`

Viscosity of gases increases with Increase in temperature.

`CGS` unit of `eta` is poise.

Consider motion of a fluid flow between two parallel plates. The bottom plate is stationary while top plate is moving with constant velocity `(vec v)` . The fluid in contact with plate surface has same velocity as that of surface so, top layer moves with velocity `(vec v)` and bottom most layer is stationary.

The flow velocity of intermediate layers of fluid increases uniformly from bottom to top.

'Viscosity is defined as the property of fluid by virtue of which it opposes the relative motion between its different layers'.

It is also called internal friction of fluid. Viscous force opposes the motion of one portion of fluid relative to the other.

According to Newton, the viscous force (F) acting tangentially to the fluid layer is

a. Directly proportional to the area (A) of the layers in contact

`F prop A`

b. Directly proportional to the velocity gradient `((dv)/(dy))` between the layers.

`F prop (dv)/(dy)`

Or, `F prop A (dv)/(dy), ` `F = - eta A (dv)/(dy)`

`eta` is constant of proportionally and is called coefficient of
viscosity.

`eta` depends on nature of fluid.

Negative sign implies that the direction of viscous force is opposite to the direction of relative velocity of layers.

`S.l.` unit of `eta` , is `N-s//m^2` It is also called pascal second (Pa-s) or decapoise.

`1` decapoise `= 1 N-s//m^2` `= 1 Pa-s = 10` `poise`

Dimensions of `eta` , are `[M^1L^-1T^-2]`

Viscosity of liquid decreases with Increase in temperature `eta prop 1/(sqrt T)`

Viscosity of gases increases with Increase in temperature.

`CGS` unit of `eta` is poise.

Poiseuille's Equation

The flow velocity profile for laminar flow of a viscous fluid in a long cylindrical pipe is shown.

The velocity of fluid layers is greatest along the axis and zero at the pipe walls.

Consider the equilibrium of cylindrical fluid element of radius (r) and thickness (dr).

Net pressure force on cylindrical element is due to Pressure difference `(Delta P)` across its ends = shear force on its periphery due to cylindrical lamina over it.

`Delta P (pi r^2) = - eta (2 pi rl)(dv)/(dr), ` `int_0^vdv = - (Delta P)/(2 eta l) int_r^R rdr`

`v = (Delta P)/(4 eta l) (R^2 -r^2)`

Where, `Delta P = P_1 - P_2` Pressure difference between two ends

The flow is always in direction of decreasing pressure.

v - r graph is parabolic.

At r = R, v = 0 (along the walls)

r = 0, `v = ((P_1-P_2)R^2)/(4 eta l) =v_(max)` (along the axis)

Volume flow rate (Q),

`Q = d/(dt) V` Where, `V =` volume of fluid

Consider a ring of radius `(r)` and thickness `(dr)`

Cross-section of ring `= dA = 2 pi r dr`

Volume flow rate through this element is `= vdA ,`

`v=` velocity of fluid

`Q = int_(r=0)^(r=R)v dA = intr_0^pi ((P_1-P_2))/(4 eta l)(R^2-r^2)(2 pi r dr))`

`Q = d/(dv) (V) = pi/8(R^4/(eta))((P_1-P_2)/l)`

`Q = (pi(P_1-P_2) R^2)/(8 eta l)`

This relation was first derived by Poiseuille and is called Poiseuille's equation.

`Q = (P_1-P_2)/R_t` where, `R_t= (8 eta l)/(pi R^4)`

`R_t ` is known as fluid resistance

The flow velocity profile for laminar flow of a viscous fluid in a long cylindrical pipe is shown.

The velocity of fluid layers is greatest along the axis and zero at the pipe walls.

Consider the equilibrium of cylindrical fluid element of radius (r) and thickness (dr).

Net pressure force on cylindrical element is due to Pressure difference `(Delta P)` across its ends = shear force on its periphery due to cylindrical lamina over it.

`Delta P (pi r^2) = - eta (2 pi rl)(dv)/(dr), ` `int_0^vdv = - (Delta P)/(2 eta l) int_r^R rdr`

`v = (Delta P)/(4 eta l) (R^2 -r^2)`

Where, `Delta P = P_1 - P_2` Pressure difference between two ends

The flow is always in direction of decreasing pressure.

v - r graph is parabolic.

At r = R, v = 0 (along the walls)

r = 0, `v = ((P_1-P_2)R^2)/(4 eta l) =v_(max)` (along the axis)

Volume flow rate (Q),

`Q = d/(dt) V` Where, `V =` volume of fluid

Consider a ring of radius `(r)` and thickness `(dr)`

Cross-section of ring `= dA = 2 pi r dr`

Volume flow rate through this element is `= vdA ,`

`v=` velocity of fluid

`Q = int_(r=0)^(r=R)v dA = intr_0^pi ((P_1-P_2))/(4 eta l)(R^2-r^2)(2 pi r dr))`

`Q = d/(dv) (V) = pi/8(R^4/(eta))((P_1-P_2)/l)`

`Q = (pi(P_1-P_2) R^2)/(8 eta l)`

This relation was first derived by Poiseuille and is called Poiseuille's equation.

`Q = (P_1-P_2)/R_t` where, `R_t= (8 eta l)/(pi R^4)`

`R_t ` is known as fluid resistance

SiteLock