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Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity

  • Nainaru Tarakaramu EMAIL logo , Bandi Reddappa , Gajjala Radha , Dilsora Abduvalieva , Narsu Sivakumar , Fuad A. Awwad , Emad A. A. Ismail and Kotte Amaranadha Reddy
Published/Copyright: November 30, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

The impact of variable thermal conductivity plays a key role in the analysis of fluid mechanics. The applications of liquid are significant in nuclear reactors, automobiles, technical and manufacturing industry, electronic appliances, and so on. The novelty of this analysis is to exhibited the variable thermal conductivity in motion of Casson liquid via porous stretching sheet. In energy equation is consider thermal radiation and heat generation. Rosseland approximation plays a key role in the current work. Under the liquid motion assumptions, BL approximation is applied on numerical model and developed partial differential equations (PDE). The similarity transportation variable is taken by transporting PDE’s to ordinary differential equations. Numerical model is explored with the help of fourth-order boundary value problem with R–K–F procedure via shooting technique. The main key points noticed are: the heat transfer rate is more effective in the presence of heat generation than that in the absence of heat generation parameter. The temperature is enhanced in presence of variable thermal radiation while comparing absence of variable thermal radiation for large numerical values of Biot Number.

Keywords: variable thermal radiation; Casson fluid; thermal radiation; heat generation/absorption; MHD; porous medium; stretching sheet

Nomenclature

a 1 , b 1

constants

f

dimensionless stream function

f ′

dimensionless velocity

k ⁎

mean absorption coefficient

K ( T ⁎ )

temperature-dependent thermal conductivity = 1 + ε θ

M = σ 1 B 0 2 a 1 ρ f

magnetic field parameter

Nu x

Nusselt number

Pr = υ ⁎ α ⁎

Prandtl number

P m = μ ⁎ k p

porous parameter

q r

radiative heat flux

R d = 16 σ ⁎ T ∞ 3 3 k ⁎ K ⁎

thermal radiation parameter

T ⁎

fluid temperature

T w ⁎

constant fluid temperature of the wall

T ∞ ⁎

fluid temperature far away from the surface

U w

stretching velocity

u 1, v 1, w 1

velocity components along x 1, y 1, z 1 directions

Greek symbols

μ ⁎

dynamic viscosity

α ⁎ = k f ( ρ 1 C 1 ) f

thermal diffusivity

η

similarity variable

σ 1

Boltzmann constant

λ = b 1 a 1

ratio parameter

υ ⁎ = μ ⁎ ρ f

kinematic viscosity

k ⁎

thermal conductivity (m 2 / s )

