Research article Topical Sections

The effect of nanocrystalline Ni-W coating on the tensile properties of copper

  • Received: 08 February 2016 Accepted: 09 March 2016 Published: 15 March 2016
  • Nanostructured Ni-W alloy coatings containing approximately 40 wt.% tungsten were electrodeposited onto copper substrates. The effect of the coatings thickness on the surface topography, microstructure and grain size was investigated with the aid of Atomic Force Microscopy (AFM), Scanning Electron Microscopy (SEM) and X-ray Diffraction (XRD) techniques respectively. In addition, this research work aims in understanding the influence and correlation between microstructure and thickness of these Ni-W coatings with the bulk mechanical properties of coated specimens. The experimental results indicated that the micro-hardness and Ultimate Tensile Strength (UTS) of the Ni-W coated copper were higher than that of bare copper, whereas both slightly increased with increasing coating thickness up to 21 μm. On the other hand, the ductility of Ni-W coated copper decreased significantly with increasing coating thickness. Thus it could be said that when applying Ni-W coatings there are certain limitations not only in terms of their composition, but their thickness, grain size and coating structure should be also taken into consideration, in order to obtain an understanding of their mechanical behavior.

    Citation: C. N. Panagopoulos, E. P. Georgiou, D.A. Lagaris, V. Antonakaki. The effect of nanocrystalline Ni-W coating on the tensile properties of copper[J]. AIMS Materials Science, 2016, 3(2): 324-338. doi: 10.3934/matersci.2016.2.324

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  • Nanostructured Ni-W alloy coatings containing approximately 40 wt.% tungsten were electrodeposited onto copper substrates. The effect of the coatings thickness on the surface topography, microstructure and grain size was investigated with the aid of Atomic Force Microscopy (AFM), Scanning Electron Microscopy (SEM) and X-ray Diffraction (XRD) techniques respectively. In addition, this research work aims in understanding the influence and correlation between microstructure and thickness of these Ni-W coatings with the bulk mechanical properties of coated specimens. The experimental results indicated that the micro-hardness and Ultimate Tensile Strength (UTS) of the Ni-W coated copper were higher than that of bare copper, whereas both slightly increased with increasing coating thickness up to 21 μm. On the other hand, the ductility of Ni-W coated copper decreased significantly with increasing coating thickness. Thus it could be said that when applying Ni-W coatings there are certain limitations not only in terms of their composition, but their thickness, grain size and coating structure should be also taken into consideration, in order to obtain an understanding of their mechanical behavior.


    Since Hirsch and Smale[1] proposed the necessity of structural stability, this topic has received sufficient attention from scholars. This type of research focuses on whether small disturbances in the coefficients, initial data, and geometric models in the equations will cause significant disturbances in the solutions. At the beginning, people were mainly keen on dealing with the continuous dependence and convergence of fluid in porous media defined in two-dimensional or three-dimensional bounded regions. Freitas et al. [2] studied the long-term behavior of porous-elastic systems and proved that solutions depend continuously on the initial data. Payne and Straughan[3] established a prior bounds and maximum principles for the solutions and obtained the structural stability of Darcy fluid in porous media, where they assumed that the temperature satisfies Newton's cooling conditions at the boundary. Scott[4] considered the situation where Darcy fluid undergoes exothermic reactions at the boundary and obtained the continuous dependence of the solutions on the boundary parameters. Li et al.[5] studied the interface connection between Brinkman–Forchheimer fluid and Darcy fluid in a bounded region, and obtained the continuous dependence on the heat source and Forchheimer coefficient. For more papers, on can see [6,7,8,9,10].

    With the continuous development of technology and progress in the field of engineering, the necessity of studying the structural stability of fluid equations on a semi-infinite cylinder is even more urgent. The semi-infinite cylinder refers to a cylinder whose generatrix is parallel to the coordinate axis and its base is located on the coordinate plane, i.e.,

    $ R = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq0\Big\}, $

    where $ D $ is a bounded domain on $ x_1Ox_2 $.

