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Exploration of magnetically influenced flow dynamics of a dusty fluid induced by the ramped movement of a thermally active plate

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Abstract

The study of magnetically influenced flows in conjunction with non-Newtonian dusty fluids finds wide-ranging applications in minerals processing, environmental engineering, and biomedical sciences. These include, but are not limited to, wastewater management, soil remediation and groundwater treatment, removal of contaminants from water or soil, magnetic separation techniques, controlled and site-specific drug delivery, and biomedical diagnostic procedures. This research focuses on the dynamic behaviour of a conducting dusty fluid, modelled using the Casson framework, in the presence of a thermally active plate with ramped movement and the effects of a magnetic field. The model also incorporates several key physical phenomena, including thermal radiation, heat generation, and Newtonian heating wall conditions. To describe the time-dependent flow behaviour mathematically, partial differential equations are employed. Analytical solutions are derived using mathematical approach, notably the Laplace transform (LT). The results, including changes in flow profiles and various physical quantities due to various influencing factors, are depicted through graphs and tables, offering a clear visual representation of the findings. The research concludes that a higher thermal relaxation time parameter significantly improves thermal characteristics. Higher levels of particle concentration parameter correspond to slower fluid flow. Both the heat transfer rate and shearing stress on the plate exhibit temporal variations for a range of physical parameters. The particle concentration parameter has a notable impact on thermal transmission. The study’s findings may be applied in climate modelling, weather prediction, contaminant transport, and studying phenomena such as dust storms, air pollution, and dispersion of volcanic ash clouds.

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Data availability

Data will be available on request.

Abbreviations

\(B_0\) :

Strength of magnetic field (T)

\(c_p\) :

Fluid-phase specific heat at constant pressure (J \(\hbox {kg}^{-1}\hbox {K}^{-1}\))

\(c_s\) :

Dust-phase specific heat at constant pressure (J \(\hbox {kg}^{-1}\hbox {K}^{-1}\))

\(e_{ij}\) :

Shear strain (Pa)

\(e_{\lambda _0}\) :

Planck’s function

g :

Acceleration due to gravity (m \(\hbox {s}^{-2}\))

Gr :

Thermal Grashof number

H :

Heaviside unit step function

\(h_T\) :

Convective heat transfer coefficient (W \(\hbox {m}^{-2}\hbox {K}^{-1}\))

k :

Thermal conductivity (W \(\hbox {m}^{-1}\hbox {K}^{-1}\))

\(k_0\) :

Stokes’ resistance (kg m \(\hbox {s}^{-2}\))

\(K_{\lambda _0}\) :

Absorption coefficient (\(\hbox {m}^{-1}\))

m :

Average mass of a dust particle (kg)

\(M^2\) :

Magnetic parameter

N :

Number density of dust particles

\(p_y\) :

Yield stress (N \(\hbox {m}^{-2}\))

Pr :

Prandtl number

\(q_r\) :

Radiative heat flux (kg \(\hbox {s}^{-3}\))

r :

Correlation coefficient

R :

Particle concentration parameter

Ra :

Radiation parameter

t :

Time (s)

\(t_0\) :

Characteristic time (s)

T :

Fluid-phase temperature (K)

\(T_p\) :

Dust-phase temperature (K)

\(T_\infty \) :

Constant ambient temperature (K)

\(u_0\) :

Reference velocity (m \(\hbox {s}^{-1}\))

u :

Fluid velocity (m \(\hbox {s}^{-1}\))

\(u_1\) :

Dimensionless fluid velocity

(x, y):

Cartesian coordinates (m)

\(\alpha _0\) :

Newtonian heating parameter

\(\beta \) :

Casson parameter

\(\beta ^*\) :

Volumetric thermal expansion coefficient (\(\hbox {K}^{-1}\))

\(\gamma _T\) :

Thermal relaxation time (s)

\(\eta \) :

Dimensionless variable

\(\theta \) :

Dimensionless fluid-phase temperature

\(\theta _p\) :

Dimensionless dust-phase temperature

\(\lambda \) :

