Sage Journals: Discover world-class research

Abstract

The prime concern of the present investigation includes a numerical approach targeting the study of heat and mass transfer for the flow of magnetic nanofluid based on hydrocarbon (C1-20B) past a rotating porous disk taking into cognizance the various parameters present. The modelled system of equations for the unsteady flow is rendered into the dimensionless non-linear system of differential equations, via suitable transformations. The modelled system of dimensionless equations is numerically solved by MATLAB bvp4c solver. The influence of involved emerging parameters, namely permeability, viscosity variation, rotation, radiation, ferrohydrodynamic interaction, thermophoresis, and Brownian motion parameters on flow regimes has been studied and shown graphically. Moreover, numerical data of the rate of heat and mass transfer and coefficients of skin friction are also mentioned in tabular form. It is found that the tangential velocity is reduced for increasing the value of viscosity variation parameters. The fluid temperature profile shows an increment when we increase the rotation of the disk.

Keywords

Geothermal viscosity boundary-layer thermal radiation Brownian motion thermophoresis

Introduction

During the last six decades, a high tolerance has been found to examine the flow of magnetic nanofluids (MNF) in rotating disk geometry with heat and mass transfer. MNF usually cited as ferrofluids, are colloidal suspension of magnetic nanoparticles of $F e$ , $C o$ , ${F e}_{3} O_{4}$ , ${F e}_{2} O_{3}$ , $C o {F e}_{2} O_{4}$ , $γ - {F e}_{2} O_{3}$ or $F e - C$ in the base liquid (kerosene, ethylene glycol, hydrocarbon, etc.) having surfactant coatings. The nanoparticles used in MNF having an approximate size of range about 3–15 nm. The characteristic study has many application in various engineering and industrial areas such as efficient electrical motors, thermal power generator system, bearing and dampers in cars and other machines, nuclear rotators, energy conversion system, rotating machinery, etc.

The originating study due to the infinite rotating disk of ordinary viscous fluid flow has conveyed by Karman¹. He introduced similarity transformations for reducing the governing modelled equations into dimensionless ordinary differential equations. After his contributions, Cochran² improved the solution using the power series approximations. Later, Benton³ gave improvement to the Cochran results and also elaborated the findings for time dependent flow. Frusteri and Osalusi⁴ analyzed the magnetohydrodynamics flow caused by a rotating disk, taking the temperature dependent fluid properties. Moreover, the research work on the fluid flow with different characteristics over a rotating disk was published by many authors^5–11

The depth and temperature dependent viscosity (geothermal viscosity) has many application in the geosciences for which a vanity of research work with geothermal viscosity have been published. Torrance and Turcotte¹² studied the impact of variable viscosity (temperature and depth dependent) on thermal convection in a fluid layer. Further, Maleque¹³described the impacts of combined depth and temperature dependent viscosity for MHD fluid flow past a rotating disk system. In this direction, many authors^14–17 reported their findings.

Flow of fluid and heat-transfer in rotator-stator system are of major significance in turbomachinery control, for example, gas turbine cooling¹⁸ where air flow is used for cooling in the space between adjacent surface and turbine disk. Herrero et al.¹⁹ analyzed heat-transfer for co-rotating disks, which has great importance in machinery namely, in the computer disk drive. By taking the variable fluid properties for the geometry of rotating disk, Osalusi and Sibanda²⁰ focused on hydrodynamic viscous fluid flow. Subsequently, by considering different physical effects, various findings are published on flow of fluid and heat transfer by many researchers^21–27.

In the last few years, on a rotating surface, the heat transfer and thermal radiation characteristics have been studied extensively due to its enormous application in engineering industries, thermal energy plants, building construction, solar energy collectors etc. Joshi et al.²⁸ discussed thermal radiation along with variable viscosity effects on the unsteady flow of MNF. Also, the effect of Powell-Eyring fluid flow was discussed by Vafai et al.²⁹ with radiation and Soret-Dufour effects.

Mass transfer finds great importance in various engineering such as chemical engineering, radiation engineering, heat transfer engineering etc. In a rotating disk and cone-disk system; Shevchuk³⁰, and Turkyilmazoglu and Senel³¹ explored the findings for both heat and mass transfer. Mushtaq and Mustafa³² described the Brownian motion and thermophoresis consequence on the flow of electric conducting nanofluid. Further, these consequences in the study of an incompressible hybrid nanofluid flow were reported by Tassaddiq et al.³³. Lately, Reddy et al.³⁴ investigated these phenomenon of unsteady hybrid nanofluid flow driven by thermal radiation.

The present work focused on the hydrocarbon based MNF flow past a rotating disk in a porous medium with heat and mass transfer. Important aspects of depth and temperature dependent viscosity, Brownian motion, thermal radiation, chemical reaction, and thermophoresis impacts are taken into consideration. The disk with magnetic field $\vec{H} (H_{r}, H_{φ}, 0)$ is managed at a uniform temperature ( $T_{w}$ ) and concentration ( $C_{w}$ ). Here, the governed flow problem is solved by bvp4c method in MATLAB software using suitable guesses.

Mathematical Formulation of Flow Problem

Consider the three-dimensional, unsteady, axi-symmetric, incompressible boundary layer MNF flow over a rotating disk. Here, we considered cylindrical coordinates $(r, φ, z)$ to formulate the desired model for flow problem and also, the magnetic-fluid is assumed for rotation about $z -$ axis with same angular velocity $Ω$ of the disk.

