Non-dimensional variables

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows  

Choose the characteristic velocity m so, that total heat flux on the wall is fully expended for liquid evaporation (the heating without any losses of heat r = 1). We conclude that 

The system of Eqs. (10.22-10.24) and relations (10.26-10.31) contains five dimensionless parametrical groups Eu, Fr, We, 7g and Jl, which completely determine the problem. 

10 Laminar Flow in a Heated Capillary with a Distinct Interface 

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Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

Figure 16.3 shows the dependence of the QY from the non-dimensional variables y and w. The increase of both acceptor and donor concentration, until back reactions become important (outside the simple model hypotheses), or the decrease of the absorbed light [Pg.359]

The hemispherical core of radius a is immediately followed by a plastic zone. The plastic-elastic boundary lies here at a radius c, where c > a. The model allows the Pm/ T ratio to be related to a single non-dimensional variable ( tan 6)/ T, where is the contact angle between the sample and the indenter = 19.7° for a Vickers indenter). For v = 0.5, Johnson s analysis leads to ... 

To determine which terms are large and which are small, we cast the equations into non-dimensional variables with scales that are governed by the dynamics of the flow. The appropriate scales for non-dimensionalizing the variables are usually found from the tube geometry (e.g., say, a characteristic length L), the boundary conditions and from detailed analysis of the equations that govern the flow [119]. 

Most of the non-dimensional variables are then formed in a straightforward manner ... 

The pressure scale is usually found indirectly by substituting all the other non-dimensional variable definitions into the momentum equation. 

After all the non-dimensional variables are substituted into the equations, several non-dimensional groups occur b Re =, Pr =, 7o =... 

For any given flow problem these parameters have specific fixed values. If they are large or small, they magnify or diminish the effect of the terms in which they appear as coefficients. In non-dimensional variables the equations are written as ... 

The final form of the mathematical problem statement in non-dimensional variables is as follows ... 

After suitable non-dimensional variables are substituted into the equations, following the same procedure as outlined in sect 1.2.5, the important dimensionless groups are obtained for the problem in question. These are the Reynolds number, the Schmidt number, the Peclet number, Pe = Re Sc = ul/D, and the Damkohler number, Daj = Ir/u. The u and I are the characteristic velocity and length scales, respectively, for the velocity field, and r denotes a characteristic chemical reaction rate. 

MPa, which has been observed in HMX combustion data, whereas the 1 model does not. The reason is that as pressure drops below 1 MPa, the burning rate for Eg 1 becomes increasingly sensitive to condensed phase reaction kinetics and does so in a continuous fashion. One trend that is obscured by using non-dimensional variables as presented here is that dimensional temperature sensitivity Gp = k/ T - Tg) is also sensitive to radiant flux qr in the range of... 

The model equations are eqs. (9.2-45). Because of the nonlinearity of the equations, they must be solved numerically and before this is done it is convenient to cast them into a nondimensional form. Let Cq be some reference concentration. We define the following non-dimensional variables and parameters ... 

With the above definitions of non-dimensional variables and parameters, the resulting non-dimensional governing equations take the following form ... 

The mass balance equation (eqs. 9.5-23) and its boundary and initial conditions (eqs. 9.5-24) are cast into the non-dimensional form for the subsequent collocation analysis. We define the following non-dimensional variables ... 

The Working Nondimensional Mass and Heat Balance Equations We define the following non-dimensional variables and parameters... 

The above equations (10.2-34 to 10.2-36) are non-linear due to the thermodynamic correction factor in the transport diffusivity term. The method we have been using in solving nonlinear partial differential equations is the orthogonal collocation method. We again apply it here, and to do so we define the following non-dimensional variables and parameters ... 

Nevertheless, the temperature and pressure variables need further consideration. The pressure scale is usually found indirectly by substituting all the other non-dimensional variable definitions into the momentum equation. Accordingly, after substituting the dimensionless variables into the momentum equation, the non-dimensional pressure is defined as p =. Ifps symbolizes the pressure scale, we... 

Non-dimensional variables These denote the relation between physical variables that are non-dimensional according to condition (b) in the above. Example Reynolds number Re = vpllt] (1) and the Fourier number Fo = Xr/gcS (1). 

The basic principle of dimensional analysis is as follows -a given set of dimensional variables describing a physical phenomenon is rewritten to a reduced set of non-dimensional variables. 

The relative degree of cooling (A / o) can be fully described as a function of only two independent non-dimensional variables... 

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