非牛顿流体中蛋白质气泡有限变形的动力学特性研究
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摘要
蛋白质气泡在生物医学、化工、材料、消防、油气田固井等领域有着广泛的应用。在上述应用中,蛋白质气泡通过与其它流体混合或一起流动,由于形成气泡的蛋白质膜具有粘弹性,气泡周围的流体大多为非牛顿流体,这使得对蛋白质气泡在流体中的动力学特性问题的研究变得十分复杂。而蛋白质气泡在流体中的动力学特性又会直接影响气泡在流体中的存活时间、分布均匀性以及整个流体的稳定性。因此,全面、深入研究蛋白质气泡在非牛顿流体中的动力学特性,对于更好地拓宽蛋白质气泡的应用领域具有十分重要的意义。
本文在研究由粘弹性蛋白质膜构成的气泡有限变形的基础上,建立了蛋白质气泡在非牛顿流体中的动力学方程,研究分析了蛋白质膜的粘弹特性、气泡的几何特性、气泡周围流体的性质以及气泡间的相互作用等因素对气泡动力学特性的影响,为蛋白质气泡动力学特性的深入研究和应用领域的拓宽提供理论依据。本文主要研究内容和结论如下:
(1)利用粘弹性材料有限变形的能量密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出了蛋白质膜变形与应力之间的非线性关系。再根据气泡变形的平衡方程,建立了气泡在内外压力差作用下内径有限变形的静态特性方程。计算结果表明,蛋白质气泡径向变形具有非线性特性,在不同初始压力差作用下,气泡内外压力达到平衡所需的时间及变形量均不相同。增加气泡膜厚和粘弹性,既可以延长气泡变形达到平衡的时间,又可以增强气泡承受载荷的能力。
(2)根据已经推导出的粘弹性蛋白质膜变形与应力之间的非线性关系,并结合气泡变形的动力学方程,建立了在内外压力差、弹性有限变形应力及粘性耗散应力共同作用下气泡膜的非线性振动方程。通过对该方程进行计算分析,结果表明,气泡在不同初始瞬态压力差的作用下,气泡的振动频率、振幅衰减速率是不同的,停止振动时的大小也不相同;增加膜厚和粘性会抑制气泡的振动幅值,增强气泡承受动载荷的能力;对于初始大小不同的气泡,小尺寸气泡的振动频率高,振动速度衰减慢。
(3)考虑蛋白质气泡周围非牛顿流体的粘性对气泡外壁面上作用力的影响,建立了Bingham流体中单个蛋白质气泡有限变形的动力学方程,通过对该方程进行计算分析,结果表明,增加Bingham流体的塑性粘度会使气泡壁面振幅衰减速度加快,频率降低,平衡时气泡变形量小。并把计算结果与Bingham流体中空化气泡的振动特性以及蛋白质气泡外壁面受压力差作用下的振动特性进行了对比分析。与Bingham流体中空化气泡相比,在流体压力和气泡几何尺寸相同的情况下,蛋白质气泡在有限变形时的振幅衰减速度慢,振动的频率低,振动停止时蛋白质气泡的变形较小;与蛋白质气泡受压力差作用下的振动特性相比,当流体的压力与外壁面作用力大小相同时,流体中的蛋白质气泡壁面的振动周期增加,频率减小,振动停止时的变形量较大。
(4)在研究了单个蛋白质气泡在Bingham流体中振动特性的基础上,研究了两个蛋白质气泡在Bingham流体中的振动特性。考虑了其中一个气泡在Bingham流体中振动产生的Bjerknes力对另一个气泡振动特性的影响,建立了两个等径蛋白质气泡在Bingham流体中的非线性振动方程。通过对该方程进行计算分析,并在相同初始和边界条件下与Bingham流体中单个蛋白质气泡的振动特性进行了比较。结果表明,Bingham流体中两个蛋白质气泡比单个气泡不但具有更高的振动频率,而且振幅衰减速率更快;当两个气泡间的距离减小时,气泡振动频率增加,振幅衰减速率加快,而且气泡的半径越小,振动频率的增加和振幅衰减的速率越大。
(5)为了验证蛋白质气泡的静态变形特性,本文以蛋白质气泡在固井低密度水泥浆体中的应用为例,通过模拟水泥浆中的蛋白质气泡处于不同井深处所承受的压力,测试水泥浆在此压力作用下密度的变化来确定气泡半径的收缩率。结果表明,蛋白质气泡在逐渐增加的压力作用下,实验测得的蛋白质气泡收缩率的变化趋势与数值模拟计算的变化趋势基本一致,说明本文对蛋白质气泡静态变形特性理论研究是正确的。
The protein bubble is being widely used in the fields of biomedicine, chemical industry, material industry, fire protection, oil and gas well cementing. In above engineering applications, the protein bubbles are immersed in fluid or flow with fluid as a whole. Because the protein film of bubble wall is viscoelastic, and the fluid immersing the protein bubble is non-Newtonian fluid in the majority of cases, the study on dynamic performance of protein bubble in fluid is complicated. Further, the dynamic performance of bubble will affect the survival time and the uniformity of bubble in fluid, and also affect the stability of system composed of fluid and protein bubbles. So the comprehensive and in-depth study on the dynamic performance of protein bubble in non-Newtonian fluid would be most meaningful for maximizing its efficiencies.