The system of equations for the considered unsteady ferrohydrodynamic boundary layer flow^28,35 is as follows:

\frac{\partial v_{r}}{\partial r} + \frac{v_{r}}{r} + \frac{\partial v_{z}}{\partial z} = 0,

(1)

\begin{aligned} ρ (\frac{\partial v_{r}}{\partial t} + v_{r} \frac{\partial v_{r}}{\partial r} - \frac{{v_{φ}}^{2}}{r} + v_{z} \frac{\partial v_{r}}{\partial z}) + \frac{\partial p}{\partial r} = \frac{\partial}{\partial r} (μ (z, T) \frac{\partial v_{r}}{\partial r}) \\ + \frac{\partial}{\partial r} (μ (z, T) \frac{v_{r}}{r}) + \frac{\partial}{\partial z} (μ (z, T) \frac{\partial v_{r}}{\partial z}) \\ + μ_{0} | \vec{M} | \frac{\partial}{\partial r} | \vec{H} | - \frac{μ_{\infty}}{k_{0}} v_{r}, \end{aligned}

(2 )

\begin{aligned} ρ (\frac{\partial v_{φ}}{\partial t} + v_{r} \frac{\partial v_{φ}}{\partial r} + \frac{v_{r} v_{φ}}{r} + v_{z} \frac{\partial v_{φ}}{\partial z}) = \frac{\partial}{\partial r} (μ (z, T) \frac{\partial v_{φ}}{\partial r}) \\ + \frac{\partial}{\partial r} (μ (z, T) \frac{v_{φ}}{r}) + \frac{\partial}{\partial z} (μ (z, T) \frac{\partial v_{φ}}{\partial z}) \\ + μ_{0} | \vec{M} | \frac{\partial}{\partial φ} | \vec{H} | - \frac{μ_{\infty}}{k_{0}} v_{φ}, \end{aligned}

(3 )

\begin{aligned} ρ (\frac{\partial v_{z}}{\partial t} + v_{r} \frac{\partial v_{z}}{\partial r} + v_{z} \frac{\partial v_{z}}{\partial z}) + \frac{\partial p}{\partial z} = \frac{\partial}{\partial r} (μ (z, T) \frac{\partial v_{z}}{\partial r}) \\ + \frac{1}{r} \frac{\partial}{\partial r} (μ (z, T) v_{z}) + \frac{\partial}{\partial z} (μ (z, T) \frac{\partial v_{z}}{\partial z}) \\ + μ_{0} | \vec{M} | \frac{\partial}{\partial z} | \vec{H} | - \frac{μ_{\infty}}{k_{0}} v_{z}, \end{aligned}

(4 )

\begin{aligned} {(ρ c)}_{f} (\frac{\partial T}{\partial t} + v_{r} \frac{\partial T}{\partial r} + v_{z} \frac{\partial T}{\partial z}) = κ (\frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^{2} T}{\partial z^{2}}) \\ - \frac{\partial q_{r}}{\partial z} + {(ρ c)}_{p} [D_{B} (\frac{\partial T}{\partial z} \frac{\partial C}{\partial z} + \frac{\partial T}{\partial r} \frac{\partial C}{\partial r})] + Q_{1} (T - T_{\infty}) \\ + {(ρ c)}_{p} [\frac{D_{T}}{T_{\infty}} ({(\frac{\partial T}{\partial z})}^{2} + {(\frac{\partial T}{\partial r})}^{2})], \end{aligned}

(5 )

\begin{aligned} \frac{\partial C}{\partial t} + v_{r} \frac{\partial C}{\partial r} + v_{z} \frac{\partial C}{\partial z} = D_{B} (\frac{\partial^{2} C}{\partial r^{2}} + \frac{1}{r} \frac{\partial C}{\partial r} + \frac{\partial^{2} C}{\partial z^{2}}) \\ + \frac{D_{T}}{T_{\infty}} [\frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^{2} T}{\partial z^{2}}] - R_{1} (C - C_{\infty}) . \end{aligned}

(6)

The corresponding boundary conditions³⁶ of the flow problem are

\begin{aligned} v_{r} = 0; v_{ϕ} = r Ω; v_{z} = 0 T = T_{w}, \\ C = C_{w} at z = 0, \end{aligned}

(7a)

\begin{aligned} v_{r} \to 0; v_{ϕ} \to 0; T \to T_{\infty}, C \to C_{\infty} as z \to \infty \\ v_{z} \to Some finite negative value as z \to \infty \end{aligned}

(7b)

Here, the viscosity for MNF is taking to be dependent on depth and temperature both^13,28 as

μ (z, T) = \frac{μ_{\infty} (1 - ξ z)}{[1 + ξ (T - T_{\infty})]} .

(8)

Assumptions and Solution of the Problem

Unsteady boundary layer flow of MNF over the rotating disk is considered. For finding the solution of flow equations (1)–(6), we need expression for magnetization $M$ ^28,37, pressure gradient²⁸, radiative heat flux $q_{r}$ ²⁸, and magnetic field²⁸. So we take the following assumptions:

Assuming that the applied magnetic field $H$ , is sufficiently strong to saturate the magnetic nanofluid and the variation of magnetization $M$ with temperature $T$ can be approximated by a linear function^28,37, $M = K (T_{c} - T)$ .

The centrifugal force on the MNF flow over rotating disk is approximated by the pressure force in radial direction²⁸, i.e. $\frac{1}{ρ} \frac{\partial p}{\partial r} = {r Ω}^{2}$ .

