Rotation Equation of a Point in Air and its Solution

Operator inner products on both sides of Combination of Boyles’ law and Chares law (“B-C law” in short), we got the “Wind Speed Equation of a Point in Air” (“Wind Speed Equation” in short). It suits for describing straight line motion, and It states that ̇ is in proportion to . Operator outer products on both sides of “Wind Speed Equation” (where T is replaced by T) , we get the “Rotation Equation of a Point in Air” (“Rotation Equation” in short). It is a vector partial differential equation (PDE), suits for describing circular motion. It states that ( ̇ ) is in proportion to T. Its solution is found by method of separating variables. The existence of vector T is proved by the existence of rotation in atmosphere and the solution of “Rotation Equation”. It reveals that the vector form of B-C law holds in rotating air. Examples of up-side-down vertical rotation and horizontal rotation are given.


Introduction
Stirring a cup of coffee by a bar, one can make a rotation. However, there is nothing in air to stir up rotation, how can the rotation occurs? What is the locomotive driven the air to rotate? This paper answers these questions and derives a "Rotation Equation of a Point (mass) in Air" ("Rotation Equation" in short) using properties of vector product, Wind Speed Equation, and B-C law with T replaced by T. The solution of Rotation Equation is found by method of separating variables.
The straight line motion and the circular motion are two basic kinds of motion. For the straight line motion, we derived a basic formula "Wind Speed Equation of a Point in Air" [1]. Which has been confirmed by weather forecast every year and agreed with motion equation of mushroom cloud [2] derived by different assumption and scientific law.

Model
A model of "zero-weighted membrane wrapping a small cube of air as " is used. Since the air exists in everywhere, it has no shape, no volume, and thus hard to establish an equation using scientific laws. Once the "zero-weighted model" has been used, the wrapped air has shape and volume, such that the scientific laws can be used to derive the equation, meanwhile the actually stress field has not been disturbed, since there is nothing wrapped the air.
In the following, bold face denotes vector, matrix, and tensor.

Deriving of Wind Speed Equation
According to combination of Boyles' law and Chares law, we have: , , accuracy is a stress tensor, with the same normal stress and zero shearing stress: , shearing stress . i, j, k are unit vector in Cartesian co-ordinates.
are the cross section areas of the element (point), respectively. c is a constant, T is the absolute temperature.

is called the "Wind Speed Equation of A Point (Mass) in Air". It states that
̇ is in proportion to . (2)(3)(4)(5)(6)(7)(8) also shows that the gradient temperature is the locomotive driven wind speed acceleration along a straight line.

Mathematical Description on Rotating Motion
By Editor Group [3], rotation is defined by outer product (or vector product) of two vectors and , and is denoted by : Laplace operator) is also a vector. The first row of the determinant is a unit vector , the second row is the components of vector , the third row is the components of vector .

Necessary and Sufficient Conditions for Existence of Rotation
If , (3-3) then, rotation exists. And vice versa. If vector parallels to vector , or parallels to vector , or , , or , or , then, . i.e., rotation disappears.  or (3)(4) are the necessary and sufficient condition for existence of rotation . This result is obvious because that two rows are the same in the determinant.

Solution of "Rotation Equation of a Point in Air" by Method of Separating Variables
(4-4) is a vector PDE. It is hard to solve. Changing  to scalar function (its component form) is an easy way to solve. Rot on both sides of (4-4), we have:

Proof of the Existence of Vector T Satisfied (4-1)
The existence of rotation in air is a fact, which can be seen in volcano spurting and mushroom raising. By the necessary and sufficient conditions of the existence of rotation (3-3), we have: ( ̇) ( ̇) ( ̇) (6-1) such that (4-3) holds.

Examples of Rotation in Air
For the case of rotation existing in xOy plane ( the horizontal rotation surrounding to the z-axis. (7-1) shows that the rotation exists in xOy plane (i.e., the horizontal plane ) surrounding the z-axis(K vector).
Similarly, for the case of rotation existing in vertical plane (up=side-down plane) surrounding the x-axis.
(7-2) shows that the rotation exists in vertical plane (up=side-down plane) surrounding the x-axis. Instead of ( ̇) by cT, the cases of (7-1) and (7-2) are also hold for T.

Discussion
Q = question, A = answer Q1: Temperature T is a scalar or a vector? A1: It can be a scalar, in this case, rotation disappears. It can be a vector, in this case, rotation happens. Q2: What is the difference between temperature T to be a scalar and to be a vector?
A2: If T is a scalar, then, T keeps the same value measured for different directions. If T is a vector, then, T's value varies from different directions.
Q3: Temperature T is well known to be a scalar as a common sense, do you have evidence to show T can be a vector?
Q4: What is/are the factor(s) caused the air rotation? A4: Since there is nothing in air to stir up the air rotated, ( ̇) or cT is the factor driving the air rotated. Q5: What is the new finding(s) ? A5: Vector T exists in air rotation. The solution of "Rotation Equation" is found. The vector form of B-C law holds. These are new findings.

Conclusion
Heat (measured by temperature) is the locomotive of air motion. For straight line motion, temperature is a scalar. For circular motion, temperature is a vector. The straight line motion obeys scalar form of B-C law, while, the circular motion obeys vector form of B-C law. Operator inner products (or scalar products) on both sides of scalar form of B-C law, we got "Wind Speed Equation", while operator outer products (or vector products) on both sides of "wind speed equation" (T replaced by T), we get "Rotation Equation".
Examples of up-side-down vertical rotation and horizontal rotation are given.

Data Availability Statement
All data are cited from open publications.