    Li et al. have already done some work on this topic. Li and Lin[11] proved the continuous dependence on the Forchheimer coefficient of the Brinkman–Forchheimer equations in $ R $. Papers [12] and [13] obtained structural stability for Forchheimer fluid and temperature-dependent bidispersive flow in $ R $, respectively.

    In this paper, we introduce a new cylinder with a disturbed base, which has been considered in [14]. Let $ D(f) $ represent the disturbed base, i.e.,

    $ D(f) = \Big\{(x_1, x_2, x_3)\Big|x_3 = f(x_1, x_2)\geq0,\ (x_1, x_2)\in D\Big\}, $

    where the given function $ f $ satisfies

    $ |f(x1,x2)|<ϵ, ϵ>0.
    $

    $ \epsilon $ is called the perturbation parameter. The cylinder with a disturbed base is defined as

    $ R(f) = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq f(x_1, x_2)\geq0\Big\}. $

    Different from [14], we study the heat conduction equation applicable to the study of layered composite materials in binary mixtures[15]

    $ b1ut=k1uγ(uv), in R×{t>0},
    $
    (1.1)
    $ b2vt=k2v+γ(uv), in R×{t>0},
    $
    (1.2)
    $ u=v=0, on D×{x3>0}×{t>0},
    $
    (1.3)
    $ u=v=0, in R×{t=0},
    $
    (1.4)

    where $ k_1, k_2, b_1, b_2 $ and $ \gamma $ are positive constants. $ u $ and $ v $ are the temperature fields in each constituent. Papers [16,17,18] further discussed and generalized the application of Eqs (1.1) and (1.2).

    In this paper, we shall also use the notations

    $ R(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3\geq z\geq 0\Big\}, $
    $ D(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3 = z\geq 0\Big\}. $

    The main work of this article investigates the continuous dependence of solutions to Eqs (1.1)–(1.4) on perturbation parameters and base data. Due to many practical constraints, it is very common for the base of the cylinder to experience minor disturbance. Therefore, studying the effects of these disturbances is essential. To this end, we assume that $ u^* $ and $ v^* $ are perturbed solutions of Eqs (1.1)–(1.4) on $ R(f) $, and then prove that the difference between the unperturbed solutions and the perturbed solutions satisfies a first-order differential inequality. By solving this inequality, we can obtain the continuous dependence of the solution.

    On the finite end $ D $, we assume that the solutions to (1.1)–(1.4) satisfy

    $ u(t,x)=L11(t,x1,x2), v(t,x)=L12(t,x1,x2),t>0, x3=0, (x1,x2)D(0),
    $
    (2.1)
    $ u(t,x)=L21(t,x1,x2), v(t,x)=L22(t,x1,x2),t>0, x3=f(x1,x2), (x1,x2)D(0).
    $
    (2.2)

    In (2.1) and (2.2), the known functions $ L_{ij} (i, j = 1, 2) $ satisfy the compatibility conditions on $ \partial D $.

    We let that $ \mathcal{H}_1(t, \boldsymbol{x}) $ and $ \mathcal{H}_2(t, \boldsymbol{x}) $ are specific functions who have the same boundary conditions as $ u^* $ and $ v^* $, respectively. That is

    $ H1(t,x)=L21(t,x1,x2)exp{σ(x3f)}, H2(t,x)=L22(t,x1,x2)exp{σ(x3f)},
    $
    (2.3)

    where $ \sigma > 0 $.

    We now derive some lemmas.