θ

dimensionless temperature

ε

variable thermal conductivity

β = μ B 2 π c P

Casson fluid parameter

Subscripts

w

wall mass transfer velocity

1 Introduction

Applications involving non-Newtonian liquids may be found in many areas of modern life. Drag-reducing agents, printing technologies, dampening and braking devices, Personal protective equipment (PPE), and food items are some of the uses for non-Newtonian liquids. If a fluid is considered to be a non-Newtonian, it disobeys Newton’s law of viscosity, which states that viscosity of a liquid must remain constant regardless of the applied stress. When subjected to a force, the viscosity of non-Newtonian liquids can change from liquid to solid and back again. A prominent example of a non-Newtonian liquid is ketchup, which gets more liquid when stirred. We are employing the Casson fluid model as a non-Newtonian liquid in our work. The Casson liquid is particularly significant because of its distinctive features. For instance, the constitutive equation contains a yield shear stress for Casson liquid because of its solid-like elastic behaviour. A range of chemical and mechanical engineering fields, as well as the food processing industry, use this kind of transport mechanism. Casson [1] initially gave the Casson fluid rheological model. Brid et al. [2] analysed the behaviour of liquid motion and rheology of viscoelastic materials. Subsequently, Subba Rao et al. [3] demonstrated that the Casson fluid model may be transformed to a Newtonian liquid at a very large wall shear stress. Mukhopadhyay [4] analysed the heat transfer properties of Casson liquid motion via nonlinearly stretching sheet (SS). Wang et al. [5] discussed Casson nanoliquid motion via SS using Buongiorno’s model with activation energy. Sivakumar et al. [6] explained about 3D magnetohydrodynamics (MHD) Casson fluid motion via a linear SS. Bhagya Lakshmi et al. [7] examined the heat and mass transfer on MHD Casson liquid motion via curved surface and they used “Runge–Kutta–Fehlberg” (R–K–F) with shooting technique to explain the various parameter emerging in it. Ram Prakash and Shaw [8] exhibited non-Newtonian liquid motion via SS with the effect of nonlinear thermal and viscous dissipation. Sivakumar et al. [9] performed a theoretical analysis for the characteristics for the motion of heat transfer (HT) on Sakiadis and Blasius motion of MHD Casson liquid with heat sink/source effects explained the Casson liquid motion, heat and mass transfer characterized with MHD via stretched curved sheet. Hafez et al. [10] discussed the heat and mass transfer phenomena of Casson liquid with hydro-magnetic peristaltic motion via rotating inclined asymmetric channel. Abdel-Gawad [11] studied the MHD Casson liquid motion and gave exact solution for the steady and unsteady motions, and dynamic temperature field equations are found via extended unified method. Mahato et al. [12] developed entropy generation on stagnation point (SP) Casson nanoliquid motion via SS. Vishalakshi et al. [13] investigated the MHD Casson liquid motion with porous medium via thermal radiation with Cattaneo–Christov heat flux model. Abdullah et al. [14] analysed the Couette motion in electrically conductive two phase viscous Casson liquid with applied electric field and inclined magnetic flux. Li et al. [15] studied the impact of activation energy and chemical reaction on MHD Darcy-Forchheimer squeezed Casson liquid movement via horizontal channel. Asjad et al. [16] investigated the transport problem of fractional operator model for Casson liquid motion via flat surface. Recently, Patnaik et al. [17] and Mohanty et al. [18] presented Slip motion of Casson hybrid nanoliquid via rotating disk. Das et al. [19] explored boundary slip and absorption on pulsatile channel motion of Casson liquid with magnetic field. Oyelakin et al. [20] exhibited stagnation Casson nanofluid (NF) motion via SS with Arrhenius activation energy. Himanshu et al. [21] examined Casson liquid flowing via Riga plate. The Characteristics of Synchrotron sources with high photon flux, small source size, and broad energy and butter joint of hot rolling were examined recently using infrared microspectroscopy [22–24].

In industrial technologies, where product quality is based on thermal control variables, thermal radiation demonstrates a critical role in regulating thermal transmission. Wahid et al. [25] explored the radiative MHD mixed convective copper-alumina, water hybrid nanoliquid motion via inclined shrinking plate. Muhammad et al. [26] exhibited the response of non-linear thermal radiation of nanoliquid with motile gyrotactic micro-organisms with Joule heating and viscous dissipation effects. Sreedevi et al. [27] studied the impact of thermal radiation and chemical reaction characteristics of heat and mass transfer via mixed convection (MC) nanoliquid motion with slip conditions. Rana et al. [28] investigated free convective oblique Casson liquid motion via SS with nonlinear thermal radiation effect. There are many researchers [29–31], who explored the thermal radiation effect with different geometries. Few investigators [32–34] presented numerical simulation with the help of thermal radiation, while others [35–37] presented Darcy-Forchheimer hybrid NF motion via porous disk. Recently, Some of Scientists [38–40] studied the behaviour of welding residual stress of SUS301L-MT stainless steel and high entropy rare earth titanates.

