Tangent equation

Note that K = 1 and Dm becomes D in aqueous media. Equation (79) suggests that under steady-state conditions the thickness of the film is a function of time. The tangent equation of the concentration curve at z = 0 is... 

If we extrapolate this slope toward low rates, where the tangent equates with the low shear viscosity and assume the Peclet number here is unity we eventually obtain a value of b = 2.55. This defines the Peclet number as unity, at a stress somewhere just after the curvature of the viscosity curve deviates from the low shear limit. This seems quite an appropriate reference system. By setting the Peclet number to the appropriate value of b we can determine the variation in packing fraction with stress between the high and low shear limits to the viscosity ... 

This is most often taken to be a nonlinear elastic relation again, a wide range of expressions have been used in CRTM studies, from power-laws to relations involving the hyperbolic tangent. Equation [11.3] can also express the viscoelastic properties of fibrous materials (and hence the possible dependence on the rate of change of volume fraction, Vf, in Eq. [11.3]) and... 

The last temi in the equation for (Ap) has been added to avoid adding a constant to doing so does not affect the principle, but makes figure A2.5.9 clearer.) Thus the connnon tangent satisfies tlie condition of equal chemical potentials, j- . (The connnon tangent also satisfies the condition that because p =... 

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

Figure A2.5.16 shows the ooexistenoe ourve obtained from equation (A2.5.16). The logaritluns (or the hyperbolio tangent) oan be expanded in a power series, yielding...

Since any straight line obeys the equation y = mx b, the equation for any tangent drawn in Fig. 2.5 will be... 

Asymptotes The hmiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that... 

These terms are obtained from die equation above by differentiation with respect to r, and setting the resultant equal to zero. This is equivalent to taking the point on the graph of the Gibbs energy of nucleus formation versus the size of the nucleus where the tangent has zero slope. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

In the limit of zero association, x — 0 the latter equation reduces to the adsorption isotherm of hard spheres, evaluated within the singlet Percus-Yevick approximation, whereas for xx 1 (i-S- the limit of complete association) one obtains the adsorption isotherm of tangent dimers... 

Tangents Method. The tangents on either side of the peak are drawn through the inflection points until the baseline. For an ideal Gaussian peak the resulting base line interval Wi, is equal to 4or. Equation (1) becomes with the width (Wb) ... 

A simple way of finding the roots of an equation, other than by divine inspiration, symmetry or guesswork is afforded by the Newton method. We start at some point denoted x l along the x-axis, and calculate the tangent to the curve at... 

Transition pressures were determined by equating enthalpies of different polymorphs via the common tangent construction. We found that ratile transforms to columbite structure at 11.8 GPa with a small volume change of 3%. These results agree well with experimental... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

The radius of curvature R of a plane curve at any point P is the distance along the normal (the perpendicular to the tangent to the curve at point P) on the concave side of the curve to the center of curvature (Figure 1-33). If the equation of the curve is y = f(x)... 

The velocity is always tangent to the path of motion, and thus the velocity vector has only one component (Equation 2-20). 

The acceleration vector has a component tangent to the path a = d v /d, which is the rate at which the magnitude of the velocity vector is changing, and a component perpendicular to the path a = v Vp, which represents the rate at which the direction of motion is changing (Equation 2-21). 

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