For the radiative heat flux $q_{r}$ , we adopt Rosseland’s diffusion approximation²⁸ which is given as $q_{r} = - \frac{4 σ}{3 k^{*}} \frac{\partial T^{4}}{\partial z}$ . Here, we take the assumption that within the flow, difference in temperature is sufficiently small so that the term $T^{4}$ due to radiation may be represented as a form of temperature itself. Therefore $T^{4}$ can be described by the Taylor’s series about ambient temperature $T_{\infty}$ as $T^{4} = {T_{\infty}}^{4} + \frac{4 {T_{\infty}}^{3}}{1!} (T - T_{\infty}) + \frac{12 {T_{\infty}}^{2}}{2!} {(T - T_{\infty})}^{2} + \dots$ . Here, if we include the first order terms only, we get the following expression for $T^{4} \approx {4 {T_{\infty}}^{3} T - 3 T}_{\infty}^{4}$ .

The magnetic scalar potential of magnetic field $H$ in the presence of the magnetic dipole $m$ is given as²⁸

Ψ_{m} = \frac{m}{2 π r} c o s φ .

Considering negligible variation in

z -

axis, the radial, tangential and axial magnetic field components are given as follows:

\begin{aligned} H_{r} = - \frac{\partial Ψ_{m}}{\partial r} = \frac{1}{2 π} \frac{m c o s φ}{r^{2}}, \\ H_{φ} = - \frac{1}{r} \frac{\partial Ψ_{m}}{\partial φ} = \frac{1}{2 π} \frac{m s i n φ}{r^{2}}, H_{z} = 0. \end{aligned}

Hence, the total magnetic field

$H = \sqrt{{(H_{r})}^{2} + {(H_{φ})}^{2} + {(H_{z})}^{2}} = \frac{m}{2 π r^{2}}$ .

We are interested to find numerical solution for the above flow problem. Therefore, for nondimensionalized the governing system of equations, the similarity transformations are defined as:

Considering the following similarity transformations:

\begin{aligned} v_{r} = r Ω U^{'} (η), v_{ϕ} = r Ω V (η), v_{z} = \frac{ν}{δ} W (η), \\ T - T_{\infty} = (T_{w} - T_{\infty}) θ (η), C - C_{\infty} = (C_{w} - C_{\infty}) ϕ (η), \end{aligned}

(9)

with

η = \frac{z}{δ}

. Making use of these transformation, the equation of continuity takes the following form:

W (η) = - 2 R U (η) .

(10)

Here, the constant of integration is equal to zero at

η = 0

. Making use of equation (9) and (10) along with assumptions 1 – 4, equations (2), (3), (5) and (6) respectively, reduced to following coupled non-linear ordinary differential equations:

\begin{aligned} (1 - ϵ_{1} η) U^{‴} - (1 - ϵ_{1} η) {(1 + ϵ θ)}^{- 1} ϵ θ^{'} U^{″} - ϵ_{1} U^{″} \\ + (1 + ϵ θ) [\frac{δ}{ν} \frac{d δ}{d t} η U^{″} - R ({U^{'}}^{2} - V^{2} + 1) - R β U^{'} \\ + 2 R U U^{″} - \frac{2 B R}{{R e}^{2}}] = 0, \end{aligned}

(11)

\begin{aligned} (1 - ϵ_{1} η) V^{″} - (1 - ϵ_{1} η) {(1 + ϵ θ)}^{- 1} ϵ θ^{'} V^{'} - ϵ_{1} V^{'} \\ + (1 + ϵ θ) [\frac{δ}{ν} \frac{d δ}{d t} η V^{'} - 2 R (U^{'} V - U V^{'}) - R β V] = 0, \end{aligned}

(12 )

\begin{aligned} (4 + 3 Q r) θ^{″} + 3 P r Q r [(η \frac{δ}{ν} \frac{d δ}{d t} + 2 R U) θ^{'} + Q R θ \\ + N b ϕ^{'} θ^{'} + N t {θ^{'}}^{2}] = 0, \end{aligned}

(13)

ϕ^{″} + \frac{N t}{N b} θ^{″} + P r L e (η \frac{δ}{ν} \frac{d δ}{d t} + 2 R U) ϕ^{'} - R_{c} R ϕ = 0.

(14)

The dimensionless quantities are as follows:

\begin{aligned} ϵ_{1} = ξ δ (t), ϵ = ξ Δ T, R = \frac{Ω δ^{2}}{ν}, R e = \frac{Ω r^{2}}{ν}, \\ B = \frac{m}{2 π} μ_{0} K (T_{c} - T) \frac{ρ}{μ_{\infty}^{2}}, β = \frac{μ_{\infty}}{k_{0} ρ Ω}, \\ P r = \frac{μ_{\infty} c_{p}}{κ}, Q r = \frac{k k^{*}}{4 σ {T_{\infty}}^{3}}, Q = \frac{Q_{1}}{2 Ω {(ρ c)}_{f}}, \\ N b = \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}} \frac{(T_{w} - T_{\infty}) D_{T}}{T_{\infty} ν}, R_{c} = \frac{{ν R}_{1}}{2 Ω D_{B}}, \\ N t = \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}} \frac{(C_{w} - C_{\infty}) D_{B}}{ν}, a n d L e = \frac{α}{D_{B}} . \end{aligned}

The term

\frac{δ}{ν} \frac{d δ}{d t}

in equations (11)–(14) should not be eliminated due to unsteady flow. Therefore we considered the usual factor

(δ)

for scaling the various time dependent boundary layer flows³⁵

δ = 2 \sqrt{v t} + L,

(15)