In general, heat may be distributed in three ways such as MC and convection is the primary mode of heat transmission in liquids, where the fluid’s natural motion transmits heat from one location to another. Another method of heat transmission is conduction, which does not need the movement of a substance but rather involves the transfer of energy inside a substance. Electromagnetic waves, which may be either absorbed or emitted, represent the third mode of energy transfer. Liu et al. [41] studied the heat transfer phenomena on synergistic of jacketed vacuum membrane distillation module of double boundary layers (BLs), which is most helpful in deducing the volume and complexity of the system. Samah et al. [42] analysed the importance of HT via lubricated stagnation point motion of hydraulic systems with NFs. Srilatha et al. [43] investigated the HT characteristics of a liquid motion via conical gap of a cone disk apparatus with rotating and stationary angular velocities. Recently, some researchers [44,45] analysed the hybrid NF motion with Casson liquid via various physical geometries. Sudarsana Reddy et al. [46] presented the heat and mass transfer (HMT) occurrences on unsteady MHD of hybrid NFs via SS or shrinking sheet with suction, chemical reaction, thermal radiation, and slip effects with hybrid NFs. Saidulu and Reddy [47] explained HMT features via SS with hydromagnetic micropolar and chemical reaction, viscous dissipation effects. Balaji et al. [48] investigated the HT properties for NFs via exponentially porous shrinking surface. Ramzan et al. [49] described homogeneous and heterogeneous reactions with MC, Hall current effects via rotating disk, and Darcy-Forchheimer’s porous medium. Harish Babu et al. [50] analysed the HT properties via Newtonian Jeffrey fluid via SS or shrinking sheet. Shaw et al. [51] examined the convection motion on entropy-optimized MHD nanoliquid. Himanshu et al. [52] exhibited NF motion in Darcy-Forchheimer porous medium. Satya Narayana et al. [53] presented the HT on MHD motion via Eyring-Powell liquid via SS with viscous dissipation. Gupta et al. [54] developed numerical study of unsteady motion of GP-MoS2/C2H6O2-H2O over a porous SS. Himanshu et al. [55] analysed numerical study for pressure and heat transfer via rotating disk. Wu et al. [56] developed insulating material for medium frequency transformers. Lyu et al. [57] examine the three-dimensional numerical model in twin water entries with two spheres side-by-side. Sun et al. [58] studied compounded shear thickening fluid which are fabricated by mixing needle like carbon fibre powder. Gong et al. [59] developed 3D models or data structures such as Wire Frame, Sections, 3D Grid, Section, Grid surface, Linear. Thammanna et al. [60] presented 3D MHD coupled stress Casson fluid via unsteady SS. Tarakaramu et al. [61] developed numerical simulation of 3D NF via SS.

The objective of current study is to develop variable thermal conductivity on Casson NF via stretching surface with heat and mass transfer effects. After highlighting the detailed research on variable thermal conductivity on Casson NFs, it is observed that different researchers have reported the heat transfer phenomenon due to interaction of NFs with various flow configurations. The major aspect of current model is summarized as follows:

  • The heat and mass transfer phenomenon due to chemically reactive Sutterby coupled Casson NFs has been investigated.

  • The flow problem is observed with the assessment of external variable thermal conductivity, convective of heat and mass transfer and thermal radiation.

  • The statistical analysis is predicted for Sutterby NF problem.

  • The impact of physical parameters on concentration phenomenon, liquid motion, thermal field, and energy rate are examined. Moreover, numerical computations are performed for skin friction coefficient for various values of magnetic field parameter.

2 Mathematical formulation

Let us consider variable thermal radiation on 3D convective Casson liquid motion via porous SS. It is represented by the physical model of the problem as predicted in Figure 1. The following considerations are taken into present analysis.

  1. The electrically conducting is considered in direction of liquid motion in presence of magnetic field B 0 can be applied in z 1 direction.

  2. The liquid motion in porous stretching sheet at z = 0 .

  3. The stretching velocities u 1 = U w ( x 1 ) = a 1 x 1 , u 2 = V w ( x 1 ) = a 1 y 1 (where a 1 is constant) are along x 1 and y 1 directions.

  4. The variable thermal radiation and heat generation/heat absorption are taken into account.

  5. The Convective boundary condition is consider along liquid motion.

  6. The rheological equation of Casson liquid for steady motion is taken by [51–53]:

(1) τ i j = 2 μ 0 ⁎ + 2 p y ⁎ π ⁎ e i j , if π ⁎ ≥ π c ⁎ 2 μ 0 ⁎ + 2 p y ⁎ π c ⁎ e i j , if π ⁎ < π c ⁎ ,

where p y ⁎ = e i j e i j and Λ = μ B ⁎ 2 π 1 ⁎ / p y ⁎ . The Basic Governing fluid flow equations continuity, momentum and concentration are presented in below [see ref. 62,63]:

(2) ∂ u 1 ∂ x ⁎ + ∂ v 1 ∂ y ⁎ + ∂ w 1 ∂ z ⁎ = 0 ,

(3) u 1 ∂ u 1 ∂ x ⁎ + v 1 ∂ u 1 ∂ y ⁎ + w 1 ∂ u 1 ∂ z ⁎ = υ ⁎ ( 1 + 1 / Λ ) ∂ 2 u 1 ∂ ( z ⁎ ) 2 − σ ⁎ M 0 2 ρ u 1 − υ ⁎ k p u 1 ,

(4) u 1 ∂ v 1 ∂ x ⁎ + v ∂ v 1 ∂ y ⁎ + w ∂ v 1 ∂ z ⁎ = υ ⁎ ( 1 + 1 / Λ ) ∂ 2 v 1 ∂ ( z ⁎ ) 2 − σ ⁎ M 0 2 ρ v 1 − υ ⁎ k p v 1 ,

(5) u 1 ∂ T ⁎ ∂ x ⁎ + v 1 ∂ T ⁎ ∂ y ⁎ + w 1 ∂ T ⁎ ∂ z ⁎ = K ⁎ ( ρ c p ) f ∂ ∂ z ⁎ K ( T ⁎ ) ∂ T ⁎ ∂ z ⁎ − 16 σ ⁎ T ∞ ⁎ 3 ρ c p k ⁎ ∂ 2 T ⁎ ∂ 2 z ⁎ − Q 0 ( ρ c p ) f ( T ⁎ − T ∞ ⁎ ) .

Figure 1 Physical model of the problem.
Figure 1

Physical model of the problem.

The Present work relevant boundary conditions as shown below:

(6) z ⁎ = 0 at u 1 = u w v 1 = 0 w 1 = 0 , − k ∂ T ⁎ ∂ z ⁎ = h f ( T f ⁎ − T ⁎ ) z ⁎ → ∞ as u 1 → 0 , v 1 → 0 , T ⁎ → T ∞ ⁎ .

The temperature dependent thermal conductivity K ( T ⁎ ) is defined [44,45] as follows:

(7) K ( T ⁎ ) = ω ∞ 1 + ε Δ T ⁎ ( T ⁎ − T ∞ ) ,

where Δ T ⁎ = T w ⁎ − T ∞ ⁎ , T w ⁎ is the sheet temperature, ω ∞ is the conductivity of the fluid far away from the sheet.

The similarity transformations are given below [44,45]:

(8) η 1 = a 1 υ f ⁎ z ⁎ , u 1 = a 1 x ⁎ f ' ( η ) , v 1 = a 1 y ⁎ g ' ( η 1 ) w 1 = − a 1 υ ⁎ ( f ( η ) + g ( η ) ) , θ ( η 1 ) ( T w ⁎ − T ∞ ⁎ ) = T ⁎ − T ∞ ⁎ .

Using Eq. (8), we convert Eqs. (3)–(5) into below format [44,45]:

(9) f ‴ ( 1 + 1 / Λ ) + f ″ ( f + g ) − ( f ′ ) 2 − ( M + P m ) f ′ = 0 ,

(10) g ‴ ( 1 + 1 / Λ ) + g ″ ( f + g ) − ( g ′ ) 2 − ( M + P m ) g ′ = 0 ,

(11) θ ″ Pr 1 + 4 3 R d ( 1 + ε ) θ + ( f + g ) θ ′ − H θ = 0 .

Corresponding BCs are as follows:

(12) f = 0 , g = 0 , f ′ = 1 , g ′ = λ , θ ′ = − B i ( 1 − θ ) , at η 1 = 0 f ′ → 0 , g ′ → 0 , θ → 0 , as η 1 → ∞ .

The surface frictional coefficients C f x , C f y and rate of heat transfer Nu x are defined as follows:

C f x = τ w x ρ f ( u w ) 2 , C f y = τ w y ρ f ( v w ) 2 , Nu x = x q w k f ( T w ⁎ − T ∞ ⁎ ) ,

where τ w x = μ f ∂ u ∂ z ⁎ z ⁎ = 0 and τ w y = μ f ∂ v ∂ z ⁎ z ⁎ = 0 are the wall shear stresses along x and y - axes of the stretching surface and q w = − k ⁎ ∂ T ⁎ ∂ z ⁎ z ⁎ = 0 is the wall flux from the stretching surface.

