Arbitrary function of time

It is interesting to consider the response of a Maxwell fluid to an arbitrary shear rate history. Denoting the shear rate as y(t), an arbitrary function of time, the equivalent of equation 3.83 is... 

The cylinder-wall circumferential velocity can be an arbitrary function of time, with the fluid velocity still subject to parallel-flow assumptions. The cylindrical analog of Stokes Second problem is to let the cylinder-wall velocity oscillate in a periodic manner. The wall velocity is specified as... 

In the derivation above, the input signal C (t) is an arbitrary function of time, without any restraint condition, but, of course, it should be known CAo is the response of the impinging stream device to CAi(r) that can be measured by sampling at the outlet of the device. 

The most important boundary condition in heat transfer problems encountered in polymer processing is the constant surface temperature. This can be generalized to a prescribed surface temperature condition, that is, the surface temperature may be an arbitrary function of time T 0, t). Such a boundary condition can be obtained by direct contact with an external temperature-controlled surface, or with a fluid having a large heat transfer coefficient. The former occurs frequently in the heating or melting step in most... 

The parameters are arbitrary functions of time ri(t), . In this case the direction of flow in each point in T-space changes continuously and the density distribution of Eq. (53) will not be stationary. 

A heat flow QH = Qn(t), that can be an arbitrary function of time, shall be added to the liquid for all t > 0. The heat flow Q(t) lost through the thin vessel walls is found using (1.87). These two quantities enter the balance equation... 

Here we must distinguish between time averages and position averages. A time average of some arbitrary function of time 4> is given by... 

Unfortunately, there is no very simple measurerrient we can make of the size, of an individual eddy, nor is there even a very diiject measurement of the average size of the eddies passing a given point. Therefore, two terms (the correlation coefficient and the scale of turbulence) have been defined as expressions of the average size of eddies. The correlation coefficient (borrowed from statistics, where it is widely used) is a measure o f how much of the time two variables coincide with each other. The correlation coefficient of two arbitrary functions of time and < 2(0 is ... 

In this expression, t is simply a dummy variable and can be replaced with any convenient symbol. Later, we shall inspect arbitrary functions of time /(t), instead of fix), so it is propitious to replace t with /3 to get... 

This form is now suitable for deducing the Fourier-MeUin inversion formula for Laplace transforms. In terms of real time as the independent variable, we can write the Fourier integral representation of any arbitrary function of time, with the provision that fit) = 0 when / < 0, so Eq. C39 becomes... 

From the definitions given in Section 4.4.2, it is apparent that the interfacial impedance can be calculated from the perturbation and response in the time domain, in which the excitation can be any arbitrary function of time. In principle, any one of several linear integral transforms can be used (Macdonald and McKubre [1981]) to convert from the time domain into the frequency domain, but the two most commonly used are the Laplace and Fourier transforms ... 

This description has the advantage that only four parameters are needed to specify the electromagnetic fields (the magnitude of (p and the three components of V), instead of the six components of E and B. The description is not unique because adding any arbitrary function of time to (p does not... 

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