How is viscosity the diffusion of momentum

Viscosity of an ideal gas

The viscosity of (ideal) gases is based on the impulse transport due to diffusion processes between the individual fluid layers

Definition of viscosity

In the article viscosity, the cause of viscosity was mainly attributed to the forces of attraction between the layers of a fluid. These act in a similar way to frictional forces, so that there is a mutual deceleration of the individual fluid layers. For a clear definition of the viscosity, one can consider a fluid between two plates. The lower plate rests and the upper plate is moved at a constant speed.

Because of the adhesion condition, both the top and bottom fluid layers adhere to the plates. The lower fluid layer thus remains at rest and the uppermost fluid layer moves at the same speed as the upper plate. In between, a linear speed profile is then formed. A considered fluid layer thus always moves more slowly than the fluid layer immediately above it. Because of the molecular forces of attraction, a lower layer is always trying to slow down the layer above.

These frictional forces between the layers must be compensated accordingly if the top plate is to be pulled at a constant speed. The more viscous (viscous) a fluid, the stronger the internal frictional forces and the greater the forces required to move the top plate. The viscosity η gives the relationship between the area-related force F / A (shear stress τ), which is necessary for shifting the layers, and the gradient of the speed profile dv / dy (speed gradient):

\ begin {align}
\ label {t}
& \ frac {F} {A} = \ boxed {\ tau = \ eta \ cdot \ frac {\ text {d} v} {\ text {d} y}} ~~~~~ \ text {Newton's law of friction} \ [5px]
\ end {align}

Viscosity of gases due to impulse transport

The creation of viscosity through frictional forces acting between the fluid layers is very clear in the case of liquids. In gases, however, the molecules exert almost no forces of attraction on one another. Frictional forces of individual fluid layers due to intermolecular forces of attraction are almost non-existent. Practice shows, however, that even gases have a considerable viscosity and that individual fluid layers are slowed down in flows. How can this behavior be explained in the absence of attraction?

The decelerating effect of the fluid layers in gases mainly comes about through the impulse transport of the gas molecules when they diffuse from a slower layer into a faster layer. This process also takes place in liquids; however, it is negligible in comparison to the braking effect due to the attractive forces.

Let us look again at the already mentioned layered flow (laminar flow), which this time consists of an ideal gas as a fluid. If one gas molecule collides with another in this gas, momentum exchange takes place, i.e. a slower molecule takes up part of the momentum of the faster molecule. However, such impulse transport does not only take place within a fluid layer. Due to the disordered Brownian molecular movement, molecules also diffuse into neighboring fluid layers. What happens if a slower molecule diffuses into a layer of faster particles? The fast molecules are slowed down on this gas particle that has diffused in, and the overall layer is slowed down.

You can visualize the situation with a cart and a ball. The car is supposed to be pulled at a constant speed when suddenly the heavy ball is placed in it while driving past. The ball stands for the slower gas molecule (in this case even standstill) that diffuses into the faster fluid layer (illustrated by the car). Since the ball has a slower speed than the carriage when it is inserted, the ball must be accelerated to the carriage speed after it has been inserted if the speed is to remain constant. This requires a force corresponding to the mass of the ball (force = mass x acceleration). The insertion of the slow ball into the faster carriage thus causes a resistance that is noticeable by an additional force that has to be applied. If this force were not applied, the car would be braked on the slower ball, similar to a frictional force.

Diffusion processes of gas molecules between the individual layers of a laminar flow lead to a transport of impulses. The viscosity of gases is largely based on this impulse transport!

Faster layers pass on part of their momentum by diffusion into slower layers. Overall, there is a pulse transport from the moving plate to the plate at rest. This momentum current, which ultimately corresponds to a force between the layers, flows, so to speak, in the direction of decreasing speed, i.e. against the speed gradient. The transport of the impulse becomes particularly clear when the upper plate is set in motion from rest. Initially, only the layer directly adhering to the upper plate is set in motion. By transporting the impulse to the layer below, it begins to move, etc. The impulse thus gradually spreads through the layers.

The resistance force when moving the plate finally comes about because the setting of the gas layers in motion is hindered by the fact that the lower plate is fixed. Ultimately, holding the plate in place requires the same force as maintaining the movement of the top plate. The impulse is introduced, so to speak, at the upper plate and is transmitted from layer to layer and finally flows out again at the lower plate. In the state of equilibrium, there is ultimately no net momentum current that is transferred to the fluid layers, so that these ultimately move at a constant (but different) speed.

