Perturbation, velocity

As a result of losses the state which occurs in fact differs from the point B on the Hugoniot adiabate calculated without losses we should have called it B. When drag is taken into account the line of state change in the reaction YB differs from a straight line. However, in the state B the basic property which distinguished the state B in the theory without losses is preserved, namely the condition of equality of the detonation velocity and perturbation velocity, D = w + c. 

If these vortices are considered as potential vortices, then the induced perturbation velocity in the inviscid part of the flow is given by,... 

The necessary condition for the creation of a forerunner is found by looking at the receptivity solutions for points C and D, with the former having a single stable mode and latter with two damped modes. Results are shown in Fig. 4.4 for the streamwise perturbation velocity, at the indicated large time. 

Further properties of spatio-temporally growing wave-front were studied in Sengupta et al. (2006a). It was investigated by exciting the Blasius boundary layer at a frequency that corresponds to the point on branch 11 of the neutral curve at wq = 0.1307 for Re = 1000. Computed streamwise perturbation velocity at different time instants are shown in Fig. 4.7. 

Imagine that a linear symmetric shear field M -° r) is applied to the system so that the velocity u r) at position r outside the particle core is given by the sum of u ° r) and a perturbation velocity. Thus, u(r) obeys the boundary condition... 

Hint. The angle between the cone and the plate is very small. You will need to rescale the problem for very flat cones to get a tractable problem. It is useful to define the dimensionless angle 9 = [(n/2) — a9, where 9 varies between 0 at the flat plate and 1 at the cone. You then take the limit as a - 0 and get rid of a lot of the nasty terms in spherical polar coordinates. You will find that, to do this properly, the perturbation velocities must be rescaled with a, with ur proportional to a2 and (from the continuity equation) u9 proportional to a3. After the rescaling all the complex stuff is ()(a2) and can be neglected. 

Kirkwood and Riseman 41) were the first to introduce this kind of perturbation into polymer kinetic theory, based on earlier hydro-dynamic studies by Burgers (/3) and Oseen (6/). The perturbation velocity v is given by ... 

The increased viscosity in a shear flow due to the introduction of particles can be associated with the increased energy dissipation due to the introduced solid boundaries. For a dilute solution the total dissipation with the particles present can be taken to be the energy dissipation per unit time without the particles plus the power dissipation associated with rotating the particles relative to the flow. From this and the fact that for a small volume fraction of spherical particles the perturbation velocity induced by a particle is u a y r where a is the particle radius, r is the radial distance, and y is the applied shear rate, show that the functional form for the relative viscosity is given by 17 = 1 -I- Be/), where B is a constant and is the particle volume fraction. 

In the next step of the analysis, the forms for the perturbation velocity and temperature are taken to be satisfied by normal modes... 

The solution of the Stokes equations with the given boundary conditions results in the following expressions for the components of perturbed velocity... 

The specific values to be used for the dielectric constants, explicitly or implicitly present in both equations (46) and (47), strongly depend on the times related to the perturbation of interest or tetter on the ratio perturbation velocity/relaxation velocity. 

The angular momentum equation is not considered in Martin s treatment. which immediately introduces an inconsistency. If the angular perturbation velocity component v is neglected, Eq. (125) indicates that the term Qu (which arises from the Coriolis force) is also negligible, even though this term is of comparable magnitude to the right-hand side of Eq. (163). 

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