    Lemma 2.1. If $ L_{21}, L_{22}\in H^1([0, \infty)\times D(f)) $, then

    $ t0exp{η1τ}[k1||u(τ)||2L2(R(f))+k2||v(τ)||2L2(R(f))]dτd1(t),
    $

    where

    $ d1(t)=t0exp{η1τ}[k1||H1||2L2(R(f))+k2||H2||2L2(R(f))]dτ+exp{η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))]+12t0exp{η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ+12γt0exp{η1τ}||(H1H2)(τ)||2L2(R(f))dτ.
    $
    (2.4)

    Proof. Using (1.1)–(1.4), we begin with

    $ t0R(f)exp{η1τ}[b1uτk1u+γ(uv)]udxdτ=0,t0R(f)exp{η1τ}[b2vτk2vγ(uv)]vdxdτ=0.
    $

    We compute

    $ 12exp{η1t}[b1||u(t)||2L2(R(f))+b2||v(t)||2L2(R(f))]+t0exp{η1τ}[b1η1||u(τ)||2L2(R(f))+b2η1||v(τ)||2L2(R(f))]dτ+t0exp{η1τ}[k1||u(τ)||2L2(R(f))+k2||v(τ)||2L2(R(f))]dτ+γt0exp{η1τ}||(uv)(τ)||2L2(R(f))dτ=t0D(f)exp{η1τ}[k1ux3u+k2vx3v]dAdτ.
    $
    (2.5)

    On the other hand, we use (2.3) to compute

    $ t0D(f)exp{η1τ}[k1ux3u+k2vx3v]dAdτ=t0D(f)exp{η1τ}[k1ux3H1+k2vx3H2]dAdτ=t0R(f)exp{η1τ}[k1(uH1)+k2(vH2)dxdτ=t0R(f)exp{η1τ}[k1uH1+k2vH2]dxdτ+exp{η1t}R(f)[b1uH1+b2vH2]dx+η1t0R(f)exp{η1τ}[b1uH1,τ+b2vH2,τ]dxdτ+γt0R(f)exp{η1τ}(uv)(H1H2)dxdτF1+F2+F3+F4.
    $
    (2.6)

    An application of the Schwarz inequality leads to

    $ F112t0exp{η1τ}[k1||u(τ)||2L2(R(f))+k2||v(τ)||2L2(R(f))]dτ+12t0exp{η1τ}[k1||H1||2L2(R(f))+k2||H2||2L2(R(f))]dτ,
    $
    (2.7)
    $ F212exp{η1t}[b1||u(t)||2L2(R(f))+b2||v(t)||2L2(R(f))]+12exp{η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))],
    $
    (2.8)
    $ F3t0exp{η1τ}[b1η1||u(τ)||2L2(R(f))+b2η1||v(τ)||2L2(R(f))]dτ+14t0exp{η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ,
    $
    (2.9)
    $ F4γt0exp{η1τ}||(uv)(τ)||2L2(R(f))dτ+14γt0exp{η1τ}||(H1H2)(τ)||2L2(R(f))dτ.
    $
    (2.10)

    Inserting Eqs (2.7)–(2.10) into (2.6) and combining (2.5), it can be obtained

    $ t0exp{η1τ}[k1||u(τ)||2L2(R(f))+k2||v(τ)||2L2(R(f))]dτt0exp{η1τ}[k1||H1||2L2(R(f))+k2||H2||2L2(R(f))]dτ+exp{η1t}[b1||H1(t)||2L2(R(f))+b2||H2(t)||2L2(R(f))]+12t0exp{η1τ}[b1η1||H1,τ(τ)||2L2(R(f))+b2η1||H2,τ(τ)||2L2(R(f))]dτ+12γt0exp{η1τ}||(H1H2)(τ)||2L2(R(f))dτ.
    $
    (2.11)

    From (2.11), we can conclude that Lemma 2.1 holds.

    We not only need a prior bounds for $ v $ and $ v^* $, but also for $ u $ and $ u^* $. Since $ u $ and $ u^* $ are undisturbed solutions of Eqs (1.1)–(1.4), in Lemma 2.1 we only need to set $ f = 0 $ and replace $ L_{21} $ and $ L_{22} $ with $ L_{11} $ and $ L_{12} $, respectively, and then we can obtain the a prior bounds for $ u $ and $ u^* $.