Derivation of the viscosity of ideal gases

Voltage as pulse current density

The force F can generally be determined from the change in momentum per time:

\ begin {align}
& F = \ frac {\ text {d} p} {\ text {d} t} = \ dot p ~~~~~ \ Rightarrow ~~~~~ \ boxed {\ text {force = momentum current}} \ [ 5px]
\ end {align}

The force on the individual fluid layers comes about through the change in the momentum, caused by the momentum transport due to diffusion processes, and can therefore also be called Impulse current understand p *. If you relate the force and thus the momentum current to the surface, you ultimately get a shear stress, which then turns out to be Momentum current density p *A. can be interpreted (change in momentum per unit of time and area):

\ begin {align}
& \ tau = \ frac {F} {A} = \ frac {\ dot p} {A} = \ dot p_ \ text {A} ~~~~~ \ Rightarrow ~~~~~ \ boxed {\ text { Voltage = pulse current density}} \ [5px]
\ end {align}

Newton's law of friction (\ ref {t}) can thus also be represented as follows:

\ begin {align}
\ label {tt}
& \ boxed {\ dot p_ \ text {A} = - \ eta \ cdot \ frac {\ text {d} v} {\ text {d} y}} \ [5px]
\ end {align}

The negative sign was inserted at this point in order to do justice to the fact that the momentum current is transferred away from faster layers to slower layers, i.e. in the direction of decreasing velocity gradients. At this point, an interesting analogy can be drawn to other transport processes such as heat transport and mass transport, which are ultimately described in the same way:

Impulse transport between the layers

Kinetic gas theory can be used to calculate the viscosity of ideal gases. To derive the formula we consider the stratified flow already mentioned, in which an ideal gas is located between two plates. The lower plate is fixed and the upper plate moves at a constant speed.

We consider the flow on a microscopic level and move with the individual fluid layers. The average distance traveled by the gas particles between two collisions is called the mean free path λ. We therefore consider gas layers that are at a distance λ from one another, so that during diffusion processes there is a collision within these layers and thus a transport of impulses. We now consider a layer at any height y. The mean velocity of the gas molecules in the x-direction with respect to a stationary coordinate system (plate at rest) is given by vx(y) denotes.

The mean speed vx(y + λ) of the gas particles at a distance λ in the layer above can be determined using the velocity gradient dv / dy:

\ begin {align}
& v_ {x} (y + \ lambda) = v_ {x} (y) + \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ [5px]
\ end {align}

The mean speed vxDetermine (y-λ) of the gas particles at a distance λ in the layer below:

\ begin {align}
& v_ {x} (y- \ lambda) = v_ {x} (y) - \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ [5px]
\ end {align}

Denotes n *A. the particle flux density, i.e. the number of particles per unit of time and area that diffuses from the upper layer or lower layer into the middle layer, then the corresponding momentum currents can be determined using the following formula. Note that the particle flux density is identical for both layers if we assume an incompressible gas flow, in which the particle density is the same at every point of the flow.

\ begin {align}
& \ dot p_ {A} (y + \ lambda) = \ dot n_ \ text {A} \ cdot \ overbrace {m \ cdot v_ {x} (y + \ lambda)} ^ {\ text {momentum of a particle}} = \ dot n_ \ text {A} \ cdot m \ cdot \ left (v_ {x} (y) + \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ right) \ \ [5px]
& \ dot p_ {A} (y- \ lambda) = \ dot n_ \ text {A} \ cdot m \ cdot v_ {x} (y- \ lambda) = \ dot n_ \ text {A} \ cdot m \ cdot \ left (v_ {x} (y) - \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ right) \ [5px]
\ end {align}

The net momentum current p *A.(y) in the layer at height y results from the sum of the two pulse currents. According to the chosen coordinate system, the pulse current directed from bottom to top (pointing in the positive y-direction) also corresponds to a positive value and the downward directed pulse current corresponds to a negative value. The following therefore applies to the net impulse current:

\ begin {align}
\ dot p_ \ text {A} (y) & = \ dot p_ {A} (y- \ lambda) ~ - ~ \ dot p_ {A} (y + \ lambda) \ [5px]
& = \ dot n_ \ text {A} \ cdot m \ cdot \ left (v_ {x} (y) - \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ right ) - \ dot n_ \ text {A} \ cdot m \ cdot \ left (v_ {x} (y) + \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ right ) \ [5px]
& = \ dot n_ \ text {A} \ cdot m \ cdot \ left (v_ {x} (y) ~ - \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} ~ - ~ v_ {x} (y) ~ - \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y} \ right) \ [5px]
\ end {align}

\ begin {align}
& \ boxed {\ dot p_ \ text {A} = - 2 ~ \ dot n_ \ text {A} \ cdot m \ cdot \ lambda \ cdot \ frac {\ text {d} v} {\ text {d} y }} ~~~ \ text {net impulse current} \ [5px]
\ end {align}