Finally, using equation (15) in equations (11)–(14), we get the coupled non-linear system of ODEs:

\begin{aligned} (1 - ϵ_{1} η) U^{‴} - (1 - ϵ_{1} η) {(1 + ϵ θ)}^{- 1} ϵ θ^{'} U^{″} - ϵ_{1} U^{″} \\ + (1 + ϵ θ) [2 η U^{″} - R ({U^{'}}^{2} - V^{2} + 1) - R β U^{'} \\ + 2 R U U^{″} - \frac{2 B R}{{R e}^{2}}] = 0, \end{aligned}

(16)

\begin{aligned} (1 - ϵ_{1} η) V^{″} - (1 - ϵ_{1} η) {(1 + ϵ θ)}^{- 1} ϵ θ^{'} V^{'} - ϵ_{1} V^{'} \\ + (1 + ϵ θ) [2 η V^{'} - 2 R (U^{'} V - U V^{'}) - R β V] = 0, \end{aligned}

(17)

\begin{aligned} (4 + 3 Q r) θ^{″} + 3 P r Q r [(2 η + 2 R U) θ^{'} + N b ϕ^{'} θ^{'} \\ + N t {θ^{'}}^{2} + Q R θ] = 0, \end{aligned}

(18)

\begin{aligned} ϕ^{″} + \frac{N t}{N b} θ^{″} + P r L e (2 η + 2 R U) ϕ^{'} - R_{c} R ϕ = 0. \end{aligned}

(19)

The boundary conditions equation (7) become in following form

\begin{aligned} U^{'} (0) = 0, V (0) = 1, W (0) = 0, θ (0) = 1, ϕ (0) = 1 \\ U^{'} (\infty) \to 0, V (\infty) \to 0, θ (\infty) \to 0, ϕ (\infty) \to 0 \\ W (\infty) approaches some value negatively \end{aligned}

(20)

For computing the results of the flow problem, modelled on the geometry of the disk, the dimensionless system, is solved by MATLAB bvp4c solver. This algorithm in MATLAB helps to convert the given problem to the initial value problem, and the initial guesses

U^{″} (η)

V^{'} (η)

θ^{'} (η)

and

ϕ^{'} (η)

are taken suitably. The accuracy of the guess values are upto six decimal places, is checked by simulating the computed data for

U^{'} (\infty)

V (\infty)

θ (\infty)

and

ϕ (\infty)

with their given values. For, reducing the equations 16–19 into the first order, we take

y_{1} = U, y_{2} = U^{'}, y_{3} = U^{″}, y_{3}^{'} = U^{‴}, y_{4} = V, y_{5} = V^{'}, y_{6} = θ, y_{7} = θ^{'}, y_{7}^{'} = θ^{″}, y_{8} = ϕ, y_{9} = ϕ^{'}, y_{9}^{'} = ϕ^{″}

. Then, the system of equation becomes

\begin{aligned} y_{3}^{'} = & \frac{1}{(1 - ϵ_{1} η)} [(1 - ϵ_{1} η) {(1 + ϵ y_{6})}^{- 1} ϵ y_{3} y_{7} + ϵ_{1} y_{3} \\ - (1 + ϵ y_{6}) \\ (2 η y_{3} - R ({y_{2}}^{2} - {y_{4}}^{2} + 1) - R β y_{2} + 2 R y_{1} y_{3} - \frac{2 B R}{{R e}^{2}})], \\ y_{5}^{'} = & \frac{1}{(1 - ϵ_{1} η)} [(1 - ϵ_{1} η) {(1 + ϵ y_{6})}^{- 1} ϵ y_{5} y_{7} + ϵ_{1} y_{5} \\ - (1 + ϵ y_{6}) (2 η y_{5} - 2 R (y_{2} y_{4} {- y}_{1} y_{5}) - R β y_{4})], \\ y_{7}^{'} = & \frac{- 3 P r Q r}{(4 + 3 Q r)} [(2 η + 2 R y_{1}) y_{7} + N b y_{7} y_{9} \\ + N t {y_{7}}^{2} + Q R y_{6}], \\ y_{9}^{'} = & \frac{3 P r Q r N t}{N b (4 + 3 Q r)} [(2 η + 2 R y_{1}) y_{7} + N b y_{7} y_{9} \\ + N t {y_{7}}^{2} + Q R y_{6}] - P r L e (2 η + 2 R y_{1}) y_{9} + R_{c} R y_{8} . \end{aligned}

Subject to the initial and boundary conditions:

y_{1} (0) = 0, y_{2} (0) = 0, y_{3} (0) = λ_{1}, y_{4} (0) = 1, y_{5} (0) = λ_{2}, y_{6} (0) = 1, y_{7} (0) = λ_{3}, y_{8} (0) = 1, y_{9} (0) = λ_{4}, y_{2} (\infty) = 0, y_{4} (\infty) = 0, y_{6} (\infty) = 0, y_{8} (\infty) = 0

; where

λ_{1}, λ_{2}, λ_{3},

and

λ_{4}

are initial guesses which are shooted with MATLAB environment.