Thermal radiation wavelength (m)

\(\mu \) :

Plastic dynamic viscosity (kg \(\hbox {m}^{-1}\hbox {s}^{-1}\))

\(\nu \) :

Kinematic viscosity (\(\hbox {m}^2\hbox {s}^{-1}\))

\(\xi \) :

Laplace transform parameter

\(\rho \) :

Fluid-phase density (kg \(\hbox {m}^{-3}\))

\(\rho _p\) :

Dust-phase density (kg \(\hbox {m}^{-3}\))

\(\sigma \) :

Electrical conductivity (\(\Omega ^{-1}\hbox {m}^{-1}\))

\(\sigma _1\) :

Particle relaxation time parameter

\(\tau \) :

Dimensionless time

\(\tau _{ij}\) :

Stress tensor (N \(\hbox {m}^{-2}\))

CC:

Correlation coefficient

DF:

Dusty fluid

LT:

Laplace transform

MHD:

Magnetodydrodynamics

PE:

Probable error

SL:

Sudden lifting

UL:

Uniform lifting

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Acknowledgements

The authors extend their gratitude to the esteemed editor and referees for their valuable insights and constructive feedback, which have significantly contributed to the enhancement of our paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Gour Banga, Malda, 732 103, India

    Sanatan Das

  2. Department of Mathematics, Triveni Devi Bhalotia College, Paschim Bardhaman, 713 347, India

    Soumitra Sarkar

  3. Department of Mathematics, Bajkul Milani Mahavidyalaya, Purba Medinipur, 721 655, India

    Asgar Ali

  4. Department of Applied Mathematics, Vidyasagar University, Midnapore, 721 102, India

    Rabindra Nath Jana

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Appendix A

Appendix A

The following constant expressions are utilized in the results.

