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Viscosity

Storyboard

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Momentum transfer between Layers of Gas

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Viscosity as momentum exchange

Equation

If the speed at a point z is v_x (z) and at a neighboring point z + dz it is v_x (z + dz) it will be taken that the particles at a distance of a free path l can redistribute the moment:

$mdv_x = m(v_x(z + dz) - v_x(z))$



The number of particles participating in this process is equal to those found in a volume of section S and height equal to the free path l :

$S l c_n$



Therefore, the force F will be equal to the moment change in dp and the time dt

$F=\displaystyle\frac{dp}{dt}$



so the slimy force is

$F=-Slc_nm\displaystyle\frac{dv_x}{dt}$

where the negative sign is because the force is opposite to the direction of flow.

ID:(3944, 0)


Microscopic viscosity model

Equation

If the force F created by the mixture of particles at different times is considered

$F=-Slc_nm\displaystyle\frac{dv_x}{dt}$



where S is the section, l the free path, c_n the concentration, m the mass of the particles and dv_x the variation in time dt . This expression can be re-formulated if the acceleration is rewritten as

$F=-S,l,c_nm\displaystyle\frac{dv_x}{dz}\displaystyle\frac{dz}{dt}$



The derivative of the z position with respect to time can be modeled using

$\displaystyle\frac{dz}{dt}=v_z=\displaystyle\frac{1}{3}\sqrt{\langle v^2\rangle}$



In this way the force created by the mixture of moments like

$F=-\displaystyle\frac{1}{3}S,l,c_nm\sqrt{\langle v^2\rangle}\displaystyle\frac{dv_x}{dz}$



If you compare this expression with the viscous force

$F=-S,\eta\displaystyle\frac{dv_x}{dz}$



it is concluded that the viscosity has to be

$\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}$

where the negative sign is because the force is opposite to the direction of flow.

ID:(3945, 0)


Viscosity as a function of temperature

Equation

If the viscosity is

$\eta=\displaystyle\frac{1}{3}lc_nm\sqrt{\langle v^2\rangle}$



with l the free path, c_n the concentration, m the mass and \ langle v ^ 2 \ rangle the expected value of the square of the velocity. With the expression for the free way

$l=\displaystyle\frac{1}{\sqrt{2}\pi d^2c_n}$



the viscosity as a function of the temperature will be:

$\eta=\displaystyle\frac{1}{6\pi d^2}\sqrt{fmkT}$

where the negative sign is because the force is opposite to the direction of flow.

ID:(3946, 0)