Based on the study on the finite deformation of bubble formed by protein film, the equation describing the dynamic performance of protein bubble in non-Newtonian fluid is developed. The effect of viscoelasticity of protein film, the size of bubble, the properties of fluid and the interaction force between bubbles on the finite deformation of bubble are analyzed. These analyses would provide valuable reference for better engineering applications and for expanding the further application fields of protein bubble. The main contents and results in current research are described below.
(1) According to the energy density function for finite deformation of viscoelastic material, to the relaxation function of Maxwell model and to the deformation gradient tensor of bubble, a nonlinear equation describing the relation between the stress and finite deformation of protein bubble is presented. And then by using above equation and equilibrium equation of bubble, another equation describing the relation between relative deformation ratio of inner radius and time is developed for finite deformation yielded by the pressure difference acted on bubble walls. The numeric results show that, under the action of different load, the radial deformation of protein bubble is nonlinear, the balance size of bubble and the time needed to reach balance state are both different. Increasing the thickness and viscosity of protein film can prolong the time needed to reach balance state, and enhance the load-bearing capacity of protein bubble obviously.
(2) According to the equation describing the relation between the stress and finite deformation of protein bubble, and to the dynamics equation of bubble, another nonlinear equation describing the bubble wall vibration, which is yielded by the actions of pressure difference, the elastic finite deformation stress and the dissipation stress, is developed for a single bubble. The numeric results show that, under the action of different initial transient pressure difference, the variations of vibration frequency of bubble wall, the decrement velocity of amplitude, and the balance sizes of bubble are all diverse. Increasing the thickness and the viscosity of protein film can prohibit the vibration of bubble wall and thus can enhance the load-bearing capacity of protein bubble. The smaller bubble will vibrate with higher frequency, and the decrement of velocity is slower than that of a bigger one.
(3) Considering the effect of viscosity of non-Newtonian fluid on the action acted on the outside wall of bubble, a nonlinear equation describing the dynamic performance in finite deformation for a single protein bubble in Bingham fluid is developed. The numeric results show that, increasing the plastic viscosity of liquid the protein bubble wall will vibrate with higher decrement velocity of amplitude, and lower frequency and lead to the bigger balance size of bubble. The dynamic performance of above single protein bubble is compared with that of a cavitation bubble in Bingham fluid, the results show that, under the condition of identical liquid pressure and bubble size, the protein bubble wall will vibrate with lower decrement velocity of amplitude, the lower frequency and the final size of bubble is bigger than that of cavitation bubble in the same fluid. The dynamic performance of single protein bubble in Bingham fluid is also compared with that of a single protein bubble, which is yielded by pressure difference. The numeric results show that, if the pressure of liquid is equal to the action on the outside bubble wall, the bubble wall in Bingham fluid will vibrate with a longer period, lower frequency and the balance bubble size is also bigger.
(4) Based on the study on the dynamic performance of a single protein bubble in Bingham fluid, further study on vibration of two protein bubbles in non-Newtonian fluid is carried out. Considering the Bjerknes force between two vibrating bubbles, the nonlinear vibration equations for two identical size protein bubbles in Bingham fluid are built. The numeric results show that, under the same initial condition and boundary condition, the two bubbles in Bingham fluid will vibrate with higher frequency and higher decrement velocity of amplitude than that of a single bubble. Shortening the distance between bubbles, the bubble wall will vibrate with a higher frequency and decrement velocity of amplitude, further, the smaller the bubble size is, the higher the increment of frequency and decrement velocity of amplitude are.
