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## Applied Mathematics and Statistical Mechanics and their Applications

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Research Article | Open Access

Volume 2021 |Article ID 6641835 | https://doi.org/10.1155/2021/6641835

Muhammad Bilal Riaz, Maryam Asgir, A. A. Zafar, Shaowen Yao, "Combined Effects of Heat and Mass Transfer on MHD Free Convective Flow of Maxwell Fluid with Variable Temperature and Concentration", Mathematical Problems in Engineering, vol. 2021, Article ID 6641835, 36 pages, 2021. https://doi.org/10.1155/2021/6641835

# Combined Effects of Heat and Mass Transfer on MHD Free Convective Flow of Maxwell Fluid with Variable Temperature and Concentration

Revised15 Mar 2021
Accepted26 Mar 2021
Published14 Apr 2021

#### Abstract

Heat and mass transfer combined effects on MHD natural convection for a viscoelastic fluid flow are investigated. The dynamics of the fluid are controlled by the motion of the plate with arbitrary velocity along with varying temperature and mass diffusion. The non-dimensional forms of the governing equations of the model are developed along with generalized boundary conditions and the resulting forms are solved by the classical integral (Laplace) transform technique/method and closed-form solutions are developed. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences; few of them are highlighted here as limiting cases. Moreover, parametric analysis of system parameters is done via graphical simulations.

#### 2. Problem Formulation

We studied here the motion of the viscoelastic, in-compressible, electronically conducting Maxwell fluid due to plate motion with arbitrary velocity . The plate is along and is considered normal on the plate. In the first instance, at the plate and fluid are at temperature and concentration . With the time , the plate starts to move in its own axis. Then, the level of temperature and concentration takes up to and where and are piecewise continuous functions that vanish at . Details of different parameters are given in Table 1. Momentum, energy, and concentration equations are formed as follows:

 Symbol Quantity Velocity of fluid Magnetic field parameter Laplace transforms parameter Mass diffusivity Thermal expansion parameter Concentration expansion coefficient Thermal conductivity Density of fluid Relaxation time Electric conductivity coefficient Dynamic viscosity Kinematic viscosity Specific heat Heat source parameter Chemical reaction coefficient Gravitational acceleration Schmidt number Parameter due to magnetic field Prandtl number Grashof number due to thermal effect Grashof number due to concentration

The imposed initial and boundary conditions are

For dimensionless problem, we use the following relations:

After non-dimensionalizing, the governing equations becomealong the following initial and boundary conditions:

#### 3. Solution of the Problem

##### 3.1. Concentration

Transforming equation (7) after applying the Laplace integral transform and utilizing the corresponding initial condition, we get

The above differential equation solution is

The solution of equation (11) with the transformed form of boundary conditions becomes

Applying the Laplace inverse on equation (12) and using the with , convolution theorem and equation (A.22), the generalized solution for concentration isand is specified in equation (A.23).

##### 3.2. Temperature Distribution

Implementing the Laplace transform on equation (5) and using the concerned the initial condition, we get

The solution isAfter implementing the boundary conditions, equation (15) becomes

The Laplace inverse on equation (16) and using the with , convolution theorem and equation (A.24), the generalized solution for temperature obtained iswhere is defined in equation (A.25).

##### 3.3. Velocity

Employing the Laplace transform on equation (4) and using the corresponding initial condition on velocity form the following differential equation:

In order to solve equation (18), we use the value of , from equation (12) and equation (16), respectively. With boundary conditions use on velocity, the following solution is obtained:

Further simplification reduces equation (19):where , .

Generalized expression for velocity field is acquired by employing the inverse Laplace transform on equation (20):wherewhere

By using equation (A.20) and equation (A.21), expressions for the and are evaluated as follows:where is given in equation (23) andwhere

Similarly,where is given in equation (23) andThe above results are obtained for generalized time-dependent boundary conditions on velocity, concentration, and temperature. These results have many applications in engineering and applied science. Now, we will consider and discuss its few applications.

#### 4. Applications

##### 4.1. Application 1:

This function value shows the motion of the fluid is because of the motion of an infinite plate in its plane with constant velocity. This function has importance in a lot of engineering problems such as signal waves, driving forces that act for a short time only, and impulsive forces acting for an instance such as a hammer blow.Substituting the value of into equation (12) and applying the Laplace inverse, the expression for concentration iswhere is known as Dirac delta function.