Physical Parameters of Practical Interest

The physical dimensionless parameters like skin-friction coefficients, Nusselt number and Sherwood number are calculated for the considered flow problem. By using Newtonian expression, the radial $τ_{r}$ and tangential $τ_{φ}$ shear stress on the surface of disk are given as:

\begin{aligned} τ_{r} = & {[μ (z, T) (\frac{\partial v_{r}}{\partial z} + \frac{\partial v_{z}}{\partial r})]}_{z = 0} \\ = \frac{μ_{\infty}}{(1 + ϵ)} \frac{r Ω}{δ} U^{″} (0), \end{aligned}

(21)

\begin{aligned} τ_{φ} = & {[μ (z, T) (\frac{\partial v_{φ}}{\partial z} + \frac{1}{r} \frac{\partial v_{z}}{\partial φ})]}_{z = 0} \\ = \frac{μ_{\infty}}{(1 + ϵ)} \frac{r Ω}{δ} V^{'} (0) . \end{aligned}

(22)

Therefore, the coefficients of radial skin-friction

C_{f_{r}}

and tangential skin-friction

C_{f_{φ}}

are calculated as:

C_{f_{r}} = \frac{(τ_{r})_{z = 0}}{ρ (r Ω)^{2}} = \frac{U^{″} (0)}{(1 + ϵ) R e_{r}}

, and

C_{f_{φ}} = \frac{(τ_{φ})_{z = 0}}{ρ (r Ω)^{2}} = \frac{V^{'} (0)}{(1 + ϵ) R e_{r}}

, respectively. The rate of heat transfer, is defined as:

q_{w} = - k {(\frac{\partial T}{\partial z})}_{z = 0} = \frac{- k Δ T}{δ} θ^{'} (0)

and hence, the mathematical relation of Nusselt number is

N u = - θ^{'} (0)

. Also, the rate of mass transfer (Fick’s law) is defined as:

q_{m} = - D {(\frac{\partial C}{\partial z})}_{z = 0} = \frac{- D Δ C}{δ} ϕ^{'} (0)

and hence, the Sherwood number is computed mathematically by the relation,

S h = - ϕ^{'} (0)

Results and Discussion

Unsteady flow of hydrocarbon based MNF (C1-20B) is considered for the current investigation. The solution for the modelled system is computed numerically in MATLAB environment using algorithm of bvp4c solver. The effects of governing physical parameters are high-lighted graphically. Here, the figures describes the consequence of involving parameters on the radial $U^{'} (η)$ , tangential $V (η)$ , and axial $W (η)$ velocity phenomenon, temperature $θ (η)$ , and concentration $ϕ (η)$ phenomenon. The entire numerical computations are done by fixing the value of parameters as $ϵ = ϵ_{1} = 0.1, R e = B = R = β = 1, P r = 128, R_{c} = Q r = Q = 1, N b = N t = L e = 0.1$ .

Table 1 demonstrates the findings of radial $(1 + ϵ) C_{f_{r}}$ and tangential ${(1 + ϵ) C}_{f_{φ}}$ skin-friction coefficients for distinct values of involved flow parameters. Also, the numerical data of dimensionless Nusselt $(N u)$ , and Sherwood $(S h)$ numbers are represented in Tables 2 and 3, respectively.

Table 1.

The radial and tangential skin-friction coefficients for various dimensionless parameters.

	$ϵ = ϵ_{1} = 0.0$	$ϵ = ϵ_{1} = 0.1$	$ϵ = ϵ_{1} = 0.2$	$ϵ = ϵ_{1} = 0.3$
$U^{″} (0)$	0.368236	0.371782	0.390709	0.415398
$V^{'} (0)$	0.493286	0.527296	0.547325	0.559937
	$R = 0.90$	$R = 0.93$	$R = 0.96$	$R = 1.00$
$U^{″} (0)$	0.553029	0.497632	0.423715	0.371782
$V^{'} (0)$	0.402194	0.443568	0.489317	0.527296
	$β = 0.0$	$β = 1.0$	$β = 2.0$	$β = 5.0$
$U^{″} (0)$	0.391253	0.371782	0.441782	0.499178
$V^{'} (0)$	0.547871	0.527296	0.492496	0.421859
	$B = 0.0$	$B = 0.4$	$B = 0.8$	$B = 1.0$
$U^{″} (0)$	0.793782	0.605173	0.480357	0.371782
$V^{'} (0)$	0.733672	0.653175	0.581472	0.527296

Table 2.

The Nusselt number for the various dimensionless parameters.

	$ϵ = ϵ_{1} = 0.0$	$ϵ = ϵ_{1} = 0.1$	$ϵ = ϵ_{1} = 0.2$	$ϵ = ϵ_{1} = 0.3$
$- θ^{'} (0)$	0.223961	0.243961	0.249198	0.250290
	$R = 0.90$	$R = 0.93$	$R = 0.96$	$R = 1.00$
$- θ^{'} (0)$	0.229035	0.235943	0.239957	0.243961
	$β = 0.0$	$β = 1.0$	$β = 2.0$	$β = 5.0$
$- θ^{'} (0)$	0.241983	0.243961	0.246363	0.248576
	$Q r = 0.7$	$Q r = 0.8$	$Q r = 0.9$	$Q r = 1.0$
$- θ^{'} (0)$	0.148576	0.181352	0.218216	0.243961
	$P r = 120$	$P r = 130$	$P r = 135$	$P r = 140$
$- θ^{'} (0)$	0.235034	0.249035	0.260034	0.262840
	$N b = 0.080$	$N b = 0.085$	$N b = 0.090$	$N b = 0.095$
$- θ^{'} (0)$	0.203263	0.215378	0.224055	0.229956
	$N t = 0.098$	$N t = 0.100$	$N t = 0.102$
$- θ^{'} (0)$	0.235578	0.243961	251302
	$Q = 0.6$	$Q = 0.8$	$Q = 0.9$
$- θ^{'} (0)$	0.251079	0.265072	0.279871

Table 3.

The Sherwood number for the various dimensionless parameters.