(5) To verify the static deformation of protein bubble, an engineering application of protein bubble in cementing slurry is taken as an example. By way of simulating the pressure yielded by the cementing slurry at any position below the wellhead, the density changes are tested and the contraction ratio of bubble is calculated according to density changes. The results show that, increasing the pressure acted on the bubble wall, the tested and numeric variation trends for contraction ratios of inner bubble wall are essentially uniform. It is proved that the theoretical study on the static deformation is correct.
本文在研究由粘弹性蛋白质膜构成的气泡有限变形的基础上,建立了蛋白质气泡在非牛顿流体中的动力学方程,研究分析了蛋白质膜的粘弹特性、气泡的几何特性、气泡周围流体的性质以及气泡间的相互作用等因素对气泡动力学特性的影响,为蛋白质气泡动力学特性的深入研究和应用领域的拓宽提供理论依据。本文主要研究内容和结论如下:
(1)利用粘弹性材料有限变形的能量密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出了蛋白质膜变形与应力之间的非线性关系。再根据气泡变形的平衡方程,建立了气泡在内外压力差作用下内径有限变形的静态特性方程。计算结果表明,蛋白质气泡径向变形具有非线性特性,在不同初始压力差作用下,气泡内外压力达到平衡所需的时间及变形量均不相同。增加气泡膜厚和粘弹性,既可以延长气泡变形达到平衡的时间,又可以增强气泡承受载荷的能力。
(2)根据已经推导出的粘弹性蛋白质膜变形与应力之间的非线性关系,并结合气泡变形的动力学方程,建立了在内外压力差、弹性有限变形应力及粘性耗散应力共同作用下气泡膜的非线性振动方程。通过对该方程进行计算分析,结果表明,气泡在不同初始瞬态压力差的作用下,气泡的振动频率、振幅衰减速率是不同的,停止振动时的大小也不相同;增加膜厚和粘性会抑制气泡的振动幅值,增强气泡承受动载荷的能力;对于初始大小不同的气泡,小尺寸气泡的振动频率高,振动速度衰减慢。
(3)考虑蛋白质气泡周围非牛顿流体的粘性对气泡外壁面上作用力的影响,建立了Bingham流体中单个蛋白质气泡有限变形的动力学方程,通过对该方程进行计算分析,结果表明,增加Bingham流体的塑性粘度会使气泡壁面振幅衰减速度加快,频率降低,平衡时气泡变形量小。并把计算结果与Bingham流体中空化气泡的振动特性以及蛋白质气泡外壁面受压力差作用下的振动特性进行了对比分析。与Bingham流体中空化气泡相比,在流体压力和气泡几何尺寸相同的情况下,蛋白质气泡在有限变形时的振幅衰减速度慢,振动的频率低,振动停止时蛋白质气泡的变形较小;与蛋白质气泡受压力差作用下的振动特性相比,当流体的压力与外壁面作用力大小相同时,流体中的蛋白质气泡壁面的振动周期增加,频率减小,振动停止时的变形量较大。
(4)在研究了单个蛋白质气泡在Bingham流体中振动特性的基础上,研究了两个蛋白质气泡在Bingham流体中的振动特性。考虑了其中一个气泡在Bingham流体中振动产生的Bjerknes力对另一个气泡振动特性的影响,建立了两个等径蛋白质气泡在Bingham流体中的非线性振动方程。通过对该方程进行计算分析,并在相同初始和边界条件下与Bingham流体中单个蛋白质气泡的振动特性进行了比较。结果表明,Bingham流体中两个蛋白质气泡比单个气泡不但具有更高的振动频率,而且振幅衰减速率更快;当两个气泡间的距离减小时,气泡振动频率增加,振幅衰减速率加快,而且气泡的半径越小,振动频率的增加和振幅衰减的速率越大。
(5)为了验证蛋白质气泡的静态变形特性,本文以蛋白质气泡在固井低密度水泥浆体中的应用为例,通过模拟水泥浆中的蛋白质气泡处于不同井深处所承受的压力,测试水泥浆在此压力作用下密度的变化来确定气泡半径的收缩率。结果表明,蛋白质气泡在逐渐增加的压力作用下,实验测得的蛋白质气泡收缩率的变化趋势与数值模拟计算的变化趋势基本一致,说明本文对蛋白质气泡静态变形特性理论研究是正确的。
The protein bubble is being widely used in the fields of biomedicine, chemical industry, material industry, fire protection, oil and gas well cementing. In above engineering applications, the protein bubbles are immersed in fluid or flow with fluid as a whole. Because the protein film of bubble wall is viscoelastic, and the fluid immersing the protein bubble is non-Newtonian fluid in the majority of cases, the study on dynamic performance of protein bubble in fluid is complicated. Further, the dynamic performance of bubble will affect the survival time and the uniformity of bubble in fluid, and also affect the stability of system composed of fluid and protein bubbles. So the comprehensive and in-depth study on the dynamic performance of protein bubble in non-Newtonian fluid would be most meaningful for maximizing its efficiencies.