    Lemma 2.2. If $ L_{11}, L_{12}\in H^1([0, \infty)\times D) $, then

    $ t0exp{η1τ}[k1||u(τ)||L2(R)+k2||v(τ)||L2(R)]dτd2(t),
    $

    where

    $ d2(t)=t0exp{η1τ}[k1||H3||2L2(R)+k2||H4||2L2(R)]dτ+exp{η1t}[b1||H3(t)||2L2(R)+b2||H4(t)||2L2(R)]+12t0exp{η1τ}[b1η1||H3,τ(τ)||2L2(R)+b2η1||H4,τ(τ)||2L2(R)]dτ+12γt0exp{η1τ}||(H3H4)(τ)||2L2(R)dτ
    $

    and

    $ \mathcal{H}_3(t, {\boldsymbol{x}}) = L_{11}(t, x_1, x_2)\exp\{-\sigma x_3\},\ \mathcal{H}_4(t, {\boldsymbol{x}}) = L_{12}(t, x_1, x_2)\exp\{-\sigma x_3\}. $

    Remark 2.1. Lemmas 2.1 and 2.2 will provide a priori estimates for the proof of the lemmas in the next section.

    Let $ w $ and $ s $ represent the difference between the perturbed solutions and the unperturbed solutions, i.e.,

    $ w=uu, s=vv,
    $
    (3.1)

    then $ w $ and $ s $ satisfy

    $ b1wt=k1wγ(ws), in R(ϵ)×{t>0},
    $
    (3.2)
    $ b2st=k2s+γ(ws), in R(ϵ)×{t>0},
    $
    (3.3)
    $ w=s=0, on D×{x3>ϵ}×{t>0},
    $
    (3.4)
    $ w=s=0, in R(ϵ)×{t=0}.
    $
    (3.5)

    To obtain the continuous dependence of the solution on the perturbation parameter, we establish a new energy function

    $ V(t,x3)=t0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ, x3ϵ.
    $
    (3.6)

    Noting the definition of $ R(x_3) $, we can obtain the derivative of $ V(t, x_3) $ as follows:

    $ x3V(t,x3)=t0[||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3))]dτ.
    $

    We introduce two auxiliary functions $ \varphi $ and $ \psi $ such that

    $ b1φτ+k1φ=w, b2ψτ+k2ψ=s, in R(x3),0<τ<t,
    $
    (3.7)
    $ φ(τ,x1,x2,x3)=ψ(τ,x1,x2,x3)=0, on D×{x3},0<τ<t,
    $
    (3.8)
    $ φ(τ,x1,x2,x3)=ψ(τ,x1,x2,x3)=0, (x1,x2)D,0<τ<t,
    $
    (3.9)
    $ φ(t,x)=ψ(t,x)=0, in R(x3),
    $
    (3.10)
    $ φ,φ,ψ,ψ0(uniformly in x1,x2,τ) as x3,
    $
    (3.11)

    where $ x_3 > \epsilon $.

    Next, we will derive some necessary properties of the auxiliary functions, which will play a crucial role in proving the continuous dependence of the solutions.

    Lemma 3.1. If $ \varphi, \psi\in H^1([0, t]\times R(x_3)) $, then

    $ t0[b1||φτ(τ)||2L2(R(x3))+b2||ψτ(τ)||2L2(R(x3))]dτa1V(t,x3), x3ϵ,
    $

    where $ a_1 = \max\{b_1^{-1}, b_2^{-1}\} $.

    Proof. We begin with

    $ t0R(x3)φτ[b1φτ+k1φ+w]dxdτ=0,t0R(x3)ψτ[b2ψτ+k2ψ+s]dxdτ=0.
    $

    Using the divergence theorem $ \oint_{\partial R(x_3)}Fds = \int_{R(x_3)}div F dx $ and (3.8)–(3.11), we have

    $ b1t0||φτ(τ)||2L2(R(x3))dτ=12k1||φ(0)||2L2(R(x3))+t0R(x3)wφτdxdτ[t0||φτ(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12,
    $
    (3.12)

    and

    $ b2t0||ψτ(τ)||2L2(R(x3))dτ[t0||ψτ(τ)||2L2(R(x3))dτt0||s(τ)||2L2(R(x3))dτ]12.
    $
    (3.13)

    Using the Schwarz inequality, (3.12) and (3.13), Lemma 3.1 can be obtained.