Embedding the value of into equation (16) and taking Laplace inverse make the expression of temperature

The equation of velocitywhere

is obtained asand for (see equation (A.1)).

After substituting the value of into equation (25)

Equation (26) takes the form after employing the value of whereand for (see equation (A.1)).

After substituting the value of into equation (29)

Similarly, equation (30) after substituting the value of andand for (see equation (A.1).

Similar result for concentration was obtained by Nehad Ali Shah  (equation (35)). Thus, our result supports the result already present in literature.

##### 4.2. Application 2:

The important concepts of engineering are based around linear functions. They are often used in engineering to explain data and evaluate the lines that best fit the given data sets. It has a lot of applications in engineering and it can be represented in a variety of ways. One of the particular interests is direct variation, which forms many engineering applications such as Hooke’s law and Ohm’s law. To learn about slope, engineers use linear functions to interpret and understand graphs that describe displacement, velocity, and acceleration. They use these functions to analyze data to learn how to design their engineering products more efficiently, reliably, and safely.For the choice of equal to in the appropriate equations and employing the Laplace inverse, the expression of , and then changes into, respectively,wherewhere (see equation (A.1)) and (see equation (A.2)).where (see equation (A.1)) and (see equation (A.3)).where (see equation (A.1)) and (see equation (A.4)).

##### 4.3. Application 3:

The choice of this function shows the fluid motion due to the oscillation of the plate. It has a lot of applications in physics such as wave motion, other oscillatory motions, and engineering. It is used to model the behavior that repeats. Trigonometric functions are used to calculate angles in many engineering problems. In civil and mechanical engineering, trigonometry is used to calculate torque and forces on objects, which help build bridges and girders. In the construction of bridges, we need to consider the forces which keep the bridges at their balance and trigonometry helps us to calculate the static force which keeps the bridges static. In engineering, trigonometry is used to decompose the forces into horizontal and vertical components that can be analyzed.The expression for concentration after putting the value of into equation (12) isthe expression for temperature become after putting the value of into equation (16) isand velocity change after substitute the value of into equation (21) iswherewhere (see equation (A.1)) and (see equation (A.5)).where (see equation (A.1)) and (see equation (A.6)).where (see equation (A.1)) and (see equation (A.7)).

##### 4.4. Application 4:

The exponent functions are used for real-world application as for calculating area, volume, determining growth or decay, and impacts of force. In engineering, it helps them to design, build, and improve the machinery, structure, and equipment. For example, in sound engineering, it is used to calculate sound waves. In basic engineering, it is used to compute the tensile strength, which finds out the amount of stress that a structure can withstand. In aeronautical engineering, it is used to predict how airplanes, rockets, and jets will perform during flight. To determine the kinetic and potential energy, pressure, heat, and airflow of waves behavior, it is very helpful. Nuclear power sources are one of the important things developed by nuclear engineers. They used the exponents to work with extremely small numbers to make the big things happen. Substituting the value of into equation (12), the concentration equation after implementing the Laplace inverse becomesand equation of temperature distribution after putting the value of into equation (16) and applying Laplace inverse

The expression for velocity iswhereand (see equation (A.1)) and (see equation (A.8)).and (see equation (A.1)) and (see equation (A.9)).

Similarly,and (see equation (A.1)) and (see equation (A.10)).

##### 4.5. Application 5:

Inserting the into equation (12) and applying the Laplace inverse, we getand insert the into equation (16) and take Laplace inverse:The takes the form after embedding the where (see equation (A.1)) and (see equation (A.11)).where (see equation (A.1)) and (see equation (A.12)).where (see equation (A.1)) and (see equation (A.13)).

##### 4.6. Application 6:

The choice of the value of makes the expressionwhere (see equation (A.1)) and (see equation (A.14)).where (see equation (A.1)) and (see equation (A.15)).

Similarly,where (see (A.1)) and (see equation (A.16)).

##### 4.7. Application 7:

By putting the value of into equation (12) and employing Laplace inverse,

By putting the value of into equation (16) and Laplace inverse gives the expressionwhereand (see equation (A.1)) and (see equation (A.17)).where (see equation (A.1)) and (see equation (A.18)).and for (see equation (A.1)) and (see equation (A.19)).These are solutions for the choice of same function for , and from the list of functions . We can consider the problem with the different choice of function for , e.g., and find its solution. For the validation of results, if we take with choice of or , in our system of equations (4)–(8), the results obtained are the same as the result obtained by Nehad Ali shah  (choosing the