Based on the study on the finite deformation of bubble formed by protein film, the equation describing the dynamic performance of protein bubble in non-Newtonian fluid is developed. The effect of viscoelasticity of protein film, the size of bubble, the properties of fluid and the interaction force between bubbles on the finite deformation of bubble are analyzed. These analyses would provide valuable reference for better engineering applications and for expanding the further application fields of protein bubble. The main contents and results in current research are described below.
(1) According to the energy density function for finite deformation of viscoelastic material, to the relaxation function of Maxwell model and to the deformation gradient tensor of bubble, a nonlinear equation describing the relation between the stress and finite deformation of protein bubble is presented. And then by using above equation and equilibrium equation of bubble, another equation describing the relation between relative deformation ratio of inner radius and time is developed for finite deformation yielded by the pressure difference acted on bubble walls. The numeric results show that, under the action of different load, the radial deformation of protein bubble is nonlinear, the balance size of bubble and the time needed to reach balance state are both different. Increasing the thickness and viscosity of protein film can prolong the time needed to reach balance state, and enhance the load-bearing capacity of protein bubble obviously.
(2) According to the equation describing the relation between the stress and finite deformation of protein bubble, and to the dynamics equation of bubble, another nonlinear equation describing the bubble wall vibration, which is yielded by the actions of pressure difference, the elastic finite deformation stress and the dissipation stress, is developed for a single bubble. The numeric results show that, under the action of different initial transient pressure difference, the variations of vibration frequency of bubble wall, the decrement velocity of amplitude, and the balance sizes of bubble are all diverse. Increasing the thickness and the viscosity of protein film can prohibit the vibration of bubble wall and thus can enhance the load-bearing capacity of protein bubble. The smaller bubble will vibrate with higher frequency, and the decrement of velocity is slower than that of a bigger one.
(3) Considering the effect of viscosity of non-Newtonian fluid on the action acted on the outside wall of bubble, a nonlinear equation describing the dynamic performance in finite deformation for a single protein bubble in Bingham fluid is developed. The numeric results show that, increasing the plastic viscosity of liquid the protein bubble wall will vibrate with higher decrement velocity of amplitude, and lower frequency and lead to the bigger balance size of bubble. The dynamic performance of above single protein bubble is compared with that of a cavitation bubble in Bingham fluid, the results show that, under the condition of identical liquid pressure and bubble size, the protein bubble wall will vibrate with lower decrement velocity of amplitude, the lower frequency and the final size of bubble is bigger than that of cavitation bubble in the same fluid. The dynamic performance of single protein bubble in Bingham fluid is also compared with that of a single protein bubble, which is yielded by pressure difference. The numeric results show that, if the pressure of liquid is equal to the action on the outside bubble wall, the bubble wall in Bingham fluid will vibrate with a longer period, lower frequency and the balance bubble size is also bigger.
(4) Based on the study on the dynamic performance of a single protein bubble in Bingham fluid, further study on vibration of two protein bubbles in non-Newtonian fluid is carried out. Considering the Bjerknes force between two vibrating bubbles, the nonlinear vibration equations for two identical size protein bubbles in Bingham fluid are built. The numeric results show that, under the same initial condition and boundary condition, the two bubbles in Bingham fluid will vibrate with higher frequency and higher decrement velocity of amplitude than that of a single bubble. Shortening the distance between bubbles, the bubble wall will vibrate with a higher frequency and decrement velocity of amplitude, further, the smaller the bubble size is, the higher the increment of frequency and decrement velocity of amplitude are.
(5) To verify the static deformation of protein bubble, an engineering application of protein bubble in cementing slurry is taken as an example. By way of simulating the pressure yielded by the cementing slurry at any position below the wellhead, the density changes are tested and the contraction ratio of bubble is calculated according to density changes. The results show that, increasing the pressure acted on the bubble wall, the tested and numeric variation trends for contraction ratios of inner bubble wall are essentially uniform. It is proved that the theoretical study on the static deformation is correct.
引文
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