$$\begin{aligned} a= & {} Pr+\frac{2}{3}\frac{R}{\gamma },\;b=\frac{Ra}{a},\; a_0=-\frac{\alpha _0}{\sqrt{a}},\; \beta _0=1+\frac{1}{\beta },\\ c= & {} \frac{1}{\beta _0}\left( 1+\frac{R}{\lambda _2}\right) ,\; c_0=\frac{a_0}{a-c},\; \lambda _1=\frac{M^2}{1+R/\lambda _2}, \;\lambda _2=\frac{1}{\sigma _1},\;\lambda _3=\frac{c_0Gr}{\beta _0},\\ \beta _1= & {} \frac{c\lambda _1-ab}{a-c},\;\beta _2=a_0^2-b,\\ \psi _1(x, e_0, e_1, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_1}\,x}}{\xi (e_0+\sqrt{\xi +e_1})}\right] \\= & {} -\frac{e_0}{e_0^2-e_1} e^{e_0x+(e_0^2-e_1)y} \textrm{efrc}\left( \frac{x}{2\sqrt{y}}+e_0\sqrt{y}\right) \nonumber \\&\quad +\,&\frac{1}{2(e_0^2-e_1)}\left[ (e_0+\sqrt{e_1})e^{x\sqrt{e_1}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}+\sqrt{e_1y}\right) \right. \nonumber \\{} & {} \left. \quad +\,(e_0-\sqrt{e_1})e^{-x\sqrt{e_1}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}-\sqrt{e_1y}\right) \right] ,\\ \psi _2(x, e_0, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_0}\,x}}{\xi ^2}\right] \\= & {} \frac{1}{2}\left[ \left( y+\frac{x}{2\sqrt{e_0}}\right) e^{x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}+\sqrt{e_0y}\right) \right. \\{} & {} \quad +\,\left. \left( y-\frac{x}{2\sqrt{e_0}}\right) e^{-x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}-\sqrt{e_0y}\right) \right] ,\\ \psi _3(x, e_0, e_1, e_2, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{(\xi +e_1)}\,x}}{\xi (\xi -e_2)(e_0+\sqrt{\xi +e_1})}\right] \\= & {} -\frac{e_0}{(e_0^2-e_1)(e_0^2-e_1-e_2)} e^{e_0x+(e_0^2-e_1)y} \textrm{efrc}(\frac{x}{2\sqrt{y}}+e_0\sqrt{y})\nonumber \\{} & {} \quad +\,\frac{e^{-e_2y}}{2e_2(e_0^2-e_1-e_2)}\left[ (e_0+\sqrt{e_1+e_2})e^{x\sqrt{e_1+e_2}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}+\sqrt{(e_1+e_2)y}\right) \right. \nonumber \\{} & {} \quad +\,\left. (e_0-\sqrt{e_1-e_2})e^{-x\sqrt{e_1+e_2}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}-\sqrt{(e_1+e_2)y}\right) \right] \nonumber \\{} & {} \quad -\,\frac{1}{2e_2(e_0^2-e_1)}\left[ (e_0+\sqrt{e_1})e^{x\sqrt{e_1}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}+\sqrt{e_1y}\right) \right. \nonumber \\{} & {} \quad +\,\left. (e_0-\sqrt{e_1})e^{-x\sqrt{e_1}}\,\textrm{efrc}\left( \frac{x}{2\sqrt{y}}-\sqrt{e_1y}\right) \right] ,\\ \psi _4(x, e_0, e_1, e_2, e_3, e_4, \tau )= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_1}\,x}}{\xi (\xi -e_3)(e_0+\sqrt{\xi +e_2})}\right] \\= & {} \left[ \frac{1}{e_3-e_4}\left( \frac{e_2}{e_3}+1\right) \psi _5(x, e_1, e_2, e_3,y)\right. \\{} & {} \quad +\,\left. \frac{1}{e_4-e_3} \left( \frac{e_2}{e_4}+1\right) \psi _5(x, e_1, e_4, e_2,y) +\frac{1}{e_3e_4} \psi _5(x, e_1, 0, e_2, y) \right] \\{} & {} \quad -\,e_0 \left[ \frac{1}{e_3(e_3-e_4)} \psi _6(x, e_1, e_3, y)+\frac{1}{e_4(e_4-e_3)} \psi _6(x, e_1, e_4, y) \right. \\{} & {} \quad +\,\left. \frac{1}{e_3e_4} \psi _6(x, e_1, 0, y) \right] ,\\ \psi _5(x, e_0, e_1, e_2, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_0}\,x}}{(\xi -e_2)\sqrt{\xi +e_1}}\right] \\= & {} \frac{\sqrt{e_0+e_2}}{2(e_3+e_1)} e^{e_2y}\left[ e^{-x\sqrt{e_0+e_2}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}-\sqrt{(e_0+e_2)y}\right\} \right. \\{} & {} \quad -\left. \,e^{-x\sqrt{e_0+e_2}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}+\sqrt{(e_0+e_2)y}\right\} \right] \\{} & {} \quad +\,\frac{i\sqrt{e_1-e_0}}{2(e_2+e_1)}e^{-e_1y} \left[ e^{ix\sqrt{e_1-e_0}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}+i\sqrt{(e_1-e_0)y}\right\} \right. \\{} & {} \quad -\,\left. e^{-ix\sqrt{e_1-e_0}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}-i\sqrt{(e_1-e_0)y}\right\} \right] ,\\ \psi _6(x, e_0, e_1, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_0}\,x}}{\xi -e_1}\right] \\= & {} \frac{1}{2}e^{e_1y}\left[ e^{x\sqrt{e_0+e_1}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}+\sqrt{(e_0+e_1)y}\right\} \right. \\{} & {} \quad +\,\left. e^{-x\sqrt{e_0+e_1}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}-\sqrt{(e_0+e_1)y}\right\} \right] ,\\ \psi _7(x, e_0, e_1, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_0}\,x}}{\xi ^2(\xi -e_1)}\right] \\= & {} \frac{1}{2e_1^2}e^{e_1\tau }\left[ e^{x\sqrt{e_0+e_1}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}+\sqrt{(e_0+e_1)y}\right\} \right. \\{} & {} \quad +\,\left. e^{-x\sqrt{e_0+e_1}}\textrm{erfc} \left\{ \frac{x}{2\sqrt{y}}-\sqrt{(e_0+e_1)y}\right\} \right] \\- & {} \frac{1}{2e_1}\left[ (y+\frac{1}{e_1}+\frac{x}{2\sqrt{e_0}})e^{x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}+\sqrt{e_0y}\right) \right. \\{} & {} \quad +\,\left. (y+\frac{1}{e_1}-\frac{x}{2\sqrt{e_0}})e^{-x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}-\sqrt{e_0y}\right) \right] ,\\ \psi _8(x, e_0, y)= & {} L^{-1}\left[ \frac{e^{-\sqrt{\xi +e_0}\,x}}{\xi }\right] \\= & {} \frac{1}{2}\left[ e^{x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}+\sqrt{e_0y}\right) +e^{-x\sqrt{e_0}}\textrm{erfc} \left( \frac{x}{2\sqrt{y}}-\sqrt{e_0y}\right) \right] ,\\ \psi ^\prime _1(0, e_0, e_1, y)= & {} \frac{e_0^2}{e_0^2-e_1} e^{(e_0^2-e_1)y}\left[ e_0\, \textrm{efrc}(e_0\sqrt{y})-\frac{1}{\sqrt{\pi y}} e^{-e_0^2y}\right] \\{} & {} \quad +\,\frac{1}{e_0^2-e_1}\left[ e_0\sqrt{e_1}\, \textrm{efr}(\sqrt{e_1y})+\frac{e_0}{\sqrt{\pi y}}e^{-e_1y}-e_1\right] \\ \psi ^\prime _2(0, e_0,y)= & {} -\left[ \left( y\sqrt{e_0}+\frac{1}{2\sqrt{e_0}}\right) \,\textrm{erf}(\sqrt{e_0y})+\sqrt{\frac{y}{\pi }} e^{-e_0y}\right] ,\\ \psi ^\prime _3(0, e_0, e_1, e_2, y)= & {} - \frac{e_0^2 e^{(e_0^2-e_1)y}}{(e_0^2-e_1)(e_0^2-e_1-e_2)} \left[ e_0\, \textrm{efrc}(e_0\sqrt{y})-\frac{1}{\sqrt{\pi y}} e^{-e_0^2y}\right] \\{} & {} \quad -\,\frac{e^{e_2y}}{e_2(e_0^2-e_1-e_2)}\left[ e_0\sqrt{e_1+e_2}\, \textrm{efr}(\sqrt{(e_1+e_2)y})\right. \\{} & {} \quad +\,\left. \frac{e_0}{\sqrt{\pi y}}e^{-(e_1+e_2)y}-(e_1+e_2)\right] +\frac{1}{e_2(e_0^2-e_1)}\left[ e_0\sqrt{e_1}\, \textrm{efr}(\sqrt{e_1y})\right. \\{} & {} \quad +\,\left. \frac{e_0}{\sqrt{\pi y}}e^{-e_1y}-e_1\right] ,\\ \psi ^\prime _5(0, e_0, e_1, e_2, y)= & {} -\frac{}{}\left[ (e_0+e_3)e^{e_3y}-(e_0-e_2)e^{-e_2y}\right] ,\\ \psi ^\prime _6(0, e_0, e_1, y)= & {} -e^{e_1y}\left[ \sqrt{e_1+e_2}\, \textrm{efr}(\sqrt{(e_1+e_2)y})+\frac{1}{\sqrt{\pi y}}e^{-(e_1+e_2)y}\right] ,\\ \psi ^\prime _8(0, e_0, y)= & {} -\left[ \sqrt{e_0}\, \textrm{efr}(\sqrt{e_0y})+\frac{1}{\sqrt{\pi y}}e^{-e_0y}\right] \end{aligned}$$

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Das, S., Sarkar, S., Ali, A. et al. Exploration of magnetically influenced flow dynamics of a dusty fluid induced by the ramped movement of a thermally active plate. Arch Appl Mech 94, 407–433 (2024). https://doi.org/10.1007/s00419-023-02531-z

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