    Lemma 3.2. If $ \varphi, \psi\in H^1(R(x_3)) $, then

    $ t0[k1||φ(τ)||2L2(R(x3))+k2||ψ(τ)||2L2(R(x3))]dτa2V(t,x3),
    $

    where $ a_2 = \frac{1}{\lambda}\max\{k_1^{-1}, k_2^{-1}\} $.

    Proof. We begin with

    $ t0R(x3)φ[b1φτ+k1φ+w]dxdτ=0,t0R(x3)φ[b2ψτ+k2ψ+s]dxdτ=0.
    $

    Using the divergence theorem and Lemma 2.2, we have

    $ k1t0||φ(τ)||2L2(R(x3))dτ=12b1||φ(0)||2L2(R(x3))+t0R(x3)wφdxdτ[t0||φ(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]121λ[t0||2φ(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12
    $
    (3.14)

    and

    $ k2t0||ψ(τ)||2L2(R(x3))dτ1λ[t0||2ψ(τ)||2L2(R(x3))dτt0||s(τ)||2L2(R(x3))dτ]12.
    $
    (3.15)

    Using the following inequality

    $ ab+cd(a+c)(b+d), for  a,b,c,d>0,
    $
    (3.16)

    the Young inequality and Lemma 3.1, we can have from (3.14) and (3.15)

    $ t0[k1||φ(τ)||2L2(R(x3))+k2||ψ(τ)||2L2(R(x3))]dτ1λ{t0[k1||2φ(τ)||2L2(R(x3))+k2||2ψ(τ)||2L2(R(x3))]dτt0[k11||w(τ)||2L2(R(x3))+k12||s(τ)||2L2(R(x3))]dτ}12.
    $
    (3.17)

    From (3.17) we can obtain Lemma 3.2.

    Lemma 3.3. If $ \varphi, \psi\in H^1(R(x_3)) $, then

    $ k1t0||φx3(τ)||2L2(D(x3))dτ+k2t0||ψx3(τ)||2L2(D(x3))dτa3V(t,x3),
    $

    where $ a_3 $ is a positive constant.

    Proof. Letting $ \delta $ be a positive constant. We compute

    $ t0R(x3)[φx3δφτ][b1φτ+k1φ+w]dxdτ=0,
    $
    (3.18)
    $ t0R(x3)[ψx3δψτ][b2ψτ+k2ψ+s]dxdτ=0.
    $
    (3.19)

    Using the divergence theorem and (3.8)–(3.10) in (3.18) and (3.19), we obtain

    $ 12k1δ||φ(0)||2L2(R(x3))dτ+b1δt0||φτ(τ)||2L2(R(x3))dτ+12k1t0||φx3(τ)||2L2(D(x3))dτ=t0R(x3)φx3φτdxdτ+t0R(x3)[φx3δφτ]wdxdτ.
    $
    (3.20)

    Using the Schwarz inequality, we obtain

    $ t0R(x3)φx3φτdxdτ[t0||φx3(τ)||2L2(R(x3))dτt0||φτ(τ)||2L2(R(x3))dτ]12,
    $
    (3.21)
    $ t0R(x3)φx3wdxdτ[t0||φx3(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12,
    $
    (3.22)
    $ δt0R(x3)φτwdxdτδ[t0||φτ(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12.
    $
    (3.23)

    Inserting (3.21)–(3.23) into (3.20) and dropping the first two terms in the left of (3.20), we have

