New results for periodic solution in Liebau phenomenon

Abstract

This paper is devoted to the existence and bifurcations of positive periodic solutions of nonlinear differential equation related to the Liebau phenomenon in rigid pipe-tank flow configuration. We obtain some more general requirements on the existence of a positive harmonic solution which improves the results in literatures. It is revealed that the system undergoes saddle node bifurcation, period doubling bifurcation and Neimark–Sacker bifurcation generating various periodic solution of different stability types. The multiplicity of both the positive harmonic solution and positive second-order subharmonic solution is first detected in this equation by numerical bifurcation analysis. Moreover, this work reveals that the positive subharmonic solutions of the nonlinear differential equation will also lead to the Liebau phenomenon in the model.

Appendices

Appendix

The Krasnosel’skiǐ-Guo fixed point theorem [11, P. 94].

Lemma A.1

Let X be a Banach space and K is a cone in X. Assume that $\Omega _1$ and $\Omega _2$ are open subsets of X with $0\in \Omega _1,~{\overline{\Omega }}_1\subset \Omega _2$. Let

$$\begin{aligned} {\mathcal {F}}: K\cap ({\overline{\Omega }}_2\backslash \Omega _1)\rightarrow K \end{aligned}$$

be a completely continuous operator such that one of the following conditions holds:

• (i) $\Vert {\mathcal {F}} x\Vert \ge \Vert x\Vert$ for $x\in K\cap \partial \Omega _1$ and $\Vert {\mathcal {F}} x\Vert \le \Vert x\Vert$ for $x\in K\cap \partial \Omega _2$;

• (ii) $\Vert {\mathcal {F}} x\Vert \le \Vert x\Vert$ for $x\in K\cap \partial \Omega _1$ and $\Vert {\mathcal {F}} x\Vert \ge \Vert x\Vert$ for $x\in K\cap \partial \Omega _2$.

Then, ${\mathcal {F}}$ has a fixed point in the set $K\cap ({\overline{\Omega }}_2\backslash \Omega _1)$.

The positivity of the associated Green functions [5, Appendix].

Lemma A.2

If $m>0$, then for any $h\in L^1({\mathbb {R}}/T{\mathbb {Z}})$ the equation

$$\begin{aligned} {} \left\{ \begin{aligned}&x''(t)+ax'(t)+m^2x(t)=h(t),\\&x(0)=x(T),~~x'(0)=x'(T), \end{aligned} \right. \end{aligned}$$

(A.1)

admits a unique T-periodic solution, which can be written as follows

$$\begin{aligned} x(t)=\int \limits _0^T G(t,s)h(s){\text {d}}s, \end{aligned}$$

where G(t, s) is called the Green’s function. Moreover, if $m\in \left( \frac{a}{2},\sqrt{\left( \frac{\pi }{T}\right) ^2 +\left( \frac{a}{2}\right) ^2}\right)$, then $0<l\le G(t,s)\le L$ for any $(t,s)\in [0,T]\times [0,T]$ and $\int ^T_0G(t,s)m^2{\text {d}}s\equiv 1$, here l and L are defined in Sect. 2.

Appendix

In the rigid 1 pipe-1 tank flow configuration, the variation in momentum of the mass of fluid inside the pipe is given by Newton second’s law

$$\begin{aligned} \rho (u(t) w(t))'=p(t)-p_r(t)- r_0 u(t) w(t), \end{aligned}$$

(B.1)

where $r_0 u(t) w(t)$ is a friction term modeled by Poiseuille’s law, $p_r(t)$ is the pressure at the end of the pipe at the entrance to the tank, w(t) is the velocity of the fluid. According to the property of the configuration, we have

$$\begin{aligned} w(t)=-\frac{A_P}{A_T}u'(t), ~~~~~V_0= u(t) A_P + h(t) A_T. \end{aligned}$$

(B.2)

In general, the fluid in the tank is assumed to be approximately at rest, and then, the hydrostatic pressure at the bottom of the tank is $\rho g h(t)$. The pressure loss at the junction of the pipe and tank is given as

$$\begin{aligned} \rho g h(t)-p_r(t)=\xi \frac{\rho }{2}w(t)^2. \end{aligned}$$

(B.3)

Combining Eqs. (B.1)–(B.3) and developing the derivative on the left-hand side, we arrive at the motion Eq. (1.1). Details of these equations also refers to [10, pp. 86].