Influence of Second Viscosity on Pressure Pulsation

Round 1

Reviewer 1 Report

This paper proposes a mathematical model of pulsating flow, with some application. Results appear to the referee original.

I have however some concerns:

Before Eq (3), the authors write V=S dx. I think that the correct expression is dV = S dx. So, the point: Does the last expression affects results in (2) and then (3)? After Eq. (12) in unclear the sentence: this velocity has a radial direction and is non-zero when the pressure change. As I can see, the radial velocity (12) is always non-zero if the pressure is present, independently from the fact that the pressure may change or not.  Eq. (16) is confusing for me (essentially due to the notation used by the authos that use \cdot in any expressions). According to (16), it should be a scalar equation, hpwever it appears \grad p which is a vector, and it is unclear also the expression for \Pi in (17): does the last term interpreted as (\div \cdot v) L or \grad (c \cdot L)? What is the function J_1 in Eqs. (23,25)? Before Eq. (31) appear abovementione that should be above mentioned. The presentation of  (31) is unclear. Maybe the auhtors should specify what P_m and P_p are (mentioneing that their explicit expressions are derived in the fiollowing analysis/discussion). 

Author Response

Thank you for your valuable comments, which help us to improve our paper. Changes are marked with red color.

 

 

Before Eq (3), the authors write V=S dx. I think that the correct expression is dV = S dx. So, the point: Does the last expression affects results in (2) and then (3)?

 

The suggested change was included into the text, into the Equation (2) and  Figure 1. The Equation 3 remains unaffected.

 

 

After Eq. (12) in unclear the sentence: this velocity has a radial direction and is non-zero when the pressure change. As I can see, the radial velocity (12) is always non-zero if the pressure is present, independently from the fact that the pressure may change or not.

 

The Equation (12) describes the wall velocity as a function of the pressure change (after the Laplace transform). So, when the liquid pressure changes, the pipe diameter also changes. When the pressure is constant the pipe diameter stays constant so the wall velocity is zero, then. When the inverse Laplace transform is applied the time derivative of the pressure occurs in the expression (12).

 

Eq. (16) is confusing for me (essentially due to the notation used by the authors that use \cdot in any expressions). According to (16), it should be a scalar equation, however it appears \grad p which is a vector, and it is unclear also the expression for \Pi in (17): does the last term interpreted as (\div \cdot v) L or \grad (c \cdot L)?

Equation (16) is a vector equation.  It has been changed into more convenient shape. The first term contains time derivation of the velocity vector, the second term contains expression (grad v).v, which also gives vector, the third term is div(\Pi) and the last term is grad p.

The round brackets have been added into the Equation (17).

 

What is the function J_1 in Eqs. (23,25)?

J_1 and J_0 are Bessel function. Now, it is mentioned below the equation.

 

Before Eq. (31) appear above mentioned that should be above mentioned. The presentation of  (31) is unclear. Maybe the authors should specify what P_m and P_p are (mentioning that their explicit expressions are derived in the following analysis/discussion). 

A short note has been added below the Equation (31).

Reviewer 2 Report

The paper is relevant in that it presents a mathematical model associated with the pulsating flow. The article is well written and well structured. The only suggestion that I think is important is to refer to works associated with one-dimensional models, for example:

 

For the Newtonian case:

 

[1] D.A. Caulk, and P.M. Naghdi, Axisymmetric motion of a viscous fluid inside a slender surface of revolution, Journal of Applied Mechanics, Vol.54, 1987, pp.190-196.

[2] A.M. Robertson, and A. Sequeira, A Director Theory Approach for Modeling Blood Flow in the Arterial System: An Alternative to Classical 1D Models, Mathematical Models & Methods in Applied Sciences, 15, nr.6, 2005, pp.871-906.

For the Non-Newtonian case:

[1] Carapau, F., and Correia, P., Numerical simulations of a third-grade fluid flow on a tube through a contraction, European Journal of Mechanics B/Fluids, V. 65, pp. 45-53, 2017.

Finally, my opinion: the article should be published taking into account these suggestions to reference other works related to one-dimensional models where the pulsating flow is studied.

Author Response

Thank you for your recommendation. The papers have been included into our article. Changes are marked with red color.