    $ 12k1t0||φx3(τ)||2L2(D(x3))dτ[t0||φx3(τ)||2L2(R(x3))dτt0||φτ(τ)||2L2(R(x3))dτ]12+[t0||φx3(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12+δ[t0||φτ(τ)||2L2(R(x3))dτt0||w(τ)||2L2(R(x3))dτ]12.
    $
    (3.24)

    Similar, we can also have from (3.19)

    $ 12k2t0||ψx3(τ)||2L2(D(x3))dτ[t0||ψx3(τ)||2L2(R(x3))dτt0||ψτ(τ)||2L2(R(x3))dτ]12+[t0||ψx3(τ)||2L2(R(x3))dτt0||s(τ)||2L2(R(x3))dτ]12+δ[t0||ψτ(τ)||2L2(R(x3))dτt0||s(τ)||2L2(R(x3))dτ]12.
    $
    (3.25)

    Using (3.16) and Lemmas 3.1 and 3.2, we obtain

    $ k1t0||φx3(τ)||2L2(D(x3))dτ+k2t0||ψx3(τ)||2L2(D(x3))dτ2a1a2{t0[b1||φx3(τ)||2L2(R(x3))+b2||ψx3(τ)||2L2(R(x3))]dτt0[k1||φτ(τ)||2L2(R(x3))+k2||ψτ(τ)||2L2(R(x3))]dτ}12+2a2{t0[b1||φx3(τ)||2L2(R(x3))+b2||ψx3(τ)||2L2(R(x3))]dτt0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ}12+2a1δ{t0[k1||φτ(τ)||2L2(R(x3))+k2||ψτ(τ)||2L2(R(x3))]dτt0[||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3))]dτ}12a3V(t,x3),
    $
    (3.26)

    where $ a_3 = 2a_1^2a_2^2+2a_2^2+2a_1^2 $.

    In the next section, we will use Lemma 3.3 to derive the continuous dependence of the solutions.

    In this section, we first derive a bound for $ V(t, \epsilon) $. To do this, we define

    $ u(t,x)=L11(t,x1,x2), v(t,x)=L12(t,x1,x2), ϵx30,(x1,x2)D,t[0,+),
    $
    (4.1)
    $ u(t,x)=L21(t,x1,x2), v(t,x)=L22(t,x1,x2), ϵx3f(x1,x2),(x1,x2)D,t[0,+).
    $
    (4.2)

    When $ -\epsilon\leq x_3\leq\epsilon $, we let

    $ w(t,x)=u(t,x)u(t,x), s(t,x)=v(t,x)v(t,x),(x1,x2)D,t[0,+).
    $
    (4.3)

    In view of (3.1) and (4.3), using the triangle inequality, it can be obtained that

    $ k1t0R(ϵ)(wx3)2dxdτ+k2t0R(ϵ)(sx3)2dxdτt0R(ϵ)[k1(ux3)2+k2(vx3)2]dxdτ+t0R(ϵ)[k1(ux3)2+k2(vx3)2]dxdτ.
    $
    (4.4)

    Using Lemmas 2.1 and 2.2, (4.1) and (4.2), from (4.4), we obtain

    $ k1t0R(ϵ)(wx3)2dxdτ+k2t0R(ϵ)(sx3)2dxdτt0R[k1(ux3)2+k2(vx3)2]dxdτ+t0R(f)[k1(ux3)2+k2(vx3)2]dxdτ.eη1t[d1(t)+d2(t)]d3(t).
    $
    (4.5)

    Now, we write the main theorem as:

    Theorem 4.1. If $ L_{11}, L_{12}\in H^1([0, \infty)\times R), L_{21}, L_{22}\in H^1([0, \infty)\times R(f)) $ and $ t < \frac{\pi}{4a_1\gamma} $, then

    $ V(t,x3)exp{d4(x3ϵ)}{32d4πmax{1k1,1k2}d3(t)ϵ+d5t0[||(L11L21)(τ)||2L2(D)+||(L12L22)(τ)||2L2(D)]dτ},x3ϵ
    $

    holds, where $ d_4 = a_3^{-1}\max\{k_1, k_2\}^{-1} $ and $ d_5 = \frac{d_4\pi}{2}+\frac{2}{d_4} $.

