Fig. 3

Illustration of the time delay (\({T}_{\mathrm{delay}}\)) between the vial concentration and the fluid element concentration as a function of time \(t\) and the position of the fluid element in cylindrical coordinates \((s,r)\). A For \(t < {T}_{1}\), the fluid element traveled the distance \(s\) at the velocity \(\overrightarrow{{v}_{1}}(r)\). \({T}_{\mathrm{delay}}\) is thus expressed as the ratio between the fluid element curvilinear abscissa \(s\) and the velocity \(\overrightarrow{{v}_{1}}(r)({T}_{\mathrm{delay}}=\frac{s}{\overrightarrow{{v}_{1}}\left(r\right) \, })\). B For \({T}_{1}<\mathrm{t }\le {T}_{2}\), there are two cases depending on the curvilinear abscissa s of the fluid element: Case 1. \(s{\le D}_{2}(r,t)\), the fluid element has traveled the distance \(s\) at the velocity \(\overrightarrow{{v}_{2}}(r)\,({T}_{\mathrm{delay}}=\frac{s}{\overrightarrow{{v}_{2}}\left(r\right) \, })\). Case 2. \(s>{D}_{2}(r,t)\), the fluid element has traveled the distance \({D}_{2}(r,t)\) at the velocity \(\overrightarrow{{v}_{2}}(r)\) and the distance \(s-{D}_{2}(r,t)\) at the velocity \(\overrightarrow{{v}_{1}}(r)\) \(({T}_{\mathrm{delay}}=\frac{{D}_{2}(r,t)}{\overrightarrow{{v}_{2}}(r) \, }+\frac{s-{D}_{2}(r,t)}{\overrightarrow{{v}_{1}}(r) \, })\). C For \(t>{T}_{2}\), there are three cases depending on the curvilinear abscissa \(s\) of the fluid element. The mathematical expression of \({T}_{\mathrm{delay}}\) is obtained by following the same method as for \({T}_{1}<t \le {T}_{2}\)