Abstract
This paper presents the analysis of two-layer cilia induced flow of Phan-Thien-Tanner (PTT) fluid with thermal and concentration effect. The Phan-Thien-Tanner fluid model has been used in the analogy of mucus present in the respiratory tract. The two-layer model approach was used due to the Peri Ciliary liquid Layer (PCL) and Airway Ciliary Layer (ACL) present on the epithelium cell in respiratory tract. The mathematical modelling of two-layer flow problem was simplified using long wavelength and small Reynold’s number approximation. The resulting differential equation with moving boundary gives exact solution for velocity, temperature and concentration profiles in two layers. The change in pressure has calculated by the results of velocity profile, also the pressure rise was evaluated by the numerical integration of pressure gradient along the channel wall. The impact of physical parameters on pressure rise, velocity, temperature and concentration profile was explained by the graphs. It can be seen from graphs that velocity and temperature profile are maximum in the inner layer of fluid (PCL) and concentration profile is maximum at outer layers of fluid (ACL).
摘要
本文分析了两层纤毛诱导的Phan-Thien-Tanner (PTT)流体的热效应和浓度效应。采用Phan-Thien-Tanner 流体模型模拟呼吸道黏液。由于呼吸道上皮细胞上存在睫状体周液层(PCL)和气道睫状体层(ACL),因此采用两层模型方法进行模拟。采用长波长和小雷诺数近似简化了两层流动问题的数学模型,所得到的带有移动边界的微分方程给出了两层中速度、温度和浓度分布的精确解。采用速度分布计算了压力的变化,并采用沿通道壁面压力梯度的数值积分计算了压力的上升。利用曲线图说明了物理参数对压力上升、速度、温度和浓度分布的影响。结果表明:流体内层(PCL)的速度和温度分布最大,流体外层(ACL)的浓度分布最大。
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Abbreviations
- V :
-
Velocity field vector
- U, V :
-
Velocity components in fixed frame
- u, v :
-
Velocity components in wave frame
- X, Y :
-
Rectangular coordinates of ciliated tube in fixed frame
- x, y :
-
Rectangular coordinates of ciliated tube in wave frame
- η (k) :
-
Viscosity of fluid in two phases
- P :
-
Pressure in fixed frame
- p :
-
Pressure in wave frame
- τ (k) :
-
Cauchy stress tensor
- S ( k) :
-
Shear rate
- c :
-
Wave speed
- c p :
-
Specific heat capacity
- ε (k) :
-
Elongation parameter
- λ (k) :
-
Material parameter
- ϵ :
-
Cilia length
- ρ (k) :
-
Density of fluid
- α :
-
Eccentricity of elliptical path
- β :
-
Wave numbe
- A 1 (k) :
-
Rivlin Erickson tensor
- T (k) :
-
Temperature profile in two layers
- T 0 :
-
Temperature at the center of the tube
- T 1 :
-
Temperature at the ciliated wall
- C (k) :
-
Concentration profile in two layers
- C 0 :
-
Concentration profile at the center of the tube
- C 1 :
-
Concentration at ciliated wall
- D B :
-
Coefficient of mass diffusivity
- D T :
-
Constant ratio due to thermal diffusion
- Re :
-
Reynolds’ number
- Br :
-
Brinkman number
- S H :
-
Schmidt number
- S T :
-
Soret number
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Khadija MAQBOOL provided the model, reviewed and edited the final draft. Sidra SHAHEEN used methodology to obtain the analytical results. Elena BOBESCU presented the validation of results. R. ELLAHI provided the formal analysis and handled the submission.
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Khadija Maqbool, Sidra Shaheen, Elena Bobescu and R. Ellahi declare that they have no conflict of interest.
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Maqbool, K., Shaheen, S., Bobescu, E. et al. Thermal and concentration analysis of Phan-Thien-Tanner fluid flow due to ciliary movement in a peripheral layer. J. Cent. South Univ. 28, 3327–3339 (2021). https://doi.org/10.1007/s11771-021-4858-8
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DOI: https://doi.org/10.1007/s11771-021-4858-8