    Proof. Let $ x_3\geq\epsilon $ be a fixed point on the coordinate axis $ x_ 3 $. Using (3.7)–(3.11) and the divergence theorem, we can have

    $ V(x3,t)=t0R(x3)w[b1φτ+k1φ]dxdτt0R(x3)s[b2ψτ+k2ψ]dxdτ=t0R(x3)[b1φτw+b2ψτs]dxdτ+t0R(x3)[k1wφ+k2sψ]dxdτ+t0D(x3)[k1wφx3+k2sψx3]dAdτ=t0R(x3)[b1φτw+b2ψτs]dxdτt0R(x3)[k1wφ+k2sψ]dxdτ+t0D(x3)[k1wφx3+k2sψx3]dAdτ=t0R(x3)[b1φτw+b2ψτs]dxdτt0R(x3)[b1φwτ+b2ψsτ]dxdτ+t0D(x3)[k1wφx3+k2sψx3]dAdτγt0R(x3)(φψ)(ws)dxdτ.
    $
    (4.6)

    In light of (1.4) and (3.10), it is clear that

    $ t0R(x3)[b1φτw+b1φwτ]dxdτ=0, t0R(x3)[b2ψτs+b2ψsτ]dxdτ=0.
    $
    (4.7)

    A combination of the Hölder inequality, (3.16) and Lemma 3.3 leads to

    $ t0D(x3)[k1wφx3+k2sψx3]dAdτk1[t0||φx3(τ)||2L2(D(x3))dτt0||w(τ)||2L2(D(x3))dτ]12+k2[t0||ψx3(τ)||2L2(D(x3))dτt0||s(τ)||2L2(D(x3))dτ]12max{k1,k2}[t0(k1||φx3(τ)||2L2(D(x3))+k2||ψx3(τ)||2L2(D(x3)))dτ]12[t0(||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3)))dτ]12a3max{k1,k2}V(t,x3)[x3V(t,x3)]12.
    $
    (4.8)

    For the fourth term in the right of (4.6), we compute

    $ γt0R(x3)(φψ)(ws)dxdτγ[t0(||φ(τ)||2L2(R(x3))+||ψ(τ)||2L2(R(x3)))dτt0(||w(τ)||2L2(R(x3))+||s(τ)||2L2(R(x3)))dτ]12.
    $
    (4.9)

    Using the inequality (see p182 in [19])

    $ 10ϕ2dx4π210(ϕ)2dx, for ϕ(0)=0,
    $
    (4.10)

    we have from (4.9)

    $ γt0R(x3)(φψ)(ws)dxdτγ2tπ[t0(||φτ(τ)||2L2(R(x3))+||ψτ(τ)||2L2(R(x3)))dτV(t,x3)]12γ2tπa1V(t,x3),
    $
    (4.11)

    where we have also used Lemma 3.1. Combining (4.6), (4.7), (4.8) and (4.11) and choosing $ t < \frac{\pi}{4a_1\gamma} $, we can have

    $ V(t,x3)1d4x3V(t,x3),x3>ϵ.
    $
    (4.12)

    Integrating (4.12) from $ \epsilon $ to $ x_3 $, we have

    $ V(t,x3)V(t,ϵ)exp{d4(x3ϵ)},x3ϵ.
    $
    (4.13)

    Equation (4.13) only indicates that the solutions to (1.1)–(1.4) decay exponentially as $ x_3\rightarrow \infty $. This decay result is not rigorous because we do not yet know whether $ V(t, \epsilon) $ depends on the perturbation parameter $ \epsilon $. Therefore, we derive the explicit bound of $ V(t, \epsilon) $ in terms of $ \epsilon $ and $ L_{ij}(ij = 1, 2) $.

