Weakly nonlinear waves in fluid-filled tapered elastic tubes

Akışkanla dolu yarıçapı değişken elastik tüplerde zayıf nonlineer dalgalar

Abstract

The striking feature of the arterial blood flow is its pulsatile character. The intermittent ejection of blood from the left ventricle produces pressure and flow pulses in the arterial tree. Experimental studies reveal that flow velocity in blood vessels largely depends on the elastic properties of the vessel wall and they propagate towards the periphery with a characteristic pattern. The studies in the existing literature treated the arteries as circularly cylindrical long thin tubes. In essence, the arteries have variable radius along the axis of the tube. In the present work, treating the arterial tree as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the reductive perturbation method, the propagation of weakly nonlinear waves in such a fluid-filled elastic tube is studied. Assuming that the problem of concern is a boundary value problem, a coordinate stretching is introduced to the field equations. Furthermore, the field quantities are expressed as some asymptotic series of a smallness parameter $epsilon$. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that these types of equations admit a solitary wave type of solution with variable wave speed. It is observed that the wave speed increases with distance for narrowing tube while it decreases for expanding tube.

Özet

Bu çalışmada, kan, sıkışmaz ve viskoz olmayan bir akışkan ve büyük damarlar, değişken yarıçaplı, ön gerilmeli, ince, dairesel konik elastik tüp olarak kabul edilerek, böyle bir tüp içerisinde zayıf nonlineer dalgaların yayılımı indirgeyici pertürbasyon yöntemi kullanılarak incelenmiş ve evolüsyon denklemi olarak değişken katsayılı Korteweg-de Vries (KdV) denklemi elde edilmiştir. Bu evolüsyon denkleminin yalnız dalga (solitary wave) tipi bir çözümü kabul ettiği gösterilmiş ve ilerleyen dalganın hızı belirlenmiştir. İlerleyen dalga hızının yarıçapı genişleyen tüpler için, orijinden uzaklaştıkça azaldığı; buna karşılık yarıçapı daralan tüpler için orijinden uzaklaştıkça arttığı gözlenmiştir.