Advertisement
Published in:
18-11-2020 | Letter to the editor
Authors’ reply to Comment on “Magneto-thermoelastic interaction in a reinforced medium with cylindrical cavity in the context of Caputo–Fabrizio heat transport law, S. Mondal, A. Sur, M. Kanoria, Acta Mech 230, 4367–4384 (2019)” by A. Pantokratoras
Published in: Acta Mechanica | Issue 1/2021
Login to get accessActivate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Excerpt
The authors would like to thank the reviewer for his critical analysis of the above-mentioned paper. Our responses are as follows:
1.
The corrected form of the third equation in (16) should be \(\frac{1}{r}\frac{\partial }{\partial r}\left( rE_{\theta } \right) = -\mu _{e}\frac{\partial H_{z}}{\partial t}\).
Therefore, it can be noted that on the L.H.S., \(\frac{1}{r}\frac{\partial }{\partial r}(rE_{\theta } )\) is of dimension \(\frac{1}{m}\times \frac{1}{m}\times \left( {m\times \hbox {kg}^{1/2}\Omega ^{1/2}\hbox {s}^{-3/2}} \right) =\hbox {kg}^{1/2}\Omega ^{1/2}\hbox {s}^{-3/2}\hbox {m}^{-1}\), since r is of dimension m, \(\frac{\partial }{\partial r}\) is of dimension 1/m, and \(E_{\theta } \)has dimension \(\hbox {kg}^{1/2}\Omega ^{1/2}\hbox {s}^{-3/2}\). Therefore, it can be clearly observed that the dimension of \(\frac{1}{r}\frac{\partial }{\partial r}(rE_{\theta } )\) is now the same as the dimension of \(\mu _{e} \frac{\partial H_{z} }{\partial t}\).
2.
In the mentioned paper, Eq. (14) is given as:
Using suitable non-dimensional variables, we arrive at
Substituting the suitable non-dimensional variables, we have
$$\begin{aligned}&K^{*}\left( {\frac{\partial ^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}} \right) +K\left( {\frac{\partial ^{2}{\dot{T}}}{\partial r^{2}}+\frac{1}{r}\frac{{\dot{\partial }}T}{\partial r}} \right) \,\,+(\hbox {other terms}) \\&\quad =\rho c_{\nu } {\ddot{T}}+T_{0} \frac{\partial ^{2}}{\partial t^{2}}\left( {\beta _{rr} \frac{\partial u}{\partial r}+\beta _{\theta \theta } \frac{u}{r}} \right) \,\,+(\hbox {other terms}). \end{aligned}$$
$$\begin{aligned} \frac{\partial }{\partial r'}\equiv \frac{1}{c_{1} \eta }\frac{\partial }{\partial r},\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial }{\partial t'}\equiv \frac{1}{c_{1}^{2}\eta }\frac{\partial }{\partial t}. \end{aligned}$$
$$\begin{aligned} \begin{aligned}&K^{*}c_{1}^{2} \eta ^{2\,\,\,}\left( {\frac{\partial ^{2}\theta }{\partial r^{2}}+\frac{1}{r}\frac{\partial \theta }{\partial r}} \right) T_{0} +Kc_{1}^{2} \eta ^{2\,\,\,}\times c_{1}^{2} \eta ^{\,\,}\frac{\partial }{\partial t}\left( {\frac{\partial ^{2}\theta }{\partial r^{2}}+\frac{1}{r}\frac{{\dot{\partial }}\theta }{\partial r}} \right) T_{0} +(\hbox {other terms}) \\&\quad = \rho c_{\nu } c_{1}^{4} \eta ^{2}\frac{\partial ^{2}\theta }{\partial t^{2}} T_{0} +T_{0} c_{1}^{4} \eta ^{2}\frac{\partial ^{2}}{\partial t^{2}}\left( {\beta _{rr} \frac{\partial u}{\partial r}+\beta _{\theta \theta } \frac{u}{r}} \right) \,\,+(\hbox {other terms}). \end{aligned} \end{aligned}$$
Hence, dividing both sides by \(\rho c_{\nu } c_{1}^{4} \eta ^{2}T_{0}\), we have, \(\frac{Kc_{1}^{4} \eta ^{3}}{\rho c_{\nu } c_{1}^{4} \eta ^{2}}=1\) since \(K\eta =\rho c_{\nu } \), and also \(C_{T}^{2} =\frac{K^{*}}{\rho c_{\nu } c_{1}^{2} }\), which is a dimensionless quantity. Earlier, there was a misprint in the published manuscript just after Eq. (28).
3.
In Eq. (28) of the manuscript, we have used the non-dimensional form of t and \(\xi \) (i.e., \(t'\) and \(\xi ')\) in the expression of the exponential kernel [1, 2] given by Caputo–Fabrizio, but the primes have been omitted for convenience. Therefore, it can be noted that t, \(\xi \) and \(\zeta \) are non-dimensional, which ensures that the expression \(\frac{\zeta \left( t-\xi \right) }{\zeta }\) is dimensionless. (Here, \(\zeta \) is always non-dimensional and takes a value between zero and one.) We may conclude that the exponential function of the kernel is in a non-dimensional form.
4.
It can be clearly noted from the manuscript that T represents the temperature profile and \(\theta \) represents the non-dimensional form of the same. Already, before Eq. (24), suitable non-dimensional variables have been used. Since, in Eq. (30), we are defining \(\theta =\theta _{1} \cos \,(\omega t)\), it clearly means that this temperature distribution indicates the temperature distribution in non-dimensional form. Actually, here t stands for \(t'\) (primes omitted for convenience). Hence, \(\omega \) is the non-dimensional form of the angular frequency. Therefore, the entire expression is non-dimensional.
5.
In Eqs. (27), (35), (37), (45), (46), (47) and (50), a parameter \(R_{M}^{2} \) is defined. The expression would be \(R_{M}^{2} =1+\frac{\mu _{e} H_{0}^{2} }{A_{11}}\), where \(A_{11} =\lambda +2\alpha +4\mu _{L} -2\mu _{T} +\beta \) [3].