Scalar Static Equations

As mentioned earlier, the approach of this book is to treat problems with bifurcations as the general case and problems without bifurcation as special cases. 

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The quantity A appears in these equations and is the vector potential of electromagnetic theory. In a very elementary discussion of the static electric field we are introduced to the theory of Coulomb. It is demonstrated that the electric field can be written as the gradient of a scalar potential E = —Vc)>, constant term to this potential leaves the electric field invariant. Where you choose to set the potential to zero is purely arbitrary. In order to describe a time-varying electric field a time dependent vector potential must be introduced A. If one takes any scalar function % and uses it in the substitutions... 

In the absence of interactions, electrons are described by the Dirac equation (1928), which rules out the quantum relativistic motion of an electron in static electric and magnetic fields E= yU and B = curl A (where U and A are the scalar and vectorial potentials, respectively) [43-45]. As the electrons involved in a solid structure are characterized by a small velocity with respect to the light celerity c (v/c 10 ) a 1/c-expansion of the Dirac equation may be achieved. More details are given in a paper published by one of us [46]. At the zeroth order, the Pauli equation (1927), in which the electronic spin contribution appears, is retrieved then conferring to this last one a relativistic origin. At first order the spin-orbit interaction arises and is described by the following Hamiltonian... 

Newtonian constitutive equation, (2 80), that the normal component of the surface force or stress acting on a fluid element at a point will generally have different values depending on the orientation of the surface. Nevertheless, it is often useful to have available a scalar quantity for a moving fluid that is analogous to static pressure in the sense that it is a measure of the local intensity of squeezing of a fluid element at the point of interest. Thus it is common practice to introduce a mechanical definition of pressure in a moving fluid as... 

We now have the necessary tools to discuss the transformation of the Dirac Hamiltonian. We consider the time-independent equation with a static scalar potential, (4.74)... 

In the present eonfiguration, air inlet boundaries are assumed to be Pressure Inlet while outflow boundaries are assumed Pressure Outlet . Pressure inlet boundary conditions were used to define the total pressure and other scalar quantities at flow inlets. Pressure outlet boundary conditions were used to define the static pressure at flow outlets. At the nozzle inlet, the air pressure was varied. At the nozzle outlet, the pressure was supposed to be the external pressure (one atmosphere). At the wall of the nozzle standard wall function boundary condition was applied. Although the high velocity of air stream was a heat source that will increase the temperature in the nozzle, the nozzle length was very short and the process oecurs in a very short time. For simplification, it was assumed that the process is adiabatic i.e. no heat transfer occurred through walls. The flow model used was viscous, compressible airflow [1, 6-10]. The following series of equations were used to solve a compressible turbulent flow for airflow simulation [1,6-12] ... 

The divergence is again taken with respect to the second index and y is an arbitrary scalar which absorbs all contributions parallel to the director n. Noting the first of EQNS (35) this is effectively the Euler-Lagrange equation of static theory with a dynamic term, g, added. We can also rewrite the first of EQNS (S) representing balance of linear momentum by substituting expression (17) for the stress tensor, and adding an inner product of EQN (38) with Vn to obtain... 

The static (and dynamic) theory of SmC will be developed using the vectors a and c which are subject to the constraints contained in equations (6.3) and (6.4). These constraints will lead to four Lagrange multipliers in the theory three scalar function multipliers arising from the three constraints in (6.3) and one vector function multiplier arising from the vector constraint in (6.4). Knowledge of the behaviour of a and c is sufficient to derive the orientation of the usual director n through the relation (6.1). 

Equation (6.78) represents a balance of forces, while, similar to the static theory of nematics, equations (6.79) and (6.80) are equivalent to a balance of moments see also Remark (i) below. Notice that the a-equations in (6.79) are coupled to the c-equations in (6.80) via the multiplier p. Recall that the Lagrange multipliers 7, p and T are scalar valued functions while is a vector function. 

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