Viscosity of ideal gases as a function of the particle flow density

According to the above equation, the net momentum flow is obviously no longer a function of the variable y and is therefore identical at every point of the flow! If one compares this formula with Newton's law of friction (\ ref {tt}), one immediately shows that the expression 2⋅n *A.⋅m⋅λ obviously corresponds to the viscosity η:

\ begin {align}
& \ dot p_ \ text {A} = - \ eta \ cdot \ frac {\ text {d} v} {\ text {d} y} \ [5px]
& \ dot p_ \ text {A} = - \ underbrace {2 ~ \ dot n_ \ text {A} \ cdot m \ cdot \ lambda} _ {\ eta} \ cdot \ frac {\ text {d} v} { \ text {d} y} \ [5px]
\ label {eta}
& \ boxed {\ eta = 2 ~ \ dot n_ \ text {A} \ cdot m \ cdot \ lambda} ~~~ \ text {Viscosity of ideal gases} \ [5px]
\ end {align}

The viscosity of an (ideal) gas is therefore only dependent on the mass of a gas particle, the mean free path and the particle flow density. The area-related particle flow n *A., which diffuses in from a layer above or below, in turn depends on how strongly the gas molecules move due to the disordered diffusion movement (Brownian molecular movement). This in turn is determined by the temperature.

If the temperature is high, more diffusion processes take place and more particles diffuse between the layers. The particle flux density is correspondingly high and thus also the momentum transport. This becomes noticeable in an increasing force that is required to maintain the macroscopic flow (movement of the plate)! The viscosity of gases therefore generally increases with temperature and not decreases as with liquids!

This also shows the fact that with ideal gases the pressure has no influence on the viscosity. Although the particle density and thus the diffusing particle flow increases proportionally with increasing pressure, the mean free path decreases to the same extent. Both effects cancel each other out.

With ideal gases the viscosity is independent of the pressure and increases with increasing temperature!

Viscosity of ideal gases as a function of temperature

At this point we would like to explicitly derive the quantitative dependence of the viscosity of ideal gases on the temperature. For this purpose, a connection must be found between the particle current density diffusing perpendicular to the flow and the temperature.

To do this, we move in thought with a layer so that it rests relative to us as an observer. However, the gas particles themselves are by no means at rest on a microscopic level. Due to Brown’s molecular movement, they move in all directions in a completely disordered manner. According to the Maxwell-Boltzmann distribution, the mean velocity of a gas particle is vT related to the temperature T of the gas as follows:

\ begin {align}
\ label {a}
& \ boxed {\ overline {v_ \ text {T}} = \ sqrt {\ frac {8 k_B T} {\ pi m}}} ​​~~~ \ text {arithmetic mean speed} \ [5px]
\ end {align}

In this equation, m denotes the mass of a gas particle and kB. the Boltzmann constant. Note that the speed vT represents the mean velocity relative to the moving fluid layers and does not include the superposition of the macroscopic flow movement. The latter has no influence on the temperature anyway; After all, the temperature of a gas does not depend on whether the gas is at rest or whether it is moving.

Let us now consider a directed flow in which all particles move in the same direction with the (mean) velocity v. The number of particles that flow per unit of time and area (particle flux density) can be determined as follows. For this purpose we consider a surface element dA through which the particles flow within a time dt with the (mean) velocity v. The particles cover the distance dl = v⋅dt. Thus the particles obviously flow through the following volume dV:

\ begin {align}
& \ text {d} V = \ text {d} A \ cdot \ text {d} l = \ text {d} A \ cdot \ overline {v} \ cdot \ text {d} t \ [5px]
\ end {align}

For a given particle density n (number of particles per unit volume), the following number of particles dN are in this volume or flow through this volume:

\ begin {align}
& \ text {d} N = n \ cdot \ text {d} V = n \ cdot \ text {d} A \ cdot \ overline {v} \ cdot \ text {d} t \ [5px]
\ end {align}

The number of particles passing through per unit of time and area (particle flux density n *A.) by a surface directed perpendicular to the flow results as follows:

\ begin {align}
& \ dot n_ \ text {A} = \ frac {\ text {d} N} {\ text {d} A \ cdot \ text {d} t} = n \ cdot \ overline {v} \ [5px]
& \ boxed {\ dot n_ \ text {A} = n \ cdot \ overline {v}} ~~~ \ text {Particle flux density of a directed movement} \ [5px]
\ end {align}

