Non-linear transport equations in gaseous medium

Theoretical studies had been undertaken [17, 18] for analysing transport equations for mass flux as well as energy flux in continuous systems although experimental studies are not available for the purpose. 

The state of a gas is characterized by the distribution function / which depends on the peculiar velocity, the radius vector and the time t. For a stationary gas, the distribution function is Maxwellian and is denoted by f , which is given by 

This is also true for mass flux. Only those terms are important which are associated with the force corresponding to symmetrical non-divergent part of the viscous pressure tensor and space derivative of X . These can become important in capillaries. It seems that linear transport equations have larger domain of validity than expected. 

An examination of the derived flux equations [18] leads to the following conclusions (i) Onsager reciprocity relation (ORR) is obeyed, (ii) All coefficients are scalar, (iii) Higher powers of single force do not occur, (iv) Space derivatives of forces occur in the transport equations, (v) In none of the X, Xj terms, the tensorial order of X and Xj term is the same, (vi) All the X,X terms have the same tensorial order as the fluxes, (vii) Non-linearity arises on account of gradient of barycentric velocity. 

The above conclusions have serious limitations, which are as follows  

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