	$ϵ = ϵ_{1} = 0.0$	$ϵ = ϵ_{1} = 0.1$	$ϵ = ϵ_{1} = 0.2$	$ϵ = ϵ_{1} = 0.3$
$- ϕ^{'} (0)$	0.233648	0.253647	0.262287	0.271997
	$R = 0.90$	$R = 0.93$	$R = 0.96$	$R = 1.00$
$- ϕ^{'} (0)$	0.258293	0.256137	0.254315	0.253647
	$β = 0.0$	$β = 1.0$	$β = 2.0$	$β = 5.0$
$- ϕ^{'} (0)$	0.257948	0.253647	0.252890	0.252145
	$Q r = 0.7$	$Q r = 0.8$	$Q r = 0.9$	$Q r = 1.0$
$- ϕ^{'} (0)$	0.268945	0.250126	0.251987	0.253647
	$P r = 120$	$P r = 130$	$P r = 135$	$P r = 140$
$- ϕ^{'} (0)$	0.250193	0.258293	0.260016	0.268393
	$N b = 0.080$	$N b = 0.085$	$N b = 0.090$	$N b = 0.095$
$- ϕ^{'} (0)$	0.223474	0.234499	0.238463	0.240826
	$R_{c} = 0.0$	$R_{c} = 2.0$	$R_{c} = 5.0$	$R_{c} = 10.0$
$- ϕ^{'} (0)$	0.249832	0.246323	0.242964	0.240093
	$N t = 0.098$	$N t = 0.100$	$N t = 0.102$
$- ϕ^{'} (0)$	0.257201	0.253692	0.251051
	$L e = 0.90$	$L e = 0.100$	$L e = 0.110$
$- ϕ^{'} (0)$	0.243672	0.253647	0.269915

To estimate the accuracy and justify computation of the numerical model, the computed values of heat transfer coefficient has been compared (Table 4) with Gregg and Sparrow³⁸, and Maleque¹³; and conclusively, a very good consent with results is observed here.

Table 4.

Comparison of heat transfer coefficient when; $ϵ = ϵ_{1} = B = β = N b = N t = L e = Q = R_{c} = 0$ , $R = 1$ .

Prandtl number	Gregg and Sparrow³⁸	Maleque¹³	Present study
1	0.39625	0.39623	0.39680
10	1.13410	1.13331	1.13542
100	2.68710	2.68682	2.68693

Here, effects of thermal radiation and pressure force in radial direction have been neglected.

Figures 2 –6 demonstrate the distribution of $U^{'} (η), V (η), W (η), θ (η)$ and $ϕ (η)$ for distinct values of viscosity parameter. Here, the case of $ϵ = ϵ_{1} = 0$ indicates the flow with uniform viscosity and the growing values of $ϵ_{1}$ and $ϵ$ characterize the relation of viscosity on depth and temperature respectively. From Figures 2 and 5, it is noted that the radial velocity and temperature rises due to increment in the viscosity parameters $ϵ & ϵ_{1}$ both, while opposite behaviour is noted for tangential and axial velocity, shown in Figures 3 and 4 respectively. Also, the dual behaviour is observed for concentration phenomenon, shown in Figure 6.

Figure 1.

Geometrical Representation of Flow Problem on a Rotating Disk.

Figure 2.

Radial velocity profile for viscosity parameters.

Figure 3.

Tangential velocity profile for viscosity parameters.

Figure 4.

Axial velocity profile for viscosity parameters.

Figure 5.

Temperature profile for viscosity parameters.

Figure 6.

Concentration profile for viscosity parameters.

The rotation parameter $R$ shows rotational influence of the disk on flow profile. Supplementary Figures 7–11 represents the impact of rotation on velocity & temperature distributions, and concentration phenomenon. Here, the parameter $R$ increase the radial velocity, temperature distributions, whereas a marginal increment for tangential velocity phenomenon is observed. However, axial velocity of the fluid decrease due to increment in rotation parameter. Also, a considerable change is seen for the radial & axial velocity distributions in contrary to the distribution of tangential velocity.

In fluid dynamics, permeability is a characteristic of the porous material to permit the fluid flow through it. Effect of permeability parameter on $U^{'} (η), V (η), W (η), θ (η)$ and $ϕ (η)$ respectively is presented in Supplementary Figures 12–16. Here, $β = 0$ indicates the case of without porous medium. It is seen from the Supplementary Figures 12–15, increasing the permeability parameter $β$ decrease the radial and azimuthal velocity, and temperature, whereas an enhancement is seen for the axial velocity. Further, the permeability parameter $β$ gives the dual behaviour about the critical point on concentration distribution in the flow dispersion, (Supplementary Figure 16). Also, the consequence of FHD interaction parameter $B$ on the radial velocity is described through Supplementary Figure 17, displaying enhancement for radial velocity of the fluid with expanding range of $B$ .

The radiation parameter $Q r$ describes the relative contribution of heat transfer by conduction to heat transfer by thermal radiation. Here, the parameter $Q r$ does not affect the velocity profile significantly, whereas a significant change is observed in both temperature and concentration profile, displayed by Supplementary Figures 18 and 19 respectively. Here, the growing the value of $Q r$ enhances the temperature distribution in fluid flow, whereas concentration distribution decreases before $η = 0.35$ , approximately and after it reverse trend is observed concluding with slow convergence.

The relative amount of kinematic viscosity to heat-diffusivity is defined as Prandtl number $P r$ . Since Prandtl number $P r$ arises for energy and concentration equation and hence, $P r$ shows negligible effects on the velocity of fluid. The consequences of $P r$ on dimensionless temperature and dimensionless concentration of magnetic nanoparticle are depicted through Supplementary Figures 20 and 21, respectively. The figures reveal that fluid temperature increases and nanoparticle concentration gives dual behaviour with growing value of Prandtl number $P r$ .