The non-dimensional form of surface friction coefficients and heat transfer coefficients are defined as follows:

(13) C f x = 1 + 1 Λ f ″ ( 0 ) Re x − 1 / 2 , C f y = 1 + 1 Λ g ″ ( 0 ) Re x − 1 / 2 , Nu x = − θ ′ ( 0 ) Re x 1 / 2 ,

where Re x = u w x υ ⁎ and Re y = V w y υ ⁎ are local Reynolds number.

3 Numerical procedure

3

4 Results and discussion

Figure 2 exhibits brief explanation about the β (Casson liquid parameter) in direction g ′ ( η ) . It perceived that the velocity falls down on g ′ ( η ) with large statistical numbers β , which means that the BL converges to approximately ( 0 ≤ η ≤ 3 ) . Physically, the liquid Casson liquid proportional to yield stress. Due to this the velocity movement is high in porous stretching sheet. Hence, the Casson liquid motion is slow on porous stretching sheet (PSS).

Figure 2 The impact of Casson parameter on transverse velocity profile.
Figure 2

The impact of Casson parameter on transverse velocity profile.

The physical character of P m (porous parameter) on g ' ( η ) as displayed in Figure 3, which shows that large statistical values of P m will lower the velocity of Casson liquid via porous sheets. This can happen by high impact of magnetic field in liquid motion on PSS. The BL of Casson liquid converges approximately at ( 0 ≤ η ≤ 2 ) . Physically, the dynamic viscosity is more effective in non-Newtonian liquid. Hence, the liquid motion is slow via porous SS.

Figure 3 The impact of porous parameter on transverse velocity profile.
Figure 3

The impact of porous parameter on transverse velocity profile.

The effect of parameter ε (variable thermal radiation parameter) on θ ( η ) (temperature profile) is shown in Figure 4. It tells about the temperature of non-Newtonian liquid. It is clear that, θ ( η ) increases for large numerical values of ε and BL converges approximately at ( 0 ≤ η ≤ 1 ) . Physically, the high impact of magnetic field can be applied in opposite direction of liquid motion. Due to this, the temperature is produces more in Casson liquid motion via porous stretching sheet SS.

Figure 4 The impact of variable thermal conductivity on temperature profile.
Figure 4

The impact of variable thermal conductivity on temperature profile.

The predicted θ ( η ) for physical parameter H (heat generation/absorption parameter) is shown in Figure 5. It shows that θ ( η ) increases for large numerical values of H and “boundary layer” converges approximately at ( 0 ≤ η ≤ 1 ) . Physically, the heat generation parameter is direct positional to heat source due to this, the heat source effect released more temperature in motion of Casson liquid motion at surface area. Then, the temperature of BL is found to be very high in Casson liquid.

Figure 5 The impact of heat generation on temperature profile.
Figure 5

The impact of heat generation on temperature profile.

Figure 6 shows the effect of Biot number ( B i ) on θ ( η ) in the presence ( ε = 0.5 ) and absence ( ε = 0.0 ) of variable thermal radiation. It is clear that the θ ( η ) produced is more in case of ( ε = 0.5 ) when compared to that in case of ( ε = 0.0 ) for large numerical values of B i and thermal BL converges approximately at ( 0 ≤ η ≤ 1 ) . Physically, at low kinetic viscosity, the temperature of BL is found to be very high.

Figure 6 The impact of Biot number on temperature profile.
Figure 6

The impact of Biot number on temperature profile.

Figure 7(a) and (b) displays the effect of Prandtl number ( Pr ) on θ ( η ) , and heat transfer rate (HTR; Re x − 1 / 2 Nu x ) in the presence ( H = 0.5 ) and absence ( H = 0.0 ) of heat generation/absorption. It is clear that the temperature of heat generated is less in Casson liquid and also, Re x − 1 / 2 Nu x (HT) is more effective in case of ( H = 0.5 ) when compared to ( H = 0.0 ) for large statistical values of Pr . Physically, when the thermal diffusivity is more in Casson liquid movement in porous medium, then temperature BL and HTR developed is low.

Figure 7 The impact of Prandtl number on temperature profile and heat transfer rate.
Figure 7

The impact of Prandtl number on temperature profile and heat transfer rate.