Let us now look again at our stratified flow, with which we move with our thoughts. From this point of view, the flow is no longer directed, but with an average velocity vT completely disordered. The particles move equally in all spatial directions. This means that only a sixth of the particles move downwards and diffuse into a gas layer below. The particle flow density directed perpendicular to the main flow is therefore only one sixth as large:

\ begin {align}
\ label {na}
& \ boxed {\ dot n_ \ text {A} = \ frac {1} {6} n \ cdot \ overline {v_ \ text {T}}} ~~~ \ text {particle flux density of a disordered movement} \ [5px ]
\ end {align}

Inserting (\ ref {a}) into equation (\ ref {na}) finally results in the following diffusing particle current density as a function of temperature:

\ begin {align}
& \ dot n_ \ text {A} = \ frac {1} {6} n \ cdot \ underbrace {\ sqrt {\ frac {8 k_B T} {\ pi m}}} ​​_ {\ overline {v_ \ text { T}}} \ [5px]
\ end {align}

Inserting this equation into the formula (\ ref {eta}) shows the following relationship between viscosity and temperature:

\ begin {align}
& \ eta = 2 ~ \ dot n_ \ text {A} \ cdot m \ cdot \ lambda \ [5px]
& \ eta = 2 ~ \ frac {1} {6} n \ cdot \ sqrt {\ frac {8 k_B T} {\ pi m}} \ cdot m \ cdot \ lambda \ [5px]
\ label {ac}
& \ boxed {\ eta = \ frac {1} {3} n \ cdot \ sqrt {\ frac {8 k_B m T} {\ pi}} \ cdot \ lambda} \ [5px]
\ end {align}

Finally, the mean free path λ can be expressed by the particle density n and the diameter of the gas molecules d (derivation see article Mean free path & impact number):

\ begin {align}
& \ boxed {\ lambda = \ frac {1} {\ sqrt {2} ~ n ~ \ pi d ^ 2}} \ [5px]
\ end {align}

If you insert this formula into equation (\ ref {ac}), the following formula for calculating the viscosity of ideal gases is shown:

\ begin {align}
& \ eta = \ frac {1} {3} n \ cdot \ sqrt {\ frac {8 k_B m T} {\ pi}} \ cdot \ lambda \ [5px]
& \ eta = \ frac {1} {3} n \ cdot \ sqrt {\ frac {8 k_B m T} {\ pi}} \ cdot \ frac {1} {\ sqrt {2} ~ n ~ \ pi d ^ 2} \ [5px]
& \ boxed {\ eta = \ sqrt {\ frac {4 k_B m ~ T} {9 \ pi ^ 3 ~ d ^ 4}}} \ [5px]
& \ boxed {\ eta \ sim \ sqrt {T}} \ [5px]
\ end {align}

In this formula it becomes obvious that the particle density and thus the pressure have no influence on the viscosity of ideal gases. Only the temperature as a variable parameter influences the viscosity. The viscosity increases proportionally with the square root of the temperature!

Note that this formula only applies to laminar flows in which the layers do not mix macroscopically and diffusion processes take place between the layers only on a microscopic level. In the case of turbulent flows, the exchange of momentum due to the turbulence is greater and the viscosity is correspondingly higher.

Comparison between viscosity and thermal conductivity

Equation (\ ref {na}) can also be inserted directly into the formula (\ ref {eta}) for the viscosity, so that it follows:

\ begin {align}
& \ eta = 2 ~ \ dot n_ \ text {A} \ cdot m \ cdot \ lambda \ [5px]
& \ eta = 2 ~ \ frac {1} {6} n \ cdot \ overline {v_ \ text {T}} \ cdot m \ cdot \ lambda \ [5px]
& \ eta = \ frac {1} {3} \ underbrace {n \ cdot m} _ {\ rho} \ cdot \ lambda \ cdot \ overline {v_ \ text {T}} \ [5px]
& \ boxed {\ eta = \ frac {1} {3} \ cdot \ rho \ cdot \ lambda \ cdot \ overline {v_ \ text {T}}} \ [5px]
\ end {align}

The derivation made use of the fact that the product of particle density and mass of a particle corresponds to the density ϱ of the gas. At this point there is an interesting analogy to the thermal conductivity k of ideal gases (in order to avoid confusion with the mean free path, the thermal conductivity was not designated with λ but with k):

\ begin {align}
& \ boxed {k = \ frac {1} {3} \ cdot \ rho \ cdot \ lambda \ cdot c_v \ cdot \ overline {v_ \ text {T}}} \ [5px]
\ end {align}

In principle, thermal conductivity follows the same principles as viscosity, i.e. it increases with increasing mean particle speed (increasing temperature). This is not surprising, since the diffusion of particles is not only associated with the transport of impulses, but also (heat) energy transport.