Supplementary Figures 22 and 23 illustrate the role of the parameter $N b$ as enhancement for the temperature of fluid and concentration of nanoparticle respectively. Here, Brownian motion of the particles of fluid takes place due to the increment in the parameter $N b$ for Brownian motion which causes to uplift the temperature distribution. The thermophoresis parameter $N t$ influence on the temperature of fluid and concentration of nanoparticle is high-lighted through Supplementary Figures 24 and 25 respectively.

Fluid temperature got increment while nanoparticle concentration diminish before $η = 0.4$ and increase after that, due to increment in thermophoresis parameter $N t$ . Here, Supplementary Figure 26 displays that the scaling fluid temperature due to the increase in parameter $Q$ for heat generation.

The influence of Lewis parameter $L e$ and chemical reaction parameter $R_{c}$ on the concentration outline are shown in Supplementary Figures 27 and 28 respectively. Supplementary Figure 27 describes the loss in nanoparticle concentration with gain in Lewis number $L e$ before $η = 0.56$ , approximately and after it reverse behaviour is noted. Further, the magnitude of the nanoparticle concentration decrease for larger value of chemical reaction parameter $R_{c}$ (Supplementary Figure 28).

Conclusion

Unsteady ferrohydrodynamic flow of MNF based on hydrocarbon (C1-20B) past a porous rotating disk has been studied. The investigations reveal many fascinating results concerning the impacts of temperature and depth dependent viscosity, Brownian motion, thermal radiation, thermophoresis and chemical reaction on MNF flow. The dominant conclusion of the current investigation are observed as:

The radial velocity profile rises and tends to its steady state when we enter into variable geothermal viscosity from the uniform viscosity $(ϵ = ϵ_{1} = 0)$ while reverse trend is observed for both azimuthal and vertical velocity profiles.

Increment in the rotation parameter enhance the radial and azimuthal velocity outlines and decrease the vertical velocity outline.

The radial flow is dominant when we increase the rotation of the disk.

A higher value of permeability parameter diminishes the radial and azimuthal velocity and enhance the vertical velocity.

Fluid temperature diminishes due to increase in permeability parameter while it rise due to all other considered parameters.

The Brownian motion $(N b)$ and thermophoresis $(N t)$ parameters take place due to nanoparticle present in the fluid and these increase the heat-transfer from disk surface towards the ambient fluid.

Concentration distribution shows dual behaviour for the viscosity-variation parameters, permeability parameter, rotation parameter, thermophoresis parameter, radiation parameter, Lewis number, and Prandtl number.

For the parameter of Brownian motion, an increasing function is seen for the nanoparticle concentration.

The concentration profile displays the decline curve for higher values of chemical reaction parameter.

The value of radial and tangential skin-friction rises coefficients when we increase the value of viscosity vaiation parameters.

Supplemental Material

sj-png-1-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-1-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-2-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-2-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-3-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-3-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-4-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-4-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-5-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-5-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-6-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-6-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-7-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-7-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-8-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-8-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-9-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-9-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-10-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-10-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-11-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-11-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-12-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-12-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-13-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-13-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-14-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-14-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-15-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-15-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-16-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-16-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-17-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-17-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-18-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-18-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-19-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-19-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-20-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-20-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-21-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-21-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-png-22-pie-10.1177_09544089221079949 - Supplemental material for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity

Supplemental material, sj-png-22-pie-10.1177_09544089221079949 for Heat and mass transfer study of hydrocarbon based magnetic nanofluid (C1-20B) with geothermal viscosity by Kushal Sharma, Neha Vijay, Sanjay Kumar and Ruchika Mehta in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Footnotes

Conflict of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iDs

Kushal Sharma

Neha Vijay

Supplemental Material

Supplementary material for this article is available online.

Nomenclature

References

Kármán

. Uber laminare und turbulente reibung. Z Angew Math Mech 1921; 1: 233–252.

Cochran

. The flow due to a rotating disc. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 30. Cambridge University Press, pp. 365–375.

Benton

. On the flow due to a rotating disk. J Fluid Mech 1966; 24: 781–800.

Frusteri

Osalusi

. On mhd and slip flow over a rotating porous disk with variable properties. Int Commun Heat Mass Transf 2007; 34: 492–501.

Ram

Kumar

. Swirling flow of field dependent viscous ferrofluid over a porous ratating disk. Int J Appl Mech 2014; 6: 1450033.

Reddy

Sreedevi

Chamkha

. Mhd boundary layer flow, heat and mass transfer analysis over a rotating disk through porous medium saturated by cu-water and ag-water nanofluid with chemical reaction. Powder Technol 2017; 307: 46–55.

Sharma

Vijay

Kumar

et al. Hydromagnetic boundary layer flow with heat transfer past a rotating disc embedded in a porous medium. Heat Trans Asian Res 2021a; 50: 4342–4353.

Sharma

Vijay

Makinde

et al. Boundary layer flow with forced convective heat transfer and viscous dissipation past a porous rotating disk. Chaos Solitons Fractals 2021b; 148: 111055.

Sharma

. Rheological effects on boundary layer flow of ferro fluid with forced convective heat transfer over an infinite rotating disk. Pramana J Phys 2021; 95: 113.

10.

Bég

Kabir

Uddin

et al. Numerical investigation of von karman swirling bioconvective nanofluid transport from a rotating disk in a porous medium with stefan blowing and anisotropic slip effects. Proc Inst Mech Eng, Part C: J Mech Eng Sci 2020; 235(19): 3933–3951.

11.