Figure 8(a) and (b) shows the effect of thermal radiation ( R d ) on θ ( η ) in the presence ( H = 0.5 ) and absence ( H = 0.0 ) of “Heat Generation/Absorption Parameter” and HTR ( Re x − 1 / 2 Nu x ) in the presence ( ε = 0.5 ) and absence ( ε = 0.0 ) of variable thermal radiation parameter for large statistical values of R d . It is clear that θ ( η ) and Re x − 1 / 2 Nu x are more effective in the presence of H = 0.5 and ε = 0.5 when compared to H = 0.0 and ε = 0.0 . Physically, thermal radiation is relation between thermal conductivity and thermal diffusivity of liquid. Due to this reason, the most effective of thermal conductive is produced high temperature and low heat transfer rate of Casson liquid motion.

Figure 8 The impact of thermal radiation parameter on temperature profile and heat transfer rate.
Figure 8

The impact of thermal radiation parameter on temperature profile and heat transfer rate.

Tables 1 and 2 show a comparison study of initial and final values of velocity gradient, and also skin friction coefficient along axial and transverse directions for various numerical values of λ . It is clear that the numerical solutions are in very good agreement with the existing solutions up to eight decimal places.

Table 1

Comparison of velocity gradient at the sheet in x and y directions of ordinary fluid in the absence of β → ∞ , P m = 0 , and M = 0 for various values of λ

λ Wang [64] f ( ∞ ) Present study f ( ∞ ) Wang [64] g ( ∞ ) Present study g ( ∞ )
0.00 1.000000 1.00000000 0.000000 0.00000000
0.10 0.95764489 0.11486959
0.20 0.92265314 0.23236124
0.25 0.907075 0.90707538 0.257986 0.25798684
0.30 0.89253159 0.30036742
0.40 0.86603394 0.37922676
0.50 0.842360 0.84236062 0.451671 0.45167117
0.60 0.82096200 0.51896019
0.70 0.80144106 0.58199182
0.75 0.792308 0.79230879 0.612049 0.61212644
0.80 0.78350008 0.64143354
0.90 0.76690922 0.69779777
1.00 0.751527 0.75152755 0.751527 0.75148557
Table 2

Comparison of “Skin friction coefficient” − f ″ ( 0 ) in the absence of β → ∞ , P m = 0 , and M = 0 for various values of λ

λ Wang [64] − f ″ ( 0 ) Wang [64] − g ″ ( 0 ) Present study − f ″ ( 0 ) Present study − g ″ ( 0 )
0.0 1.000000 0.000000 1.0000000 0.00000000
0.1 1.0202666 0.06684975
0.2 1.039511 1.148745 1.0394989 0.14873990
0.3 1.0579567 0.24336270
0.4 1.075795 1.349214 1.0757889 0.34921139
0.5 1.0930951 0.46520737
0.6 1.109951 1.590532 1.1099481 0.59053070
0.7 1.1263982 0.72453342
0.8 1.142491 0.866685 1.1424891 0.86668491
0.9 1.1582544 0.01654017
1.0 1.173722 1.173722 1.1737215 0.17372159

5 Conclusion

The main findings of the present study are as follows:

  • The heat transfer rate is low in the presence of variable thermal conductivity when compared with that in the absence of variable thermal conductivity for large values of thermal radiation.

  • The impact of heat transfer rate is high in the presence of heat generation parameter when compared with that in the absence of heat generation parameter for large values of Prandtl number.

  • The temperature is high in Casson fluid flow when large numerical values of variable thermal conductivity.

Acknowledgments

Researchers Supporting Project number (RSPD2023R1060), King Saud University, Riyadh, Saudi Arabia. We all thanks to M. Ijaz Khan mechanical engineering, Lebanese American University for supporting technically and P.V. Satya Narayana, department of mathematics, Vellore Institute of Technology, Vellore for supporting framing mathematical equations and coding of the problem.

  1. Funding information: This research received funding from King Saud University through Researchers Supporting Project Number (RSPD2023R1060), King Saud University, Riyadh, Saudi Arabia.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-05-15
Revised: 2023-10-09
Accepted: 2023-10-19
Published Online: 2023-11-30

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2023-0137
Keywords for this article
variable thermal radiation; Casson fluid; thermal radiation; heat generation/absorption; MHD; porous medium; stretching sheet
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Articles in the same Issue

  1. Regular Articles
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  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
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