    After letting $ x_3 = \epsilon $ in (4.12), we have

    $ V(ϵ,t)1d4t0[||w(τ)||2L2(D(ϵ))+||s(τ)||2L2(D(ϵ))]dAdτ=2d4t0ϵϵD(x3)[wwx3+ssx3]dxdτ+2d4t0[||(L11L21)(τ)||2L2(D)+||(L12L22)(τ)||2L2(D)]dτ2d4[t0||w(τ)||2L2(D(x3)×[ϵ,ϵ])dτt0||wx3(τ)||2L2(D(x3)×[ϵ,ϵ])dτ]12+2d4[t0||s(τ)||2L2(D(x3)×[ϵ,ϵ])dτt0||sx3(τ)||2L2(D(x3)×[ϵ,ϵ])dτ]12+2d4t0[||(L11L21)(τ)||2L2(D)+||(L12L22)(τ)||2L2(D)]dτ.
    $
    (4.14)

    Using (4.10) again, we have

    $ t0||w(τ)||2L2(D(x3)×[ϵ,ϵ])dτ16ϵ2π2t0||wx3(τ)||2L2(D(x3)×[ϵ,ϵ])dτ+2ϵt0||(L11L21)(τ)||2L2(D)dτ,
    $
    (4.15)
    $ t0||s(τ)||2L2(D(x3)×[ϵ,ϵ])dτ16ϵ2π2t0||sx3(τ)||2L2(D(x3)×[ϵ,ϵ])dτ+2ϵt0||(L12L22)(τ)||2L2(D)dτ.
    $
    (4.16)

    Inserting (4.15) into (4.16) and combining the Schwarz inequality, we obtain

    $ V(ϵ,t)32d4πϵt0[||wx3(τ)||2L2(D(x3)×[ϵ,ϵ])+||sx3(τ)||2L2(D(x3)×[ϵ,ϵ])]dτ+[d4π2+2d4]t0[||(L11L21)(τ)||2L2(D)+||(L12L22)(τ)||2L2(D)]dτ32d4πmax{1k1,1k2}ϵt0[k1||wx3(τ)||2L2(R(ϵ))+k2||sx3(τ)||2L2(R(ϵ))]dτ+[d4π2+2d4]t0[||(L11L21)(τ)||2L2(D)+||(L12L22)(τ)||2L2(D)]dτ.
    $
    (4.17)

    In view of (4.5) and (4.13), from (4.17) we have Theorem 4.1.

    Remark 4.1. Theorem 4.1 indicates that $ V(t, x_3) $ continuously depends on $ \epsilon $ and the base data. That is, when $ \epsilon $ approaches 0, then $ u(t, x_3) $ and $ v(t, x_3) $ approach 0. If $ \epsilon = 0 $, Theorem 4.1 is the Saint-Venant's principle type decay result.

    Remark 4.2. In any cross-section of $ R $, the continuous dependence result can still be obtained. We compute

    $ t0[||w(τ)||2L2(D(x3))+||s(τ)||2L2(D(x3))]dτ=2t0R(x3)[wwx3+ssx3]dxdτ2V(x3)[t0[k1||wx3(τ)||2L2(R(ϵ))+k2||sx3(τ)||2L2(R(ϵ))]dτ]12.
    $
    (4.18)

    Using (4.18) and Theorem 4.1, we can obtain the continuous dependence result.

    This article adopts the methods of the a prior estimates and energy estimate to obtain the continuous dependence of the solution on the base. This method can be further extended to other linear partial differential equation systems, such as pseudo-parabolic equation

    $ u_t = \Delta u+\delta \Delta u_t, $

    where $ \delta $ is a positive constant. However, for nonlinear equations (e.g., the Darcy equations), due to the inability to control nonlinear terms and derive a prior bounds for nonlinear terms, Lemma 3.3 will be difficult to obtain. This is a difficult problem we need to solve next.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the Research Team Project of Guangzhou Huashang College (2021HSKT01).

    The author declares there is no conflict of interest.

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