Hafeez

Khan

Ahmed

. Oldroyd-b fluid flow over a rotating disk subject to soret–dufour effects and thermophoresis particle deposition. Proc Inst Mech Eng, Part C: J Mech Eng Sci 2021; 235: 2408–2415.

12.

Torrance

Turcotte

. Thermal convection with large viscosity variation. J Fluid Mech 1971; 47: 113–125.

13.

Maleque

. Effects of combined temperature and depth-dependent viscosity and hall current on an unsteady mhd laminar convective flow due to a rotating disk. Chem Eng Commun 2009; 197: 506–521.

14.

Ellahi

. The effects of mhd and temperature dependent viscosity on the flow of non-newtonian nanofluid in a pipe: Analytical solutions. Appl Math Model 2013; 37: 1451–1467.

15.

Ram

Kumar

. Ferrofluid flow with magnetic field dependent viscosity due to a rotating disk in porous medium. Int J Appl Mech 2012; 4: 1250041.

16.

Ram

Joshi

Sharma

et al. Variable viscosity effects on time dependent magnetic nanofluid flow past a stretchable rotating plate. Open Phys 2016; 14: 651–658.

17.

Ram

Pop

Joshi

et al. Polarization force and geothermal viscosity driven unsteady bodewadt transport phenomenon over a ferrofluid saturated disk. Phys Scr 2020; 96: 015202.

18.

Owen

Rogers

. Flow and heat transfer in rotating-disc systems. New York, USA: John Wiley and Sons, 1989.

19.

Herrero

Humphrey

JAC

Giralt

. Comparative analysis of coupled flow and heat transfer between corotating discs in rotating and fixed cylindrical enclosures. Am Soc Mech Eng Heat Transfer Div 1994; 300: 111–121.

20.

Osalusi

Sibanda

. On variable laminar convective flow properties due to a porous rotating disk in a magnetic field. Rom J Phys 2006; 51: 937–950.

21.

Attia

. Numerical study of flow and heat transfer of a non-newtonian fluid on a rotating porous disk. Nonlinear Anal: Model Contr 2009; 14: 21–26.

22.

Turkyilmazoglu

. Nanofluid flow and heat transfer due to a rotating disk. Comput Fluids 2014; 94: 139–146.

23.

Mustafa

Junaid

Hayat

et al. On bodewadt flow and heat transfer of nanofluids over a stretching stationary disk. J Mol Liq 2015; 211: 119–125.

24.

Eid

Mabood

. Thermal analysis of higher-order chemical reactive viscoelastic nanofluids flow in porous media via stretching surface. Proc Inst Mech Eng, Part C: J Mech Eng Sci 2021. 09544062211008481.

25.

Said

Sundar

Tiwari

et al. Recent advances on the fundamental physical phenomena behind stability, dynamic motion, thermophysical properties, heat transport, applications, and challenges of nanofluids. Phys Rep 2021.

26.

Said

Hachicha

Aberoumand

et al. Recent advances on nanofluids for low to medium temperature solar collectors: energy, exergy, economic analysis and environmental impact. Prog Energy Combust Sci 2021; 84: 100898.

27.

Mahian

Bellos

Markides

et al. Recent advances in using nanofluids in renewable energy systems and the environmental implications of their uptake. Nano Energy 2021; 86: 106069.

28.

Joshi

Ram

Tripathi

et al. Numerical investigation of magnetic nanofluids flow over rotating disk embedded in porous medium. Therm Sci 2018; 22: 2883–2895.

29.

Vafai

Khan

Fatima

et al. Dufour, soret and radiation effects with magnetic dipole on powell-eyring fluid flow over a stretching sheet. Int J Numer Methods Heat Fluid Flow 2020; 31(4): 1085–1103.

30.

Shevchuk

. Convective heat and mass transfer in rotating disk system. Berlin, Heidelberg: Springer New York, 2009.

31.

Turkyilmazoglu

Senel

. Heat and mass transfer of the flow due to a rotating rough and porous disk. Int J Thermal Sci 2013; 63: 146–158.

32.

Mushtaq

Mustafa

. Computations for nanofluid flow near a stretchable rotating disk with axial magnetic field and convective conditions. Results Phys 2017; 7: 3137–3144.

33.

Tassaddiq

Khan

Bilal

et al. Heat and mass transfer together with hybrid nanofluid flow over a rotating disk. AIP Adv 2020; 10: 055317.

34.

Reddy

Sreedevi

Chamkha

. Heat and mass transfer analysis of unsteady hybrid nanofuid fow over a stretching sheet with thermal radiation. SN Appl Sci 2020; 2: 1–15.

35.

Schlichting

Gersten

. Boundary-layer theory. New York: Springer Science & Business Media, 2003.

36.

Mangey

. Mathematics in engineering sciences: Novel theories, technologies, and applications. New York, USA: CRC Press, Taylor and Francis Group, 2019.

37.

Neuringer

. Some viscous flows of a saturated ferro-fluid under the combined influence of thermal and magnetic field gradients. Int J Non Linear Mech 1966; 1: 123–137.

38.

Gregg

Sparrow

. Heat transfer from a rotating disk to fluids of any prandtl number. Washington DC, USA: NASA, 1959.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.87 MB

0.79 MB

0.86 MB

0.70 MB

0.75 MB

0.70 MB

0.78 MB

0.71 MB

0.75 MB

0.64 MB

0.71 MB

0.87 MB

0.58 MB

0.66 MB

0.72 MB

0.78 MB

0.71 MB

0.76 MB

0.88 MB

0.81 MB

0.78 MB

0.72 MB