Stretching a Surface in a Rotating Fluid through Porous Medium

1. INTRODUCTION

 Flow through porous media is an important class of small Reynolds number (Re) laminar flow. This type of flow is found in the filtration of fluids and the seepage of water in canal and river banks. Some other examples of this flow are the movements of underground water and oils [1-3].

The slip condition is also an important aspect which has not been given proper attention in the study of fluid dynamics. Navier [4] described shear stress-based slip boundary condition. Saqib et al. [5] used fractional derivatives in Caputo sense. In another paper, Saqib et al. [6] discussed Cu – Al2O3 – H2O hybrid nanofluid. Hussan et al. [7] investigated  a viscoplastic Casson fluid in a two dimension flow, with a stretching surface  taken into account. Some of the recent advancements regarding flow over a stretching sheet and slip effects have been referenced in the literatue [8-14]. Nadeem et al. [15] developed the Caputo fractional model for Casson fluid with the help of  Flick’s and Fourier’s laws. Farhad et al. [16] analyzed blood flow using Casson fluid model through a horizontal cylinder in the presence of magnetic particles. Nadeem et al. [17] discussed the  Brinkman type fluid flow in a channel.

We used the Homotopy Analysis Method (HAM) [18-22] to obtain the analytic series outcomes in this paper. Crane [23] explored the stretching of a surface. Brady and Acrivos [24] and Wang [25] provided deep insight into axisymmetric and three-dimensional cases. They expressed the effects of different parameters in to and three dimensional flows. Wang [26]  discussed the case of stretching a surface in rotating fluid.

Keeping all the above-mentioned ponts, the arrangement  of the  paper is as fallows.

Section 1 includes introduction, section 2 includes mathematical formulation equations and Homotopy Analysis Method, section 3 includes discussion, and finally, section 4 includes graphical representation.

2. MATHEMATICAL FORMULATION

The velocity field is defined as

In this method, we use the initial guesses, satisfying the given boundary conditions

The equations of zeroth order are defined below,

We note that deformation equations of the zero-order contain the auxiliary parameters ℏ1, ℏ₂. Note that ℏ1 and ℏ₂ are assumed, so that the problem of zero-order may have a solution for all p∈ [0,1]

Now taking mth derivative of zero order deformation equations with respect to p, then putting p = 0 and dividing it by m!, we get

So the problem is

where

3. DISCUSSION

In this study, the analytical solution for stretching a surface in a rotating fluid through a porous medium with partial slip is constructed. Figures (1) to (3) show the effect of the porosity parameter R, keeping slip parameter β and variation parameter λ fixed on the similarity velocity profile in the x-direction. The effect of the porosity parameter remains negligible. Figures (1), (4), and (5) show the effect of the slip parameter β, keeping porosity parameter R and variation parameter λ fixed on the similarity velocity profile in the x-direction. Velocity decreases as the value of β increases. Figures (6) to (8) show the effect of slip parameter β, keeping porosity parameter R and variation parameter λ fixed on the similarity velocity profile in the y-direction. Velocity increases with an increase in β. Figures (9) and (10) show the effect of porosity parameter R, keeping slip parameters β and variation parameter λ fixed on the similarity velocity profile in the y-direction. Noticeably, an increase in R causes an increase in h.

Figure 1. Effects of R, β, and λ on f ′(η) taking ℏ₁ =  0.7, β  =  0.1, and R = 0.0.

Figure 2. Effects of R, β, and λ on  f ′(η) taking ℏ₁ =  0.7,  β  =  0.1, and R = 0.3.

Figure 3. Effects of R , β, and λ on f ′(η) when ℏ₁ =  0.7,  β  =  0.1, and R = 0.5.

Figure 4. Effects of  β, R, and λ on  f ′(η) when ℏ₁ =  0.7,  β  =  0.3, and  R = 0.1.

Figure 5. Effects of  β, R, and λ on  f ′(η) when ℏ₁ =  0.7,  β  =  0.5, and  R = 0.1.

Figure 6. Effects of β, R, and λ on h(η) when ℏ₂ = – 0.3,  β  =  0.1, and  R = 0.2.

Figure 7. Effects of β, R, and λ on h(η) when ℏ₂ = – 0.3,  β  =  0.3, and R = 0.2.

Figure 8. Effects of β, R, and λ on h(η) when ℏ₂ = – 0.3,  β  =  0.6, and  R = 0.2.

Figure 9. Effects of  R, β, and λ on h(η) when ℏ₂ = – 0.3,  β  =  0.3, and R = 0.4.

Figure 10. Effects of R , β, and λ on h(η) when ℏ₂ = – 0.3,  β  =  0.3, and R = 0.7.

In this study, the rotating flow of viscous fluid caused by the stretching of the surface is investigated. The governing equations after reducing into ODEs are solved by using HAM. The results are presented by employing graphs and the influence of the involved parameters is discussed in detail. It is noticed that the velocity of the rotating fluid increases with the corresponding increase in the porosity parameter.

Author Contribution

Shafqat Ali: conceptualization & supervision. Muhammad Shahzad Shabbir: formal analysis, methodology. Sajid Hussain: project administration, formal analysis, methodology, visualization, investigation. Ayesha Mahmood: writing - original draft. Samer Perveen: validation. Muhammad Sajid Rashid: writing - review & editing

Conflict of Interest

The authors of the manuscript have no financial or non-financial conflict of interest in the subject matter or materials discussed in this manuscript.

Data Availability Statement

Data availability is not applicable as no new data was created.

Funding Details

No funding was received for this research.

Generative AI Disclosure Statement

The authors did not use any type of generative artificial intelligence software for this research.